Daily Papers

We introduce Motif-2-12.7B, a new open-weight foundation model that pushes the efficiency frontier of large language models by combining architectural innovation with system-level optimization. Designed for scalable language understanding and robust instruction generalization under constrained compute budgets, Motif-2-12.7B builds upon Motif-2.6B with the integration of Grouped Differential Attention (GDA), which improves representational efficiency by disentangling signal and noise-control attention pathways. The model is pre-trained on 5.5 trillion tokens spanning diverse linguistic, mathematical, scientific, and programming domains using a curriculum-driven data scheduler that gradually changes the data composition ratio. The training system leverages the MuonClip optimizer alongside custom high-performance kernels, including fused PolyNorm activations and the Parallel Muon algorithm, yielding significant throughput and memory efficiency gains in large-scale distributed environments. Post-training employs a three-stage supervised fine-tuning pipeline that successively enhances general instruction adherence, compositional understanding, and linguistic precision. Motif-2-12.7B demonstrates competitive performance across diverse benchmarks, showing that thoughtful architectural scaling and optimized training design can rival the capabilities of much larger models.

Recently, the Muon optimizer based on matrix orthogonalization has demonstrated strong results in training small-scale language models, but the scalability to larger models has not been proven. We identify two crucial techniques for scaling up Muon: (1) adding weight decay and (2) carefully adjusting the per-parameter update scale. These techniques allow Muon to work out-of-the-box on large-scale training without the need of hyper-parameter tuning. Scaling law experiments indicate that Muon achieves sim!2times computational efficiency compared to AdamW with compute optimal training. Based on these improvements, we introduce Moonlight, a 3B/16B-parameter Mixture-of-Expert (MoE) model trained with 5.7T tokens using Muon. Our model improves the current Pareto frontier, achieving better performance with much fewer training FLOPs compared to prior models. We open-source our distributed Muon implementation that is memory optimal and communication efficient. We also release the pretrained, instruction-tuned, and intermediate checkpoints to support future research.

The Muon optimizer has demonstrated promising performance in pre-training large language models through gradient (or momentum) orthogonalization. In this work, we propose a simple yet effective enhancement to Muon, namely Muon+, which introduces an additional normalization step after orthogonalization. We demonstrate the effectiveness of Muon+ through extensive pre-training experiments across a wide range of model scales and architectures. Our evaluation includes GPT-style models ranging from 130M to 774M parameters and LLaMA-style models ranging from 60M to 1B parameters. We comprehensively evaluate the effectiveness of Muon+ in the compute-optimal training regime and further extend the token-to-parameter (T2P) ratio to an industrial level of approx 200. Experimental results show that Muon+ provides a consistent boost on training and validation perplexity over Muon. We provide our code here: https://github.com/K1seki221/MuonPlus.

Matrix-based preconditioned optimizers, such as Muon, have recently been shown to be more efficient than scalar-based optimizers for training large-scale neural networks, including large language models (LLMs). On the other hand, recent benchmarks on optimizers for LLM pre-training have demonstrated that variance-reduction techniques such as MARS can achieve substantial speedups over standard optimizers that do not employ variance reduction. In this paper, to achieve the best of both worlds, we introduce MARS-M, a new optimizer that integrates the variance reduction technique in MARS with Muon. Under standard regularity conditions, we prove that Muon-M converges to a first-order stationary point at a rate of mathcal{O}(T^{-1/3}), which improves upon mathcal{O}(T^{-1/4}) rate attained by Muon. Our empirical results on language modeling and computer vision tasks demonstrate that MARS-M consistently yields lower losses and improved performance across various downstream benchmarks. The implementation of MARS-M is available at https://github.com/AGI-Arena/MARS/MARS_M.

We demonstrate that Muon, the simplest instantiation of a second-order optimizer, explicitly expands the Pareto frontier over AdamW on the compute-time tradeoff. We find that Muon is more effective than AdamW in retaining data efficiency at large batch sizes, far beyond the so-called critical batch size, while remaining computationally efficient, thus enabling more economical training. We study the combination of Muon and the maximal update parameterization (muP) for efficient hyperparameter transfer and present a simple telescoping algorithm that accounts for all sources of error in muP while introducing only a modest overhead in resources. We validate our findings through extensive experiments with model sizes up to four billion parameters and ablations on the data distribution and architecture.

Preconditioned adaptive methods have gained significant attention for training deep neural networks, as they capture rich curvature information of the loss landscape. The central challenge in this field lies in balancing preconditioning effectiveness with computational efficiency of implementing the preconditioner. Among recent advances, Muon stands out by using Newton-Schulz iteration to obtain preconditioned updates without explicitly constructing the preconditioning matrix. Despite its advantages, the efficiency of Muon still leaves room for further improvement. In this paper, we introduce RMNP (Row Momentum Normalized Preconditioning), an optimizer that replaces Newton-Schulz iteration with a simple row-wise (d_{in}) ell_2 normalization operation, motivated by the empirically observed diagonal block structure of the Transformer layerwise Hessian. We empirically verified that orthogonalization and row-wise (on input dim) ell_2 normalization are asymptotically equivalent in the case of the transformer. This substitution reduces the per-iteration computational complexity from {O}(mncdotmin(m,n)) to {O}(mn) for an mtimes n weight matrix while maintaining comparable optimization performance. Theoretically, we establish convergence guarantees for RMNP in the non-convex setting that match recent results for Muon optimizers, achieving the minimax optimal complexity. Extensive experiments on large language model pretraining show that RMNP delivers competitive optimization performance compared with Muon while substantially reducing preconditioning wall-clock time. Our code is available at https://github.com/Dominator-Index/RMNP.

The Muon optimizer has rapidly emerged as a powerful, geometry-aware alternative to AdamW, demonstrating strong performance in large-scale training of neural networks. However, a critical theory-practice disconnect exists: Muon's efficiency relies on fast, approximate orthogonalization, yet all prior theoretical work analyzes an idealized, computationally intractable version assuming exact SVD-based updates. This work moves beyond the ideal by providing the first analysis of the inexact orthogonalized update at Muon's core. We develop our analysis within the general framework of Linear Minimization Oracle (LMO)-based optimization, introducing a realistic additive error model to capture the inexactness of practical approximation schemes. Our analysis yields explicit bounds that quantify performance degradation as a function of the LMO inexactness/error. We reveal a fundamental coupling between this inexactness and the optimal step size and momentum: lower oracle precision requires a smaller step size but larger momentum parameter. These findings elevate the approximation procedure (e.g., the number of Newton-Schulz steps) from an implementation detail to a critical parameter that must be co-tuned with the learning schedule. NanoGPT experiments directly confirm the predicted coupling, with optimal learning rates clearly shifting as approximation precision changes.

The Muon optimizer has received considerable attention for its strong performance in training large language models, yet the design principle behind its matrix-gradient orthogonalization remains largely elusive. In this paper, we introduce a surrogate model that not only sheds new light on the design of Muon, but more importantly leads to a new optimizer. In the same spirit as the derivation of Newton's method, the surrogate approximates the loss as a quadratic function of the perturbation to a weight matrix W using only three matrices: the gradient G, an output-space curvature matrix H, and the data matrix Z that stacks the layer inputs. By minimizing this surrogate in one step and adopting a certain isotropic assumption on the weights, we obtain the closed-form update rule (up to momentum and weight decay) W leftarrow W - ηcdot msgn(G(ZZ^top)^{-1}), where η is the learning rate and msgn(X)=UV^top if X=USV^top is a compact singular value decomposition. This new optimization method, which we refer to as Newton-Muon, shows that standard Muon can be interpreted as an implicit Newton-type method that neglects the right preconditioning induced by the input second moment. Empirically, on a reproduction of the earliest publicly released Modded-NanoGPT speedrun configuration using Muon for GPT-2 pretraining, Newton-Muon reaches the target validation loss in 6\% fewer iteration steps and reduces wall-clock training time by about 4\%.

Muon optimizer has demonstrated robust results in pretraining of language models but its performance in finetuning of existing public pretrained models is not yet explored. Currently, Muon is used along with AdamW introducing a scope of improvement for adopting all parameters inside Muon. We introduce MuonAll, which incorporates all the parameters inside Muon by transforming into 2D matrices. We conduct extensive finetuning experiments across publicly available language models with model sizes upto half billion parameters. Muon and MuonAll perform at par with AdamW across major benchmarks, highlighting their effectiveness as alternative optimizers. We open-source the distributed implementations of Muon and MuonAll, available at https://github.com/Saurabh750/optimizer

We present a comprehensive theoretical and empirical study of the Muon optimizer for training transformers only with a small to medium decoder (30M - 200M parameters), with an emphasis on its mathematical foundations, convergence properties and synergistic interactions with modern architectural optimizations. Building on recent work showing Muon's scalability, we provide rigorous theoretical analysis including: (i)showing the convergence rate under standard assumptions, (ii) spectral regularization properties that prevent gradient explosion, (iii) connection to natural gradient descent on the Stiefel manifold, and (iv) equivalence to steepest gradient descent under the spectral norm. Crucially, we demonstrate that Muon expands the Pareto frontier in the compute-time trade-off by maintaining superior data efficiency at large batch sizes, a key finding of~essentialai2025muon that we validate across our model scales. Empirically, Muon reaches the target loss with 48-52\% of the training calculated by AdamW while maintaining or improving the final perplexity, consistent with larger-scale results. When combined with Multi-Head Latent Attention (MLA) and Mixture-of-Experts (MoE), we observe multiplicative efficiency gains: MLA+MoE+Muon achieves 68\% memory reduction and 3.2times inference speedup, while improving perplexity by 8-12\%. We provide detailed procedures on 15 architectural and optimizer components, stability analyzes across 100+ training runs, and practical implementation guidelines including Newton-Schulz coefficients (3.4445, -4.7750, 2.0315) optimized by~su2024muonblog. Our theoretical analysis and comprehensive experiments establish Muon as a principled, robust alternative to AdamW that particularly excels when combined with modern efficiency techniques and large-batch training regimes.

The Muon optimizer, based on matrix orthogonalization, has recently shown faster convergence and up to 2x computational efficiency over AdamW in LLM pretraining. Like AdamW, Muon is stateful, requiring storage of both model weights and accumulated gradients. While 8-bit AdamW variants mitigate this overhead using blockwise quantization, they are typically stable only under dynamic quantization - which improves stability on linear quantization for extreme values. In this paper, we introduce the 8-bit Muon optimizer using blockwise quantization, supporting both linear and dynamic schemes. We demonstrate that 8-bit Muon maintains stability under both, while delivering sim74\% reduction in memory footprint compared to full-precision Muon. In extensive experiments, 8-bit Muon closely matches the performance of Muon while outperforming AdamW and 8-bit AdamW in pre-training a 1.6B model on 4B FineWeb tokens. It also shows competitive results when fine-tuning the Llama 3.2 3B model on post-training data. We also provide a theoretical perspective to help explain this robustness under quantization.

Modern large-scale LLM pretraining benefits from utilizing Pipeline Parallelism; however, synchronous implementations leave GPUs idle during pipeline bubbles, wasting computational resources. Asynchronous Pipeline Parallelism eliminates these bubbles, maximizing throughput at the cost of gradient staleness. Among asynchronous schedules, PipeDream-2BW is particularly appealing: unlike the original PipeDream schedule, it ensures a constant one-step gradient delay regardless of pipeline depth. However, its adoption remains limited due to the common belief that optimizing under staleness is fundamentally unstable. In this work, we challenge this assumption, demonstrating that degradation under one-step delay depends strongly on optimizer choice rather than being an intrinsic limitation. We provide the first comprehensive empirical analysis showing that while AdamW, the predominant optimizer at the time when PipeDream-2BW was introduced, indeed suffers from severe degradation, recent methods like Muon exhibit strong robustness under a one-step delay. We introduce an optimizer-agnostic Error Feedback-inspired correction to further mitigate delay effects. We provide supporting theoretical analysis demonstrating convergence for Muon with and without this correction. Extensive evaluation on models up to 10B parameters confirms that our strategies bridge the performance gap with synchronous training, highlighting the practical potential of asynchronous pipeline parallelism at scale.

Muon is an increasingly widely used optimizer that replaces a gradient G=USV^top with its polar factor UV^top, thereby flattening the singular spectrum. However, full flattening discards singular-value information that may matter for adaptation. We introduce Muon^p, a Muon-style optimizer that instead uses fractional spectral-power updates US^pV^top for rational pin(0,1), interpolating between Muon and gradient descent. To make it practical, we prove that fractional spectral powers cannot be computed by any fixed univariate polynomial iteration, and furthermore derive low-degree odd bivariate recurrences that approximate US^pV^top using only matrix multiplications, preserving Muon's matrix-multiplication-only structure and compute complexity. We show that Muon^p maximizes the linear improvement in loss under the Schatten q-norm for q=1+1{p}. Empirically, Muon^p is especially effective for finetuning: on billion-scale models, Muon^p improves validation perplexity and downstream task performance. We further analyze when Muon^p is less suitable, through the lens of spectral geometry. Our results reveal important insights on when preserving the singular spectrum can bring significant gains, and introduce a principled way to achieve them.

The Muon optimizer has recently offered a promising alternative to AdamW for large language model training, leveraging matrix orthogonalization to produce geometry-aware updates. However, like all first-order methods, Muon can become trapped in sharp local minima. In this work, we present MONA, an optimizer that bridges Muon's orthogonalization framework with curvature-aware acceleration. MONA adds an acceleration term directly into Muon's gradient processing pipeline. This term is calculated from the exponential moving average of gradient differences. We provide a detailed convergence analysis for MONA, showing that the acceleration term enables escape from sharp minima while preserving Muon's spectral-norm regularization. Empirically, MONA achieves better convergence and downstream task performance compared to both Muon and AdamW across three scales of Mixture-of-Experts pretraining, spanning from 1B to 68B parameters, with the largest model trained on 1 trillion tokens. Furthermore, we conduct supervised fine-tuning on the MOE-68B-A3B model and evaluate it on general capability, mathematical reasoning, and code generation benchmarks, where MONA achieves SOTA performance.

Muon orthogonalizes the momentum buffer before each update, replacing its singular values with ones via Newton-Schulz iterations. This simple change lets Muon tolerate far larger learning rates and converge faster than other optimizers, but why? We show that the mechanism is spectral flattening, and develop two results around it. First, we prove that Muon's maximal stable step size scales with the average singular value of the gradient rather than the largest, which bottlenecks standard gradient descent. Second, we recast Muon as a preconditioned gradient method and show, under a Kronecker-factored curvature model, that it improves the effective convergence factor, with the improvement controlled by the spectrum of the gradient covariance. Extensive experiments validate both results: Muon remains stable at learning rates that cause SGD to diverge within the first few iterations, and reaches accuracy milestones several epochs earlier even at identical step sizes. Taken together, our results offer a principled, geometric explanation for Muon's empirical success.

Large Language Models (LLMs) achieve competitive performance across diverse natural language processing (NLP) tasks, yet pretraining is computationally demanding, making optimizer efficiency an important practical consideration. Muon accelerates LLM pretraining via orthogonal momentum updates that serve as a matrix analogue of the element-wise sign operator. Motivated by the recent perspective that Adam is a variance-adaptive sign update algorithm, we propose two variants of Muon, Muon-NSR and Muon-VS, which apply variance-adaptive normalization to momentum before orthogonalization. Muon-NSR applies noise-to-signal ratio (NSR) modulation, while Muon-VS performs variance-based scaling without introducing additional hyperparameters. Experiments on GPT-2 and LLaMA pretraining demonstrate that our proposed methods accelerate convergence and consistently achieve lower validation loss than both competitive, well-tuned AdamW and Muon baselines. For example, on the LLaMA-1.2B model, Muon-NSR and Muon-VS reduce the iterations required to reach the target validation loss by 1.36times relative to the well-tuned Muon following the recent benchmark.

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions typically either rescale updates after orthogonalization or use heavier whitening-based preconditioners before it. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon with three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). By rebalancing the momentum matrix before finite-step Newton--Schulz orthogonalization, {\method} improves the geometry seen by orthogonalization. We show that finite-step orthogonalization is governed by the input spectrum, especially stable rank and condition number, and that row/column normalization acts as a zeroth-order surrogate for whitening. For hidden matrix weights, R is the default variant. Theoretically, {\method} (R) retains the standard mathcal O(T^{-1/4}) Muon-type nonconvex stationarity guarantee with decoupled weight decay and a horizon-free diminishing learning-rate schedule, and extends it to finite-step NS5 up to an explicit inexactness constant. In LLaMA2 pretraining on C4, {\method} (R) consistently outperforms Muon on 130M, 350M, and 1B models, with faster convergence and lower validation perplexity. The code is available at the https://github.com/MaeChd/muon-eq{MuonEq codebase}.

The μ-parameterization (μP) provides a principled foundation for large language model (LLM) training by prescribing width-independent learning dynamics, which in turn enables predictable scaling behavior and robust hyperparameter transfer across model sizes. A central requirement of μP is the satisfaction of certain spectral conditions on weight matrices, which ensure consistent feature learning and optimization behavior as model width grows. While these conditions are well understood in theory, guaranteeing their validity in practical training for matrix-based optimizers such as Muon is still under studied. Existing works that study Muon under μP exhibit important limitations: they either do not ensure that the spectral conditions hold throughout the entire training horizon, or require repeated spectral normalization (or Newton-Schulz iterations) applied to both weights and updates, leading to significant computational overhead and reduced practicality. In this work, we show how to reliably guarantee the spectral conditions required by μP for Muon during the entire training process. Our key insight is that for moderately large models, maintaining spectral control at the level of optimizer updates alone is sufficient to preserve μP-compatible scaling, eliminating the need for explicit spectral normalization of the weights. Based on this principle, we develop a variant of Muon, namely Muon++, that satisfies spectral condition throughout the training process. Our results bridge the gap between the theoretical promises of μP and the practical deployment of matrix-based optimizers in long-horizon training. We also take the first step towards an adaptive spectral condition by incorporating data-dependent effects, making it better suited for long-horizon LLM training.

In large-scale optimization, the cheapness and effectiveness of update steps are the most crucial factors for a successful optimizer. Sign-based optimizers like Lion or Signum produce cheap per-step updates, whereas Muon's spectral matrix-sign update gives a much stronger direction at a substantially higher per-step cost. In this work, we propose LionMuon, which retains the effectiveness of Muon steps while considerably cutting the averaged iteration cost, similar to sign-based methods. It alternates between Lion's and Muon's updates on a fixed period P, sharing a single dual-EMA momentum buffer between them. The optimizer state memory therefore matches Lion and is exactly half of AdamW's. A simpler single-EMA variant, SignMuon, by itself already outperforms pure Muon. At P = 2, LionMuon Pareto-dominates Muon, Lion, Signum, and AdamW on every dataset and architecture we tested at 124M model size, reaching lower validation loss at lower compute, and the same advantage persists at 355M and 720M scale. On the theory side, we prove sharp complexity bounds under heavy-tailed noise which are governed by period-averaged smoothness and noise that interpolate between Muon's and Lion's constants. These bounds predict the compute-optimal period and the conditions under which LionMuon outruns Muon and Lion. Code: https://github.com/brain-lab-research/lion-muon

Orthogonality-based optimizers, such as Muon, have recently shown strong performance across large-scale training and community-driven efficiency challenges. However, these methods rely on a costly gradient orthogonalization step. Even efficient iterative approximations such as Newton-Schulz remain expensive, typically requiring dozens of matrix multiplications to converge. We introduce a preconditioning procedure that accelerates Newton-Schulz convergence and reduces its computational cost. We evaluate its impact and show that the overhead of our preconditioning can be made negligible. Furthermore, the faster convergence it enables allows us to remove one iteration out of the usual five without degrading approximation quality. Our publicly available implementation achieves up to a 2.8x speedup in the Newton-Schulz approximation. We also show that this has a direct impact on end-to-end training runtime with 5-10% improvement in realistic training scenarios across two efficiency-focused tasks. On challenging language or vision tasks, we validate that our method maintains equal or superior model performance while improving runtime. Crucially, these improvements require no hyperparameter tuning and can be adopted as a simple drop-in replacement. Our code is publicly available on github.

The Muon optimizer has emerged as a compelling alternative to Adam for training large language models, achieving remarkable computational savings through gradient orthogonalization. However, Muon's optimizer state is more sensitive to quantization errors: because the orthogonalization discards the magnitudes of singular values and retains only directional information, even small quantization errors in singular vector directions are amplified in the update. In this work, we propose MuonQ, a low-bit Muon training framework built on the principle of directional fidelity optimization. First, we apply a pre-quantization normalization so that each step introduces quantization errors of the same magnitude, preventing the accumulated error from developing a preferred direction. Second, we introduce a structural decomposition that separately quantizes the dominant singular components via power iteration, ensuring that quantization errors perturb only singular value magnitudes rather than rotating singular vector directions. Third, we adopt μ-law companding quantization to allocate higher resolution to densely packed momentum values, shifting the quantization objective from outlier preservation to dense-region distinguishability. Together, these techniques enable stable 4-bit quantization of Muon's optimizer states. Pre-training experiments on GPT-style and LLaMA-style models demonstrate that MuonQ at 4-bit precision closely matches full-precision Muon in both training loss and downstream task accuracy, while reducing optimizer state memory by up to 7.3 times. Our code is available at https://github.com/YupengSu/MuonQ.

Memory-efficient optimization is critical for training increasingly large language models (LLMs). A popular strategy involves gradient low-rank projection, storing only the projected optimizer states, with GaLore being a representative example. However, a significant drawback of many such methods is their lack of convergence guarantees, as various low-rank projection approaches introduce inherent biases relative to the original optimization algorithms, which contribute to performance gaps compared to full-parameter training. Aiming to tackle this problem, this paper investigates the layerwise sampling technique for debiasing low-rank projection mechanisms. In particular, an instantiation of the paradigm gives rise to a novel and unbiased low-rank optimization method built upon GaLore's mechanism and the Muon algorithm, named GaLore Unbiased with Muon (GUM). We theoretically prove our method matches the convergence guarantees of the base Muon algorithm while preserving the memory efficiency of low-rank techniques. Empirical experiments on LLM fine-tuning and pretraining also demonstrate non-trivial improvements over GaLore and even better performance than full-parameter training. Further investigation shows that the improvement of this technique comes from a more uniform distribution of knowledge inside layers, leading to more efficient utilization of the model parameter space and better memorization.

Modern deep learning commonly relies on AdamW with prescribed learning rate schedules, but recent works challenge both components: Schedule-Free optimization removes explicit schedules via iterate averaging, and Muon improves the update geometry by orthogonalizing momentum for matrix parameters. Despite Muon's strong empirical performance, its underlying mechanism remains partially understood. We study Muon through the river-valley loss landscape, where useful training progress occurs along a flat, low-curvature bulk subspace (the river), while high-curvature dominant directions form steep valley walls that induce oscillations. We empirically show that while Muon's orthogonalization accelerates river progress by increasing the bulk component, it also amplifies dominant-direction noise, causing oscillatory trajectories. Building on this, we propose Anytime MUon with Stable gradient Evaluation (AMUSE), which integrates Muon's rapid bulk progress with the stabilizing effect of Schedule-Free averaging. AMUSE uses a time-varying interpolation coefficient that initially evaluates gradients near the fast Muon sequence for rapid adaptation, then gradually shifts toward the stable averaged sequence to suppress valley-wall oscillations. As a result, AMUSE requires no learning rate schedules and supports anytime training. Across vision tasks and large language model pretraining, AMUSE consistently improves the performance-iteration Pareto frontier over (Schedule-Free) AdamW and Muon.

In this work, we present a new approach for fast tracking on multiwire proportional chambers with neural networks. The tracking networks are developed and adapted for the first-level trigger at hadron collider experiments. We use Monte Carlo samples generated by Geant4 with a custom muon chamber, which resembles part of the thin gap chambers from the ATLAS experiment, for training and performance evaluations. The chamber has a total of seven gas gaps, where the first and last gas gaps are displaced by ~1.5 m. Each gas gap has 50 channels with a size of 18-20 mm. Two neural network models are developed and presented: a convolutional neural network and a neural network optimized for the detector configuration of this study. In the latter network, a convolution layer is provided for each of three groups formed from 2-3 gas gaps of the chamber, and the outputs are fed into multilayer perceptrons in sequence. Both networks are transformed into hardware description language and implemented in Virtex UltraScale+ FPGA. The angular resolution is 2 mrad, which is comparable to the maximum resolution of the detector estimated by the minimum chi2 method. The latency achieved by the implemented firmware is less than 100 ns, and the throughput rate is 160 MHz.

The rapid progress of large language models (LLMs) is increasingly constrained by memory and deployment costs, motivating compression methods for practical deployment. Many state-of-the-art compression pipelines leverage the low-rank structure of trained weight matrices, a phenomenon often associated with the properties of popular optimizers such as Adam. In this context, Muon is a recently proposed optimizer that improves LLM pretraining via full-rank update steps, but its induced weight-space structure has not been characterized yet. In this work, we report a surprising empirical finding: despite imposing full-rank updates, Muon-trained models exhibit pronounced low-rank structure in their weight matrices and are readily compressible under standard pipelines. Motivated by this insight, we propose NuMuon, which augments Muon with a nuclear-norm constraint on the update direction, further constraining the learned weights toward low-rank structure. Across billion-parameter-scale models, we show that NuMuon increases weight compressibility and improves post-compression model quality under state-of-the-art LLM compression pipelines while retaining Muon's favorable convergence behavior.

Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.

Most first-order optimizers treat matrix-valued parameters as vectors, ignoring the intrinsic geometry of hidden-layer weights in neural networks. Muon addresses this mismatch by updating along the polar factor of a momentum matrix, but its theoretical understanding has lagged behind practice. In particular, practical implementations incorporate Nesterov momentum, compute the polar factor only approximately, and operate with stochastic gradients that may be heavy-tailed. We close this gap by developing a convergence theory for Muon with Nesterov momentum and inexact polar decomposition in non-convex matrix optimization under heavy-tailed noise. Our analysis builds on a unified framework for inexact polar decomposition that captures practical iterative approximations such as Newton-Schulz and quantifies how their errors propagate through the optimization dynamics. Under this framework, we establish an optimal iteration and sample complexity of O left(varepsilon^{-(3α-2){(α-1)}} right) for finding an varepsilon-stationary point, where αin(1,2] denotes the heavy-tail index. For the inexact-polar setting with σ_1=0, we also provide guarantees that do not require prior knowledge of α. We analyze a randomized low-rank polar decomposition that is substantially more efficient than full-space methods while remaining compatible with our theory. Numerical experiments further demonstrate the effectiveness of the proposed inexact and randomized variants.

Muon has recently demonstrated strong empirical performance in large language model training, but the theoretical role of momentum in Muon remains unclear. Existing analyses of Muon either remove momentum to study spectral updates in isolation, or retain momentum without explaining why it improves empirical performance. Our work bridges this gap by showing momentum in Muon acts as a spectral filter. Under a structured signal-plus-perturbation gradient model, we prove that momentum suppresses perturbations while preserving the dominant signal, thereby enlarging the spectral gap between them. This enlarged gap stabilizes the singular subspaces of the matrix passed to Muon's orthogonalization step, making the resulting update more reliable. We further show that applying momentum before orthogonalization achieves provably stronger alignment with the signal component of the gradient than either reversing this order or simply removing momentum. Experiments across diverse tasks, including LLM pretraining, support our theoretical analysis. More broadly, our theory offers a starting point for understanding the benefits of momentum in other matrix-based optimizers.

Muon is a matrix-aware optimizer that leverages Newton-Schulz (NS) iterations to enforce spectral gradient orthogonalization by driving all singular values of the momentum matrix toward 1. While this uniform spectral whitening enhances exploration and outperforms AdamW in LLM pretraining, we show it could lead to fundamental limitations beyond pretraining in two regimes: (i) cross-modality vision-language-action (VLA) training, where inherently low-rank action-module gradients cause amplification of noisy tail directions, and (ii) reinforcement learning with verifiable rewards (RLVR), where low-SNR gradients and the need to preserve per-head specialization from prior training make whitening unstable. To address these challenges, we propose Pion, a drop-in replacement for Muon that preserves its computational efficiency while replacing uniform spectral whitening with a two-stage Promotion+Suppression mechanism, which we call the high-pass NS iteration. This design induces a sharp spectral high-pass effect, anchoring dominant singular values at 1 while suppressing noisy tail components toward 0, with controllable filter strength. To preserve pretrained per-head heterogeneity, Pion also supports a per-head mode that applies updates independently across attention heads via a simple reshape, at no extra cost. In VLA training on LIBERO and LIBERO-Plus, Pion consistently outperforms both baselines across l_1-regression (VLA-Adapter) and flow-matching (VLANeXt) architectures, e.g., reaching 100% success rate on LIBERO Object after 1,500 training steps with VLA-Adapter, vs. 97.0% for Muon and only 32.2% for AdamW. The advantage of Pion further extends to a real Franka Research 3 robot with a pi_0.5 backbone under the DROID setup on three grasp-and-place tasks. In RLVR post-training on Qwen3-1.7B/4B with GRPO and GMPO, Pion also outperforms AdamW on MATH and GSM8K while Muon collapses to zero.

Orthogonal gradient updates have emerged as a promising direction in optimization for machine learning. However, traditional approaches such as SVD/QR decomposition incur prohibitive computational costs of O(n^3) and underperform compared to well-tuned SGD with momentum, since momentum is applied only after strict orthogonalization. Recent advances, such as Muon, improve efficiency by applying momentum before orthogonalization and producing semi-orthogonal matrices via Newton-Schulz iterations, reducing complexity to O(n^2). Nevertheless, quadratic costs remain a bottleneck. In this work, we study the semi-orthogonal properties of momentum-based updates and develop a method to bound momentum updates under a spectral-norm trust region, preserving directional information without requiring explicit semi-orthogonalization. We propose AuON (Alternative Unit-norm momentum updates by Normalized nonlinear scaling), a linear-time optimizer that achieves strong performance without constructing semi-orthogonal matrices, while preserving structural alignment and reconditioning ill-posed updates. Our approach combines hyperbolic-cosine RMS scaling transformations with normalization, demonstrating both effectiveness and computational efficiency compared to Newton-Schulz methods. We further introduce a hybrid variant (Hybrid-AuON) that applies a single Newton-Schulz iteration. Experiments across vision and language benchmarks show that AuON and its hybrid variant achieve performance comparable to strong baselines such as AdamW and Muon. Code is available at: https://github.com/ryyzn9/AuON

We analyze Muon as originally proposed and used in practice -- using the momentum orthogonalization with a few Newton-Schulz steps. The prior theoretical results replace this key step in Muon with an exact SVD-based polar factor. We prove that Muon with Newton-Schulz converges to a stationary point at the same rate as the SVD-polar idealization, up to a constant factor for a given number q of Newton-Schulz steps. We further analyze this constant factor and prove that it converges to 1 doubly exponentially in q and improves with the degree of the polynomial used in Newton-Schulz for approximating the orthogonalization direction. We also prove that Muon removes the typical square-root-of-rank loss compared to its vector-based counterpart, SGD with momentum. Our results explain why Muon with a few low-degree Newton-Schulz steps matches exact-polar (SVD) behavior at a much faster wall-clock time and explain how much momentum matrix orthogonalization via Newton-Schulz benefits over the vector-based optimizer. Overall, our theory justifies the practical Newton-Schulz design of Muon, narrowing its practice-theory gap.

The recently proposed Muon optimizer updates weight matrices via orthogonalized momentum and has demonstrated strong empirical success in large language model training. However, it remains unclear how to determine the learning rates for such orthogonalized updates. AdaGrad, by contrast, is a widely used adaptive method that scales stochastic gradients by accumulated past gradients. We propose a new algorithm, AdaGO, which combines a norm-based AdaGrad-type stepsize with an orthogonalized update direction, bringing together the benefits of both approaches. Unlike other adaptive variants of Muon, AdaGO preserves the orthogonality of the update direction, which can be interpreted as a spectral descent direction, while adapting the stepsizes to the optimization landscape by scaling the direction with accumulated past gradient norms. The implementation of AdaGO requires only minimal modification to Muon, with a single additional scalar variable, the accumulated squared gradient norms, to be computed, making it computationally and memory efficient. Optimal theoretical convergence rates are established for nonconvex functions in both stochastic and deterministic settings under standard smoothness and unbiased bounded-variance noise assumptions. Empirical results on CIFAR-10 classification and function regression demonstrate that AdaGO outperforms Muon and Adam.

The problem of computing optimal orthogonal approximation to a given matrix has attracted growing interest in machine learning. Notable applications include the recent Muon optimizer or Riemannian optimization on the Stiefel manifold. Among existing approaches, the Newton-Schulz iteration has emerged as a particularly effective solution, as it relies solely on matrix multiplications and thus achieves high computational efficiency on GPU hardware. Despite its efficiency, the method has inherent limitations - its coefficients are fixed and thus not optimized for a given matrix. In this paper we address this issue by proposing a Chebyshev-optimized version of Newton-Schulz (CANS). Based on the Chebyshev's alternance theorem, we theoretically derive optimal coefficients for the 3-rd order Newton-Schulz iteration and apply a Remez algorithm to compute optimal higher-degree polynomials. We leverage these polynomials to construct controlled approximate orthogonalization schemes, which is of interest in deep learning applications. Practically, we demonstrate the method's effectiveness in two key applications: orthogonalization in the Muon optimizer, and providing an efficient retraction alternative for Riemannian optimization on the Stiefel manifold.

This work investigates the production of high-resolution images of typical support elements in concrete structures by means of the muon tomography (muography). By exploiting detailed Monte Carlo radiation-matter simulations, we demonstrate the feasibility of the reconstruction of 1 cm--thick iron tubes inside 30 cm--deep concrete blocks, regarded as an important testbed within the structural diagnostics community. In addition, we present a new method for integrating simulated data with advanced deep learning techniques in order to improve the muon imaging of concrete structures. Through deep learning enhancement techniques, this results into a dramatic improvement of the image quality, as well as into a significant reduction of the data acquisition time, which are two critical limitations within the usual practice of muography for civil engineering diagnostics.

Orthonormalized update rules have rapidly become a leading choice of optimizer for training large language models, with recent open-source state-of-the-art models adopting Muon. To keep these updates tractable, Muon performs the orthonormalization with the Newton--Schulz (NS) iteration. Since NS is only approximate, directions with small singular values fail to be orthonormalized. In Muon, NS is applied to the momentum matrix at every step, yet little is known about how the singular value spectrum of these momentum matrices behaves during training, or how that behavior changes with model size. We present the first systematic study of this question. Tracking singular value quantiles of the momentum buffer across layers in models ranging from 77M to 2.8B parameters, we observe a consistent picture: after a short burn-in, the quantiles stabilize at a value determined by the layer type and model size. These stabilization values follow remarkably clean power laws in model size, with layer-dependent exponents. Layers up to mid-late depth scale very mildly with model size M (around M^{-0.25}), so the standard 5-step NS configuration used at academic scale will continue to orthonormalize them at much larger scales. Some of the late layers, however, scale much more aggressively (up to M^{-0.96}) and will fall into the NS failure regime at frontier scale unless one uses more NS iterations or better-tuned coefficients. NS iterations are computationally expensive at scale; our laws give practitioners a principled, layer-aware recipe for choosing the minimum NS configuration that still orthonormalizes the directions that matter -- avoiding unnecessary computation without sacrificing update quality.

Optimizers are crucial for the efficient training of Large Language Models (LLMs). While AdamW is the de facto standard, recent structure-aware optimizers like Muon have emerged, which regularize gradient updates by operating on entire weight matrices. The Muon optimizer balances the gradient updates along all the directions. However, Muon's reliance on the matrix sign function can lead to training instability, exhibits incompatibility when fine-tuning models pre-trained with AdamW. To address these limitations, we propose REG, a novel optimizer that replaces Muon's aggressive matrix sign operator with the Row-and-Column-Scaling (RACS) operator. Theoretically grounded in balancing a matrix, the RACS operator regularizes the update steps in a less drastic manner, making it simpler to implement and more compatible with established training dynamics. Through extensive empirical experiments on LLM training, we demonstrate that our REG optimizer not only achieves superior performance and stability over AdamW, but also maintains consistency with the AdamW training paradigm. This consistency is particularly evident during the fine-tuning stage, where REG optimizer avoids the performance degradation observed with Muon.

The choice of optimizer significantly impacts the training efficiency and computational costs of large language models (LLMs). Recently, the Muon optimizer has demonstrated promising results by orthogonalizing parameter updates, improving optimization geometry through better conditioning. Despite Muon's emergence as a candidate successor to Adam, the potential for jointly leveraging their strengths has not been systematically explored. In this work, we bridge this gap by proposing NorMuon (Neuron-wise Normalized Muon), an optimizer that synergistically combines orthogonalization with neuron-level adaptive learning rates. Our analysis reveals that while Muon effectively reduces condition numbers, the resulting updates exhibit highly non-uniform neuron norms, causing certain neurons to dominate the optimization process. NorMuon addresses this imbalance by maintaining second-order momentum statistics for each neuron and applying row-wise normalization after orthogonalization, ensuring balanced parameter utilization while preserving Muon's conditioning benefits. To enable practical deployment at scale, we develop an efficient distributed implementation under the FSDP2 framework that strategically distributes orthogonalization computations across devices. Experiments across multiple model scales demonstrate that NorMuon consistently outperforms both Adam and Muon, achieving 21.74% better training efficiency than Adam and 11.31% improvement over Muon on 1.1 B pretraining setting, while maintaining a comparable memory footprint to Muon. Our findings suggest that orthogonalization and adaptive learning rates are complementary rather than competing approaches, opening new avenues for optimizer design in large-scale deep learning.

Computing the polar decomposition and the related matrix sign function has been a well-studied problem in numerical analysis for decades. Recently, it has emerged as an important subroutine within the Muon algorithm for training deep neural networks. However, the requirements of this application differ sharply from classical settings: deep learning demands GPU-friendly algorithms that prioritize high throughput over high precision. We introduce Polar Express, a new method for computing the polar decomposition. Like Newton-Schulz and other classical polynomial methods, our approach uses only matrix-matrix multiplications, making it very efficient on GPUs. Inspired by earlier work of Chen & Chow and Nakatsukasa & Freund, Polar Express adapts the update rule at each iteration by solving a minimax optimization problem. We prove that this strategy minimizes error in a worst-case sense, allowing Polar Express to converge as rapidly as possible both in the early iterations and asymptotically. We also address finite-precision issues, making it practical to use in bfloat16. When integrated into the Muon training framework, our method leads to consistent improvements in validation loss when training a GPT-2 model on one billion tokens from the FineWeb dataset, outperforming recent alternatives across a range of learning rates.

In recent years, Muon has emerged as the dominant method for training large language models, and transformers more broadly. The essential difference, when compared to standard gradient descent methods, is to replace the usual update matrix M=UΣV^top with its polar factor UV^top. In this work, we consider a class of Muon-like updates, where we replace the update M with UΣ^p V^top for some parameter p. We call this a "spectral-shaping" operation, and develop a theory of how to pick p which depends on (a) local curvature of the loss function, (b) noise stemming from stochastic gradients and label noise, and (c) training stage. Our theory and experimentation reveal a previously overlooked behavior: positive p helps early by emphasizing high-curvature directions and accelerating signal contraction, while mildly negative p helps later by reallocating update strength toward low-curvature directions that still contain useful training signals. Building on the insight, we propose DynMuon, an efficient dynamic spectral shaping method that schedules p from positive to mildly negative over training. Extensive experiments across model sizes, architectures, and training settings show that DynMuon consistently achieves lower validation loss than Muon, while requiring 10.6-26.5% fewer steps to reach the same target loss.

We here propose a machine learning approach for monitoring particle detectors in real-time. The goal is to assess the compatibility of incoming experimental data with a reference dataset, characterising the data behaviour under normal circumstances, via a likelihood-ratio hypothesis test. The model is based on a modern implementation of kernel methods, nonparametric algorithms that can learn any continuous function given enough data. The resulting approach is efficient and agnostic to the type of anomaly that may be present in the data. Our study demonstrates the effectiveness of this strategy on multivariate data from drift tube chamber muon detectors.

Conventional wisdom in deep learning optimization dictates updating all layers at every step-a principle followed by all recent state-of-the-art optimizers such as Muon. In this work, we challenge this assumption, showing that full-network updates can be fundamentally suboptimal, both in theory and in practice. We introduce a non-Euclidean Randomized Progressive Training method-Drop-Muon-a simple yet powerful framework that updates only a subset of layers per step according to a randomized schedule, combining the efficiency of progressive training with layer-specific non-Euclidean updates for top-tier performance. We provide rigorous convergence guarantees under both layer-wise smoothness and layer-wise (L^0, L^1)-smoothness, covering deterministic and stochastic gradient settings, marking the first such results for progressive training in the stochastic and non-smooth regime. Our cost analysis further reveals that full-network updates are not optimal unless a very specific relationship between layer smoothness constants holds. Through controlled CNN experiments, we empirically demonstrate that Drop-Muon consistently outperforms full-network Muon, achieving the same accuracy up to 1.4times faster in wall-clock time. Together, our results suggest a shift in how large-scale models can be efficiently trained, challenging the status quo and offering a highly efficient, theoretically grounded alternative to full-network updates.

We introduce Post-Optimization Model Edit (POME), a new algorithm that enhances the performance of fine-tuned large language models using only their pretrained and fine-tuned checkpoints, without requiring extra data or further optimization. The core idea is to apply a muon-style projection to ΔW, the difference between the fine-tuned and pretrained weights. This projection uses truncated singular value decomposition (SVD) to equalize the influence of dominant update directions and prune small singular values, which often represent noise. As a simple post-processing step, POME is completely decoupled from the training pipeline. It requires zero modifications and imposes no overhead, making it universally compatible with any optimizer or distributed framework. POME delivers consistent gains, boosting average performance by +2.5\% on GSM8K and +1.0\% on code generation. Its broad applicability -- from 7B foundation models to 72B RLHF-instructed models -- establishes it as a practical, zero-cost enhancement for any fine-tuning pipeline. Code is available at https://github.com/NUS-HPC-AI-Lab/POME.

Scaling large models requires optimization strategies that ensure rapid convergence grounded in stability. Maximal Update Parametrization (boldsymbolμP) provides a theoretical safeguard for width-invariant Θ(1) activation control, whereas emerging optimizers like Muon are only ``half-aligned'' with these constraints: they control updates but allow weights to drift. To address this limitation, we introduce the Spectral Sphere Optimizer (SSO), which enforces strict module-wise spectral constraints on both weights and their updates. By deriving the steepest descent direction on the spectral sphere, SSO realizes a fully boldsymbolμP-aligned optimization process. To enable large-scale training, we implement SSO as an efficient parallel algorithm within Megatron. Through extensive pretraining on diverse architectures, including Dense 1.7B, MoE 8B-A1B, and 200-layer DeepNet models, SSO consistently outperforms AdamW and Muon. Furthermore, we observe significant practical stability benefits, including improved MoE router load balancing, suppressed outliers, and strictly bounded activations.

Efficient stochastic optimization typically integrates an update direction that performs well in the deterministic regime with a mechanism adapting to stochastic perturbations. While Adam uses adaptive moment estimates to promote stability, Muon utilizes the weight layers' matrix structure via orthogonalized momentum, showing superior performance in large language model training. We propose a new optimizer and a diagonal extension, NAMO and NAMO-D, providing the first principled integration of orthogonalized momentum with norm-based Adam-type noise adaptation. NAMO scales orthogonalized momentum using a single adaptive stepsize, preserving orthogonality while improving upon Muon at negligible additional cost. NAMO-D instead right-multiplies orthogonalized momentum by a diagonal matrix with clamped entries. This design enables neuron-wise noise adaptation and aligns with the common near block-diagonal Hessian structure. Under standard assumptions, we establish optimal convergence rates for both algorithms in the deterministic setting and show that, in the stochastic setting, their convergence guarantees adapt to the noise level of stochastic gradients. Experiments on pretraining GPT-2 models demonstrate improved performance of both NAMO and NAMO-D compared to the AdamW and Muon baselines, with NAMO-D achieving further gains over NAMO via an additional clamping hyperparameter that balances the competing goals of maintaining a well-conditioned update direction and leveraging fine-grained noise adaptation.

We propose a preconditioning (PC) layer, a weight parameterization via polynomial preconditioner that ensures stable weight conditioning throughout LLM training. The PC module reshapes the singular-value spectrum of weight matrices via low-degree polynomial preconditioning. After training, the preconditioned weights can be merged back into the original architecture, incurring no inference overhead. We demonstrate the advantage of the proposed PC layer over standard transformers in Llama-1B pre-training, for both the AdamW and Muon optimizers. Theoretically, we justify this spectrum-control principle by proving that uniformly bounding each layer's singular values ensures geometric convergence of gradient descent to global minima, for certain deep linear networks. Our code is available at https://github.com/Empath-aln/PC-layer.

While large language models (LLMs) have emerged as a significant advancement in artificial intelligence, the hardware and computational costs for training LLMs are also significantly burdensome. Among the state-of-the-art optimizers, AdamW relies on diagonal curvature estimates and ignores structural properties, while Muon applies global spectral normalization at the expense of losing curvature information. In this study, we restriked manifold optimization methods for training LLMs, which may address both optimizers' limitations, while conventional manifold optimization methods have been largely overlooked due to the poor performance in large-scale model optimization. By innovatively projecting the momentum onto the tangent space of model parameters and constraining it on a rotational Oblique manifold, we propose a novel, powerful, and efficient optimizer **Mano** that is the first to bridge the performance gap between manifold optimization and modern optimizers. Extensive experiments on the LLaMA and Qwen3 models demonstrate that Mano consistently and significantly outperforms AdamW and Muon even with less memory consumption and computational complexity, respectively, suggesting an expanded Pareto frontier in terms of space and time efficiency.

The Muon optimizer is consistently faster than Adam in training Large Language Models (LLMs), yet the mechanism underlying its success remains unclear. This paper demystifies this mechanism through the lens of associative memory. By ablating the transformer components optimized by Muon, we reveal that the associative memory parameters of LLMs, namely the Value and Output (VO) attention weights and Feed-Forward Networks (FFNs), are the primary contributors to Muon's superiority. Motivated by this associative memory view, we then explain Muon's superiority on real-world corpora, which are intrinsically heavy-tailed: a few classes (tail classes) appear far less frequently than others. The superiority is explained through two key properties: (i) its update rule consistently yields a more isotropic singular spectrum than Adam; and as a result, (ii) on heavy-tailed data, it optimizes tail classes more effectively than Adam. Beyond empirical evidence, we theoretically confirm these findings by analyzing a one-layer associative memory model under class-imbalanced data. We prove that Muon consistently achieves balanced learning across classes regardless of feature embeddings, whereas Adam can induce large disparities in learning errors depending on embedding properties. In summary, our empirical observations and theoretical analyses reveal Muon's core advantage: its update rule aligns with the outer-product structure of linear associative memories, enabling more balanced and effective learning of tail classes in heavy-tailed distributions than Adam.

Training large neural networks typically requires sharing gradients between accelerators through specialized high-speed interconnects. Drawing from the signal processing principles of frequency decomposition and energy compaction, we demonstrate that synchronizing full optimizer states and model parameters during training is unnecessary. By decoupling momentum updates and allowing controlled divergence in optimizer states across accelerators, we achieve improved convergence compared to state-of-the-art optimizers. We introduce {De}coupled {Mo}mentum (DeMo), a fused optimizer and data parallel algorithm that reduces inter-accelerator communication requirements by several orders of magnitude. This enables training of large neural networks even with limited network bandwidth and heterogeneous hardware. Our method is topology-agnostic and architecture-independent and supports scalable clock-synchronous distributed training with negligible compute and memory overhead. Empirical results show that models trained with DeMo match or exceed the performance of equivalent models trained with AdamW, while eliminating the need for high-speed interconnects when pre-training large scale foundation models. An open source reference PyTorch implementation is published on GitHub at https://github.com/bloc97/DeMo

The training of diffusion models is often absent in the evaluation of new optimization techniques. In this work, we benchmark recent optimization algorithms for training a diffusion model for denoising flow trajectories. We observe that Muon and SOAP are highly efficient alternatives to AdamW (18% lower final loss). We also revisit several recent phenomena related to the training of models for text or image applications in the context of diffusion model training. This includes the impact of the learning-rate schedule on the training dynamics, and the performance gap between Adam and SGD.

This paper presents a detailed analysis of the scalability and parallelization of local search algorithms for the Satisfiability problem. We propose a framework to estimate the parallel performance of a given algorithm by analyzing the runtime behavior of its sequential version. Indeed, by approximating the runtime distribution of the sequential process with statistical methods, the runtime behavior of the parallel process can be predicted by a model based on order statistics. We apply this approach to study the parallel performance of two SAT local search solvers, namely Sparrow and CCASAT, and compare the predicted performances to the results of an actual experimentation on parallel hardware up to 384 cores. We show that the model is accurate and predicts performance close to the empirical data. Moreover, as we study different types of instances (random and crafted), we observe that the local search solvers exhibit different behaviors and that their runtime distributions can be approximated by two types of distributions: exponential (shifted and non-shifted) and lognormal.

Sequential models, such as Recurrent Neural Networks and Neural Ordinary Differential Equations, have long suffered from slow training due to their inherent sequential nature. For many years this bottleneck has persisted, as many thought sequential models could not be parallelized. We challenge this long-held belief with our parallel algorithm that accelerates GPU evaluation of sequential models by up to 3 orders of magnitude faster without compromising output accuracy. The algorithm does not need any special structure in the sequential models' architecture, making it applicable to a wide range of architectures. Using our method, training sequential models can be more than 10 times faster than the common sequential method without any meaningful difference in the training results. Leveraging this accelerated training, we discovered the efficacy of the Gated Recurrent Unit in a long time series classification problem with 17k time samples. By overcoming the training bottleneck, our work serves as the first step to unlock the potential of non-linear sequential models for long sequence problems.

We introduce Kimi K2, a Mixture-of-Experts (MoE) large language model with 32 billion activated parameters and 1 trillion total parameters. We propose the MuonClip optimizer, which improves upon Muon with a novel QK-clip technique to address training instability while enjoying the advanced token efficiency of Muon. Based on MuonClip, K2 was pre-trained on 15.5 trillion tokens with zero loss spike. During post-training, K2 undergoes a multi-stage post-training process, highlighted by a large-scale agentic data synthesis pipeline and a joint reinforcement learning (RL) stage, where the model improves its capabilities through interactions with real and synthetic environments. Kimi K2 achieves state-of-the-art performance among open-source non-thinking models, with strengths in agentic capabilities. Notably, K2 obtains 66.1 on Tau2-Bench, 76.5 on ACEBench (En), 65.8 on SWE-Bench Verified, and 47.3 on SWE-Bench Multilingual -- surpassing most open and closed-sourced baselines in non-thinking settings. It also exhibits strong capabilities in coding, mathematics, and reasoning tasks, with a score of 53.7 on LiveCodeBench v6, 49.5 on AIME 2025, 75.1 on GPQA-Diamond, and 27.1 on OJBench, all without extended thinking. These results position Kimi K2 as one of the most capable open-source large language models to date, particularly in software engineering and agentic tasks. We release our base and post-trained model checkpoints to facilitate future research and applications of agentic intelligence.

The rise of GPU-based high-performance computing (HPC) has driven the widespread adoption of parallel programming models such as CUDA. Yet, the inherent complexity of parallel programming creates a demand for the automated sequential-to-parallel approaches. However, data scarcity poses a significant challenge for machine learning-based sequential-to-parallel code translation. Although recent back-translation methods show promise, they still fail to ensure functional equivalence in the translated code. In this paper, we propose QiMeng-MuPa, a novel Mutual-Supervised Learning framework for Sequential-to-Parallel code translation, to address the functional equivalence issue. QiMeng-MuPa consists of two models, a Translator and a Tester. Through an iterative loop consisting of Co-verify and Co-evolve steps, the Translator and the Tester mutually generate data for each other and improve collectively. The Tester generates unit tests to verify and filter functionally equivalent translated code, thereby evolving the Translator, while the Translator generates translated code as augmented input to evolve the Tester. Experimental results demonstrate that QiMeng-MuPa significantly enhances the performance of the base models: when applied to Qwen2.5-Coder, it not only improves Pass@1 by up to 28.91% and boosts Tester performance by 68.90%, but also outperforms the previous state-of-the-art method CodeRosetta by 1.56 and 6.92 in BLEU and CodeBLEU scores, while achieving performance comparable to DeepSeek-R1 and GPT-4.1. Our code is available at https://github.com/kcxain/mupa.

The scaling of Large Language Models (LLMs) drives interest in matrix-based optimizers (e.g., Shampoo, Muon, SOAP) for their convergence efficiency; yet their requirement for holistic updates conflicts with the tensor fragmentation in distributed frameworks like Megatron. Existing solutions are suboptimal: synchronous approaches suffer from computational redundancy, while layer-wise partitioning fails to reconcile this conflict without violating the geometric constraints of efficient communication primitives. To bridge this gap, we propose Canzona, a Unified, Asynchronous, and Load-Balanced framework that decouples logical optimizer assignment from physical parameter distribution. For Data Parallelism, we introduce an alpha-Balanced Static Partitioning strategy that respects atomicity while neutralizing the load imbalance. For Tensor Parallelism, we design an Asynchronous Compute pipeline utilizing Micro-Group Scheduling to batch fragmented updates and hide reconstruction overhead. Extensive evaluations on the Qwen3 model family (up to 32B parameters) on 256 GPUs demonstrate that our approach preserves the efficiency of established parallel architectures, achieving a 1.57x speedup in end-to-end iteration time and reducing optimizer step latency by 5.8x compared to the baseline.

A range of recent optimizers have emerged that approximate the same "matrix-whitening" transformation in various ways. In this work, we systematically deconstruct such optimizers, aiming to disentangle the key components that explain performance. Across tuned hyperparameters across the board, all flavors of matrix-whitening methods reliably outperform elementwise counterparts, such as Adam. Matrix-whitening is often related to spectral descent -- however, experiments reveal that performance gains are *not explained solely by accurate spectral normalization* -- particularly, SOAP displays the largest per-step gain, even though Muon more accurately descends along the steepest spectral descent direction. Instead, we argue that matrix-whitening serves two purposes, and the variance adaptation component of matrix-whitening is the overlooked ingredient explaining this performance gap. Experiments show that variance-adapted versions of optimizers consistently outperform their sign-descent counterparts, including an adaptive version of Muon. We further ablate variance adaptation strategies, finding that while lookahead style approximations are not as effective, low-rank variance estimators can effectively reduce memory costs without a performance loss.

Mu3e is an experiment currently under construction at the Paul Scherrer Institute in Switzerland, designed to search for the Lepton Flavor Violating (LFV) decay mu^+ rightarrow e^+e^-e^+. In extensions of the Standard Model (SM) that account for neutrino masses, this decay is theoretically allowed but occurs only through extremely rare loop processes, with a predicted branching ratio of approximately O(10^{-54}). Such a small probability implies that any observation of this decay would provide clear evidence for physics beyond the SM. The Mu3e experiment aims to probe the mu^+ rightarrow e^+e^-e^+ decay with a sensitivity of approximately O(10^{-15}) in its Phase-1 and plans to achieve a sensitivity of O(10^{-16}) after future upgrades. To reach its Phase-1 ambitious goals, Mu3e is going to use the most intense continuous muon beam in the world, generating 10^{8} muon stops per second in the target placed at the center of the Mu3e. Mu3e will use three main technologies for particle detection. The tracking will done through ultra-thin (50 - 70 mu m) pixel detectors based on MuPix11 sensors. These are high-voltage monolithic active pixel sensors (HV-MAPS) with a sim 23~mum spatial resolution. The timing will be done through scintillating fibres (sim 250 ps) and tiles (sim 40 ps), coupled to silicon photomultipliers and read out by MuTRiG3 ASICs. A triggerless DAQ system based on FPGAs will collect data from the detectors, which will then undergo reconstruction in a GPU filter farm. The assembly of the detectors has started, with a detector commissioning beam time planned for 2025. This document reports on the status of the construction, installation, and data-taking plans for the near future.

AdamW has long been the dominant optimizer in language model pretraining, despite numerous claims that alternative optimizers offer 1.4 to 2x speedup. We posit that two methodological shortcomings have obscured fair comparisons and hindered practical adoption: (i) unequal hyperparameter tuning and (ii) limited or misleading evaluation setups. To address these two issues, we conduct a systematic study of ten deep learning optimizers across four model scales (0.1B-1.2B parameters) and data-to-model ratios (1-8x the Chinchilla optimum). We find that fair and informative comparisons require rigorous hyperparameter tuning and evaluations across a range of model scales and data-to-model ratios, performed at the end of training. First, optimal hyperparameters for one optimizer may be suboptimal for another, making blind hyperparameter transfer unfair. Second, the actual speedup of many proposed optimizers over well-tuned baselines is lower than claimed and decreases with model size to only 1.1x for 1.2B parameter models. Thirdly, comparing intermediate checkpoints before reaching the target training budgets can be misleading, as rankings between two optimizers can flip during training due to learning rate decay. Through our thorough investigation, we find that all the fastest optimizers such as Muon and Soap, use matrices as preconditioners -- multiplying gradients with matrices rather than entry-wise scalars. However, the speedup of matrix-based optimizers is inversely proportional to model scale, decreasing from 1.4x over AdamW for 0.1B parameter models to merely 1.1x for 1.2B parameter models.

Training large-scale neural networks requires solving nonconvex optimization where the choice of optimizer fundamentally determines both convergence behavior and computational efficiency. While adaptive methods like Adam have long dominated practice, the recently proposed Muon optimizer achieves superior performance through orthogonalized momentum updates that enforce isotropic geometry with uniform singular values. However, this strict isotropy discards potentially valuable curvature information encoded in gradient spectra, motivating optimization methods that balance geometric structure with adaptivity. We introduce FISMO (Fisher-Structured Momentum-Orthogonalized) optimizer, which generalizes isotropic updates to incorporate anisotropic curvature information through Fisher information geometry. By reformulating the optimizer update as a trust-region problem constrained by a Kronecker-factored Fisher metric, FISMO achieves structured preconditioning that adapts to local loss landscape geometry while maintaining computational tractability. We establish convergence guarantees for FISMO in stochastic nonconvex settings, proving an O(1/T) rate for the expected squared gradient norm with explicit characterization of variance reduction through mini-batching. Empirical evaluation on image classification and language modeling benchmarks demonstrates that FISMO achieves superior training efficiency and final performance compared to established baselines.

We propose two medium-baseline, kiloton-scale neutrino experiments to study neutrinos from LHC proton-proton collisions: SINE, a surface-based scintillator panel detector observing muon neutrinos from the CMS interaction point, and UNDINE, a water Cherenkov detector submerged in lake Geneva observing all-flavor neutrinos from LHCb. Using a Monte Carlo simulation, we estimate millions of neutrino interactions during the high-luminosity LHC era. We show that these datasets can constrain neutrino cross sections, charm production in pp collisions, and strangeness enhancement as a solution to the cosmic-ray muon puzzle. SINE and UNDINE thus offer a cost-effective medium-baseline complement to the proposed short-baseline forward physics facility.

The ever-growing scale of deep learning models and training data underscores the critical importance of efficient optimization methods. While preconditioned gradient methods such as Adam and AdamW are the de facto optimizers for training neural networks and large language models, structure-aware preconditioned optimizers like Shampoo and Muon, which utilize the matrix structure of gradients, have demonstrated promising evidence of faster convergence. In this paper, we introduce a unifying framework for analyzing "matrix-aware" preconditioned methods, which not only sheds light on the effectiveness of Muon and related optimizers but also leads to a class of new structure-aware preconditioned methods. A key contribution of this framework is its precise distinction between preconditioning strategies that treat neural network weights as vectors (addressing curvature anisotropy) versus those that consider their matrix structure (addressing gradient anisotropy). This perspective provides new insights into several empirical phenomena in language model pre-training, including Adam's training instabilities, Muon's accelerated convergence, and the necessity of learning rate warmup for Adam. Building upon this framework, we introduce PolarGrad, a new class of preconditioned optimization methods based on the polar decomposition of matrix-valued gradients. As a special instance, PolarGrad includes Muon with updates scaled by the nuclear norm of the gradients. We provide numerical implementations of these methods, leveraging efficient numerical polar decomposition algorithms for enhanced convergence. Our extensive evaluations across diverse matrix optimization problems and language model pre-training tasks demonstrate that PolarGrad outperforms both Adam and Muon.

Optimization with matrix gradient orthogonalization has recently demonstrated impressive results in the training of deep neural networks (Jordan et al., 2024; Liu et al., 2025). In this paper, we provide a theoretical analysis of this approach. In particular, we show that the orthogonalized gradient method can be seen as a first-order trust-region optimization method, where the trust-region is defined in terms of the matrix spectral norm. Motivated by this observation, we develop the stochastic non-Euclidean trust-region gradient method with momentum, which recovers the Muon optimizer (Jordan et al., 2024) as a special case, along with normalized SGD and signSGD with momentum (Cutkosky and Mehta, 2020; Sun et al., 2023). In addition, we prove state-of-the-art convergence results for the proposed algorithm in a range of scenarios, which involve arbitrary non-Euclidean norms, constrained and composite problems, and non-convex, star-convex, first- and second-order smooth functions. Finally, our theoretical findings provide an explanation for several practical observations, including the practical superiority of Muon compared to the Orthogonal-SGDM algorithm of Tuddenham et al. (2022) and the importance of weight decay in the training of large-scale language models.

Triangle centrality is introduced for finding important vertices in a graph based on the concentration of triangles surrounding each vertex. It has the distinct feature of allowing a vertex to be central if it is in many triangles or none at all. We show experimentally that triangle centrality is broadly applicable to many different types of networks. Our empirical results demonstrate that 30% of the time triangle centrality identified central vertices that differed with those found by five well-known centrality measures, which suggests novelty without being overly specialized. It is also asymptotically faster to compute on sparse graphs than all but the most trivial of these other measures. We introduce optimal algorithms that compute triangle centrality in O(mbarδ) time and O(m+n) space, where barδle O(m) is the average degeneracy introduced by Burkhardt, Faber, and Harris (2020). In practical applications, barδ is much smaller than m so triangle centrality can be computed in nearly linear time. On a Concurrent Read Exclusive Write (CREW) Parallel Random Access Machine (PRAM), we give a near work-optimal parallel algorithm that takes O(log n) time using O(mm) CREW PRAM processors. In MapReduce, we show it takes four rounds using O(mm) communication bits and is therefore optimal. We also derive a linear algebraic formulation of triangle centrality which can be computed in O(mbarδ) time on sparse graphs.

This work introduces a highly efficient implementation of the transformer architecture on a Field-Programmable Gate Array (FPGA) by using the hls4ml tool. Given the demonstrated effectiveness of transformer models in addressing a wide range of problems, their application in experimental triggers within particle physics becomes a subject of significant interest. In this work, we have implemented critical components of a transformer model, such as multi-head attention and softmax layers. To evaluate the effectiveness of our implementation, we have focused on a particle physics jet flavor tagging problem, employing a public dataset. We recorded latency under 2 mus on the Xilinx UltraScale+ FPGA, which is compatible with hardware trigger requirements at the CERN Large Hadron Collider experiments.

Large Language Models (LLMs) enable advanced natural language processing but face deployment challenges on resource-constrained edge devices due to high computational, memory, and energy demands. Optimizing these models requires addressing three key challenges: acquiring task-specific data, fine-tuning for performance, and compressing models to accelerate inference while reducing resource demands. We propose an integrated framework combining GPTQ-based quantization, low-rank adaptation (LoRA), and a specialized data distillation process to significantly reduce model size and complexity while preserving or enhancing task-specific performance. By leveraging data distillation, knowledge distillation via Kullback-Leibler divergence, Bayesian hyperparameter optimization, and the Muon optimizer, our pipeline achieves up to 2x memory compression (e.g., reducing a 6GB model to 3GB) and enables efficient inference for specialized tasks. Empirical results demonstrate superior performance on standard LLM benchmarks compared to GPTQ quantization alone, with the Muon optimizer notably enhancing fine-tuned models' resistance to accuracy decay during quantization.

We generalize the Hamiltonian Monte Carlo algorithm with a stack of neural network layers and evaluate its ability to sample from different topologies in a two dimensional lattice gauge theory. We demonstrate that our model is able to successfully mix between modes of different topologies, significantly reducing the computational cost required to generated independent gauge field configurations. Our implementation is available at https://github.com/saforem2/l2hmc-qcd .

An ultra high energy electromagnetic cascade, a purely leptonic process and initiated by either photons or e^pm, can be a source of high energy neutrinos. We present a public python3 code, MUNHECA, to compute the neutrino spectrum by taking into account various QED processes, with the cascade developing either along the propagation in the cosmic microwave background in the high-redshift universe or in a predefined photon background surrounding the astrophysical source. The user can adjust various settings of MUNHECA, including the spectrum of injected high energy photons, the background photon field and the QED processes governing the cascade evolution. We improve the modeling of several processes, provide examples of the execution of MUNHECA and compare it with some earlier and more simplified estimates of the neutrino spectrum from electromagnetic cascades.

Fast, scalable decoding architectures that operate in a block-wise parallel fashion across space and time are essential for real-time fault-tolerant quantum computing. We introduce a scalable AI-based pre-decoder for the surface code that performs local, parallel error correction with low decoding runtimes, removing the majority of physical errors before passing residual syndromes to a downstream global decoder. This modular architecture is backend-agnostic and composes with arbitrary global decoding algorithms designed for surface codes, and our implementation is completely open source. Integrated with uncorrelated PyMatching, the pipeline achieves end-to-end decoding runtimes of order O(1 μs) per round at large code distances on NVIDIA GB300 GPUs while reducing logical error rates (LERs) relative to global decoding alone. In a block-wise parallel decoding scheme with access to multiple GPUs, the decoding runtime can be reduced to well below O(1 μs) per round. We observe further LER improvements by training a larger model, outperforming correlated PyMatching up to distance-13. We additionally introduce a noise-learning architecture that infers decoding weights directly from experimentally accessible syndrome statistics without requiring an explicit circuit-level noise model. We show that purely data-driven graph weight estimation can nearly match uncorrelated PyMatching and exceed correlated PyMatching in certain regimes, enabling highly-optimized decoding when hardware noise models are unknown or time-varying, as well as training pre-decoders with realistic noise models. Together, these results establish a practical, modular, and high-throughput decoding framework suitable for large-distance surface-code implementations.

Artificial neural networks are a popular and effective machine learning technique. Great progress has been made parallelizing the expensive training phase of an individual network, leading to highly specialized pieces of hardware, many based on GPU-type architectures, and more concurrent algorithms such as synthetic gradients. However, the training phase continues to be a bottleneck, where the training data must be processed serially over thousands of individual training runs. This work considers a multigrid reduction in time (MGRIT) algorithm that is able to parallelize over the thousands of training runs and converge to the exact same solution as traditional training would provide. MGRIT was originally developed to provide parallelism for time evolution problems that serially step through a finite number of time-steps. This work recasts the training of a neural network similarly, treating neural network training as an evolution equation that evolves the network weights from one step to the next. Thus, this work concerns distributed computing approaches for neural networks, but is distinct from other approaches which seek to parallelize only over individual training runs. The work concludes with supporting numerical results for two model problems.

Mixture of Experts (MoE) models enhance neural network scalability by dynamically selecting relevant experts per input token, enabling larger model sizes while maintaining manageable computation costs. However, efficient training of large-scale MoE models across thousands of GPUs presents significant challenges due to limitations in existing parallelism strategies. We introduce an end-to-end training framework for large-scale MoE models that utilizes five-dimensional hybrid parallelism: Tensor Parallelism, Expert Parallelism, Context Parallelism, Data Parallelism, and Pipeline Parallelism. Central to our approach is MoE Parallel Folding, a novel strategy that decouples the parallelization of attention and MoE layers in Transformer models, allowing each layer type to adopt optimal parallel configurations. Additionally, we develop a flexible token-level dispatcher that supports both token-dropping and token-dropless MoE training across all five dimensions of parallelism. This dispatcher accommodates dynamic tensor shapes and coordinates different parallelism schemes for Attention and MoE layers, facilitating complex parallelism implementations. Our experiments demonstrate significant improvements in training efficiency and scalability. We achieve up to 49.3% Model Flops Utilization (MFU) for the Mixtral 8x22B model and 39.0% MFU for the Qwen2-57B-A14B model on H100 GPUs, outperforming existing methods. The framework scales efficiently up to 1,024 GPUs and maintains high performance with sequence lengths up to 128K tokens, validating its effectiveness for large-scale MoE model training. The code is available in Megatron-Core.

Muon has emerged as an efficient alternative to Adam for pretraining, yet remains underused for fine-tuning. A key obstacle is that most open models are pretrained with Adam, and naively switching to Muon for fine-tuning leads to degraded performance due to an optimizer mismatch. We investigate this mismatch through controlled experiments and relate it to the distinct implicit biases of Adam and Muon. We provide evidence that the mismatch disrupts pretrained knowledge, and that this disruption scales with update strength. This leads us to hypothesize that constraining updates should mitigate the mismatch. We validate this with LoRA: across language and vision tasks, LoRA reduces the performance gap between Adam and Muon observed under full fine-tuning. Studies on LoRA rank, catastrophic forgetting, and LoRA variants further confirm that mismatch severity correlates with update strength. These results shed light on how optimizer mismatch affects fine-tuning and how it can be mitigated. Our code is available at https://github.com/XingyuQu/muon-finetune.

Future collider experiments require unprecedented precision in measurements of Higgs, electroweak, and flavour observables, placing stringent demands on event reconstruction. The achievable precision on Higgs couplings scales directly with the resolution on visible final state particles and their invariant masses. Current particle flow algorithms rely on detector specific clustering, limiting flexibility during detector design. Here we present an end-to-end global event reconstruction approach that maps charged particle tracks and calorimeter and muon hits directly to particle level objects. The method combines geometric algebra transformer networks with object condensation based clustering, followed by dedicated networks for particle identification and energy regression. Our approach is benchmarked on fully simulated electron positron collisions at FCC-ee using the CLD detector concept. It outperforms the state-of-the-art rule-based algorithm by 10--20\% in relative reconstruction efficiency, achieves up to two orders of magnitude reduction in fake-particle rates for charged hadrons, and improves visible energy and invariant mass resolution by 22\%. By decoupling reconstruction performance from detector-specific tuning, this framework enables rapid iteration during the detector design phase of future collider experiments.

Applying weight decay (WD) to matrix layers is standard practice in large-language-model pretraining. Prior work suggests that stochastic gradient noise induces a Brownian-like expansion of the weight matrices W, whose growth is counteracted by WD, leading to a WD-noise equilibrium with a certain weight norm ||W||. In this work, we view the equilibrium norm as a harmful artifact of the training procedure, and address it by introducing learnable multipliers to learn the optimal scale. First, we attach a learnable scalar multiplier to W and confirm that the WD-noise equilibrium norm is suboptimal: the learned scale adapts to data and improves performance. We then argue that individual row and column norms are similarly constrained, and free their scale by introducing learnable per-row and per-column multipliers. Our method can be viewed as a learnable, more expressive generalization of muP multipliers. It outperforms a well-tuned muP baseline, reduces the computational overhead of multiplier tuning, and surfaces practical questions such as forward-pass symmetries and the width-scaling of the learned multipliers. Finally, we validate learnable multipliers with both Adam and Muon optimizers, where it shows improvement in downstream evaluations matching the improvement of the switching from Adam to Muon.

Invertible convolutions have been an essential element for building expressive normalizing flow-based generative models since their introduction in Glow. Several attempts have been made to design invertible k times k convolutions that are efficient in training and sampling passes. Though these attempts have improved the expressivity and sampling efficiency, they severely lagged behind Glow which used only 1 times 1 convolutions in terms of sampling time. Also, many of the approaches mask a large number of parameters of the underlying convolution, resulting in lower expressivity on a fixed run-time budget. We propose a k times k convolutional layer and Deep Normalizing Flow architecture which i.) has a fast parallel inversion algorithm with running time O(n k^2) (n is height and width of the input image and k is kernel size), ii.) masks the minimal amount of learnable parameters in a layer. iii.) gives better forward pass and sampling times comparable to other k times k convolution-based models on real-world benchmarks. We provide an implementation of the proposed parallel algorithm for sampling using our invertible convolutions on GPUs. Benchmarks on CIFAR-10, ImageNet, and CelebA datasets show comparable performance to previous works regarding bits per dimension while significantly improving the sampling time.

Combinatorial inverse problems in high energy physics span enormous algorithmic challenges. This work presents a new deep learning driven clustering algorithm that utilizes a space-time non-local trainable graph constructor, a graph neural network, and a set transformer. The model is trained with loss functions at the graph node, edge and object level, including contrastive learning and meta-supervision. The algorithm can be applied to problems such as charged particle tracking, calorimetry, pile-up discrimination, jet physics, and beyond. We showcase the effectiveness of this cutting-edge AI approach through particle tracking simulations. The code is available online.

Large language models (LLMs) have demonstrated impressive generalization and emergent capabilities, yet their pre-training remains computationally expensive and sensitive to optimization dynamics. While Adam-based optimizers offer fast convergence by adapting learning rates coordinate-wise, recent studies reveal that their updates often suffer from poor spectral conditioning and low-rank structures, hindering efficiency. Muon addresses this issue via global spectral normalization but lacks the per-coordinate adaptivity of Adam. In this work, we propose Column-Normalized Adam (Conda), a novel optimizer that bridges the strengths of both approaches. Conda projects updates into an orthogonal subspace and applies column-wise second moment normalization based on the projected gradients, thereby achieving both improved spectral conditioning and maintaining coordinate-wise adaptivity. This design alleviates the spectral pathologies of Adam while preserving its fast convergence behavior. Extensive experiments on the LLaMA and GPT-2 series show that Conda consistently outperforms AdamW, Muon, and other baselines in pre-training. Remarkably, on the LLaMA series, Conda achieves 2-2.5 the convergence speed of AdamW, measured in both training steps and training time. Further ablations demonstrate its robustness under diverse training setups. These results collectively highlight Conda as an effective and broadly applicable optimizer for large-scale LLM training. The code is released on https://github.com/jie040109/Conda

The field of plasma physics heavily relies on simulations to model various phenomena, such as instabilities, turbulence, and nonlinear behaviors that would otherwise be difficult to study from a purely theoretical approach. Simulations are fundamental in accurately setting up experiments, which can be extremely costly and complex. As high-fidelity tools, gyrokinetic simulations play a crucial role in discovering new physics, interpreting experimental results, and improving the design of next-generation devices. However, their high computational costs necessitate the use of acceleration platforms to reduce execution time. This work revolves around the TRIangular MEsh based Gyrokinetic (TRIMEG) code, which performs high-accuracy particle-in-cell plasma simulations in tokamak geometries, leveraging a novel finite element approach. The rise of graphical processing units (GPUs) constitutes an occasion to satisfy such computational needs, by offloading the most expensive portion of the code to the accelerators. The chosen approach features GPU offloading with the OpenMP API, which grants portability of the code to different architectures, namely AMD and NVIDIA. The particle pushing as well as the grid-to-particle operations have been ported to GPU platforms. Compiler limitations had to be overcome, and portions of the code were restructured to be suitable for GPU acceleration. Kernel performance was evaluated by carrying out GPU grid size exploration, as well as scalability studies. In addition, the efficiency of hybrid MPI-OpenMP offloading parallelization was assessed. The speedup of the GPU implementation was calculated by comparing it with the pure CPU version using different rationales. The Ion Temperature Gradient (ITG) mode was simulated using the GPU-accelerated version, and its correctness was verified in terms of the energy growth rate and the two-dimensional mode structures.

We show evidence of particle acceleration at GEV energies associated directly with protons from the prompt emission of a long-duration M6-class solar flare on July 17, 2023, rather than from protons acceleration by shocks from its associated Coronal Mass Ejection (CME), which erupted with a speed of 1342 km/s. Solar Energetic Particles (SEP) accelerated by the blast have reached Earth, up to an almost S3 (strong) category of a radiation storm on the NOAA scale. Also, we show a temporal correlation between the fast rising of GOES-16 proton and muon excess at ground level in the count rate of the New-Tupi muon detector at the central SAA region. A Monte Carlo spectral analysis based on muon excess at New-Tupi is consistent with the acceleration of electrons and protons (ions) up to relativistic energies (GeV energy range) in the impulsive phase of the flare. In addition, we present another two marginal particle excesses (with low confidence) at ground-level detectors in correlation with the solar flare prompt emission.

Matrix based optimizers such as Muon can substantially speed up language model pretraining, but their gains over AdamW are observed to shrink as model size and data scale grow when using standard constant decoupled weight decay. We propose Hyperball, a simple optimizer wrapper that addresses this issue. Given a base optimizer such as Adam or Muon, Hyperball sets the Frobenius norms of weight matrices and their corresponding optimizer updates to fixed constants. On Qwen3 style models up to 1.2B parameters, Muon Hyperball achieves 20--30% token equivalent speedup over weight decay baselines. Hyperball also improves learning rate transfer across widths and depths compared to decoupled weight decay. This method is motivated by prior theory showing that training with weight decay leads to an equilibrium weight norm that only depends on the training hyperparameters. Through this mechanism, the weight decay then decides the angular learning rate, i.e. how fast the direction of the weight matrix changes.

We present the technical report for Arcee Trinity Large, a sparse Mixture-of-Experts model with 400B total parameters and 13B activated per token. Additionally, we report on Trinity Nano and Trinity Mini, with Trinity Nano having 6B total parameters with 1B activated per token, Trinity Mini having 26B total parameters with 3B activated per token. The models' modern architecture includes interleaved local and global attention, gated attention, depth-scaled sandwich norm, and sigmoid routing for Mixture-of-Experts. For Trinity Large, we also introduce a new MoE load balancing strategy titled Soft-clamped Momentum Expert Bias Updates (SMEBU). We train the models using the Muon optimizer. All three models completed training with zero loss spikes. Trinity Nano and Trinity Mini were pre-trained on 10 trillion tokens, and Trinity Large was pre-trained on 17 trillion tokens. The model checkpoints are available at https://huggingface.co/arcee-ai.

Voronoi diagrams are essential geometrical structures with numerous applications, particularly astrophysics-driven finite volume methods. While serial algorithms for constructing these entities are well-established, parallel construction remains challenging. This is especially true in distributed memory systems, where each host manages only a subset of the input points. This process requires redistributing points across hosts and accurately computing the corresponding Voronoi cells. In this paper, we introduce a new distributed construction algorithm, which is implemented in our open-source C++ 3-dimensional Voronoi construction framework. Our approach leverages Delaunay triangulation as an intermediate step, which is then transformed into a Voronoi diagram. We introduce the algorithms we implemented for the precise construction and our load-balancing approach and compare the running time with other state-of-the-art frameworks. MadVoro is a versatile tool that can be applied in various scientific domains, such as mesh decomposition, computational physics, chemistry, and machine learning.

Large language models deliver strong reasoning and tool-use skills, yet their computational demands make them impractical for edge or cost-sensitive deployments. We present Xmodel-2.5, a 1.3-billion-parameter small language model designed as a drop-in agent core. Training with maximal-update parameterization (μP) allows hyper-parameters tuned on a 20M-parameter proxy to transfer directly to the full model, even under the parameter-tied tie-word-embedding architecture. A 1.4T-token Warmup--Stable--Decay curriculum is used, and we further show that switching from AdamW to Muon during the decay phase improves the 13-task reasoning average by 4.58\,\% while keeping every other hyper-parameter fixed, verifying that early AdamW stability can be paired with late Muon sharpening for better downstream performance. FP8-mixed-precision training balances accuracy and throughput. All checkpoints, recipes, and evaluation code are released under the Apache-2.0 license.https://huggingface.co/XiaoduoAILab/Xmodel-2.5 and https://huggingface.co/XiaoduoAILab/Xmodel-2.5-history (training checkpoints). Training code and evaluation harness: https://github.com/XiaoduoAILab/Xmodel-2.5.

Large language models (LLMs) suffer from low efficiency as the mismatch between the requirement of auto-regressive decoding and the design of most contemporary GPUs. Specifically, billions to trillions of parameters must be loaded to the GPU cache through its limited memory bandwidth for computation, but only a small batch of tokens is actually computed. Consequently, the GPU spends most of its time on memory transfer instead of computation. Recently, parallel decoding, a type of speculative decoding algorithms, is becoming more popular and has demonstrated impressive efficiency improvement in generation. It introduces extra decoding heads to large models, enabling them to predict multiple subsequent tokens simultaneously and verify these candidate continuations in a single decoding step. However, this approach deviates from the training objective of next token prediction used during pre-training, resulting in a low hit rate for candidate tokens. In this paper, we propose a new speculative decoding algorithm, Clover, which integrates sequential knowledge into the parallel decoding process. This enhancement improves the hit rate of speculators and thus boosts the overall efficiency. Clover transmits the sequential knowledge from pre-speculated tokens via the Regressive Connection, then employs an Attention Decoder to integrate these speculated tokens. Additionally, Clover incorporates an Augmenting Block that modifies the hidden states to better align with the purpose of speculative generation rather than next token prediction. The experiment results demonstrate that Clover outperforms the baseline by up to 91% on Baichuan-Small and 146% on Baichuan-Large, respectively, and exceeds the performance of the previously top-performing method, Medusa, by up to 37% on Baichuan-Small and 57% on Baichuan-Large, respectively.

Large language models (LLMs) have demonstrated remarkable success as foundational models, benefiting various downstream applications through fine-tuning. Recent studies on loss scaling have demonstrated the superior performance of larger LLMs compared to their smaller counterparts. Nevertheless, training LLMs with billions of parameters poses significant challenges and requires considerable computational resources. For example, training a one trillion parameter GPT-style model on 20 trillion tokens requires a staggering 120 million exaflops of computation. This research explores efficient distributed training strategies to extract this computation from Frontier, the world's first exascale supercomputer dedicated to open science. We enable and investigate various model and data parallel training techniques, such as tensor parallelism, pipeline parallelism, and sharded data parallelism, to facilitate training a trillion-parameter model on Frontier. We empirically assess these techniques and their associated parameters to determine their impact on memory footprint, communication latency, and GPU's computational efficiency. We analyze the complex interplay among these techniques and find a strategy to combine them to achieve high throughput through hyperparameter tuning. We have identified efficient strategies for training large LLMs of varying sizes through empirical analysis and hyperparameter tuning. For 22 Billion, 175 Billion, and 1 Trillion parameters, we achieved GPU throughputs of 38.38%, 36.14%, and 31.96%, respectively. For the training of the 175 Billion parameter model and the 1 Trillion parameter model, we achieved 100% weak scaling efficiency on 1024 and 3072 MI250X GPUs, respectively. We also achieved strong scaling efficiencies of 89% and 87% for these two models.

Training multi-billion to trillion-parameter language models efficiently on GPU clusters requires leveraging multiple parallelism strategies. We present Galvatron, a novel open-source framework (dubbed 'Optimus-Megatron' in the implementation) that dynamically combines data parallelism, tensor model parallelism, and pipeline parallelism to optimize training throughput. Built atop PyTorch and integrating NVIDIA's Megatron-LM and Microsoft's DeepSpeed, Galvatron automatically selects and adjusts parallelism strategies in real time based on model architecture, hardware, and training dynamics. This paper details Galvatron's key features -- automatic hybrid parallelism selection, layer-wise and phase-wise strategy optimization, and runtime adaptation -- and contrasts them with existing static frameworks. We describe the system's technical stack, including its use of DeepSpeed's ZeRO and NCCL communication, and provide an in-depth implementation overview of its core modules (profilers, strategy selector, parallelism manager). We then illustrate how Galvatron can be seamlessly integrated into existing training pipelines with minimal code modifications, providing companies a plug-and-play solution for efficient large-model training. Finally, we situate Galvatron in context with related efforts (NVIDIA Megatron-LM, Microsoft DeepSpeed, Google GShard, Meta FairScale, etc.), highlighting how it advances the state of the art in distributed deep learning. References to the GitHub repository and relevant literature are provided throughout.

Scaling laws for large language models depend critically on the optimizer and parameterization. Existing hyperparameter transfer laws are mainly developed for first-order optimizers, and they do not structurally prevent training instability at scale. Recent hypersphere optimization methods constrain weight matrices to a fixed-norm hypersphere, offering a promising alternative for more stable scaling. We introduce HyperP (Hypersphere Parameterization), the first framework for transferring optimal learning rates across model width, depth, training tokens, and Mixture-of-Experts (MoE) granularity under the Frobenius-sphere constraint with the Muon optimizer. We prove that weight decay is a first-order no-op on the Frobenius sphere, show that Depth-μP remains necessary, and find that the optimal learning rate follows the same data-scaling power law with the "magic exponent" 0.32 previously observed for AdamW. A single base learning rate tuned at the smallest scale transfers across all compute budgets under HyperP, yielding 1.58times compute efficiency over a strong Muon baseline at 6times10^{21} FLOPs. Moreover, HyperP delivers transferable stability: all monitored instability indicators, including Z-values, output RMS, and activation outliers, remain bounded and non-increasing under training FLOPs scaling. We also propose SqrtGate, an MoE gating mechanism derived from the hypersphere constraint that preserves output RMS across MoE granularities for improved granularity scaling, and show that hypersphere optimization enables substantially larger auxiliary load-balancing weights, yielding both strong performance and good expert balance. We release our training codebase at https://github.com/microsoft/ArchScale.

The transition from sequential to parallel computing is essential for modern high-performance applications but is hindered by the steep learning curve of concurrent programming. This challenge is magnified for irregular data structures (such as sparse graphs, unbalanced trees, and non-uniform meshes) where static scheduling fails and data dependencies are unpredictable. Current Large Language Models (LLMs) often fail catastrophically on these tasks, generating code plagued by subtle race conditions, deadlocks, and sub-optimal scaling. We bridge this gap with ParEVO, a framework designed to synthesize high-performance parallel algorithms for irregular data. Our contributions include: (1) The Parlay-Instruct Corpus, a curated dataset of 13,820 tasks synthesized via a "Critic-Refine" pipeline that explicitly filters for empirically performant algorithms that effectively utilize Work-Span parallel primitives; (2) specialized DeepSeek, Qwen, and Gemini models fine-tuned to align probabilistic generation with the rigorous semantics of the ParlayLib library; and (3) an Evolutionary Coding Agent (ECA) that improves the "last mile" of correctness by iteratively repairing code using feedback from compilers, dynamic race detectors, and performance profilers. On the ParEval benchmark, ParEVO achieves an average 106x speedup (with a maximum of 1103x) across the suite, and a robust 13.6x speedup specifically on complex irregular graph problems, outperforming state-of-the-art commercial models. Furthermore, our evolutionary approach matches state-of-the-art expert human baselines, achieving up to a 4.1x speedup on specific highly-irregular kernels. Source code and datasets are available at https://github.com/WildAlg/ParEVO.

The emergence of Large Language Models (LLMs) has necessitated the adoption of parallel training techniques, involving the deployment of thousands of GPUs to train a single model. Unfortunately, we have found that the efficiency of current parallel training is often suboptimal, largely due to the following two main issues. Firstly, hardware failures are inevitable, leading to interruptions in the training tasks. The inability to quickly identify the faulty components results in a substantial waste of GPU resources. Secondly, since GPUs must wait for parameter synchronization to complete before proceeding to the next round of computation, network congestions can greatly increase the waiting time for GPUs. To address these challenges, this paper introduces a communication-driven solution, namely the C4. The key insights of C4 are two folds. First, in parallel training, collective communication exhibits periodic and homogeneous characteristics, so any anomalies are certainly due to some form of hardware malfunction. By leveraging this feature, C4 can rapidly identify the faulty components, swiftly isolate the anomaly, and restart the task, thereby avoiding resource wastage caused by delays in anomaly detection. Second, the predictable communication model of collective communication, involving few large flows, allows C4 to efficiently execute traffic planning, substantially reducing network congestion. C4 has been extensively implemented across our production systems, cutting error-induced overhead by roughly 30% and enhancing runtime performance by about 15% for certain applications with moderate communication costs.

Large language models have led to state-of-the-art accuracies across a range of tasks. However, training these models efficiently is challenging for two reasons: a) GPU memory capacity is limited, making it impossible to fit large models on even a multi-GPU server, and b) the number of compute operations required to train these models can result in unrealistically long training times. Consequently, new methods of model parallelism such as tensor and pipeline parallelism have been proposed. Unfortunately, naive usage of these methods leads to fundamental scaling issues at thousands of GPUs, e.g., due to expensive cross-node communication or devices spending significant time waiting on other devices to make progress. In this paper, we show how different types of parallelism methods (tensor, pipeline, and data parallelism) can be composed to scale to thousands of GPUs and models with trillions of parameters. We survey techniques for pipeline parallelism and propose a novel interleaved pipeline parallelism schedule that can improve throughput by 10+% with memory footprint comparable to existing approaches. We quantitatively study the trade-offs between tensor, pipeline, and data parallelism, and provide intuition as to how to configure distributed training of a large model. Our approach allows us to perform training iterations on a model with 1 trillion parameters at 502 petaFLOP/s on 3072 GPUs with achieved per-GPU throughput of 52% of theoretical peak. Our code is open sourced at https://github.com/nvidia/megatron-lm.

This paper explores the problem of collision-free motion generation for manipulators by formulating it as a global motion optimization problem. We develop a parallel optimization technique to solve this problem and demonstrate its effectiveness on massively parallel GPUs. We show that combining simple optimization techniques with many parallel seeds leads to solving difficult motion generation problems within 50ms on average, 60x faster than state-of-the-art (SOTA) trajectory optimization methods. We achieve SOTA performance by combining L-BFGS step direction estimation with a novel parallel noisy line search scheme and a particle-based optimization solver. To further aid trajectory optimization, we develop a parallel geometric planner that plans within 20ms and also introduce a collision-free IK solver that can solve over 7000 queries/s. We package our contributions into a state of the art GPU accelerated motion generation library, cuRobo and release it to enrich the robotics community. Additional details are available at https://curobo.org

We present a comprehensive study of the Muonic Portal to Vector Dark Matter (MPVDM), a minimal yet phenomenologically rich extension of the Standard Model featuring a new SU(2)_D gauge symmetry and vector-like muons. In this framework the dark sector interacts with the Standard Model only through these heavy leptons, linking dark matter and the muon sector. The MPVDM can simultaneously explain the observed relic abundance and the muon anomalous magnetic moment a_mu under both the "tension" and "compatibility" scenarios motivated by recent (g-2)_mu results. A key finding is a generic off-resonance velocity suppression mechanism that allows light (<1 GeV) vector dark matter to evade CMB limits near 2*m_DM ~ m_H_D. Unlike scenarios based on ultra narrow Breit-Wigner resonances and early kinetic decoupling, the suppression follows from the temperature evolution of the annihilation cross section in a moderately detuned near resonant regime, where being 10-20 percent below resonance gives the required CMB era suppression without fine tuning. A five dimensional parameter scan shows that the tension scenario requires sub GeV dark matter with g_D ~ 1e-3 and TeV scale vector like muons, while the compatibility scenario admits a broad mass range up to multi TeV. Recasting ATLAS and CMS searches for mu+ mu- + E_T^miss sets a lower bound of about 850 GeV on vector like muons. The MPVDM thus offers a unified, predictive, and experimentally accessible framework linking dark matter and muon physics across cosmological and collider frontiers.

While most autoregressive LLMs are constrained to one-by-one decoding, diffusion LLMs (dLLMs) have attracted growing interest for their potential to dramatically accelerate inference through parallel decoding. Despite this promise, the conditional independence assumption in dLLMs causes parallel decoding to ignore token dependencies, inevitably degrading generation quality when these dependencies are strong. However, existing works largely overlook these inherent challenges, and evaluations on standard benchmarks (e.g., math and coding) are not sufficient to capture the quality degradation caused by parallel decoding. To address this gap, we first provide an information-theoretic analysis of parallel decoding. We then conduct case studies on analytically tractable synthetic list operations from both data distribution and decoding strategy perspectives, offering quantitative insights that highlight the fundamental limitations of parallel decoding. Building on these insights, we propose ParallelBench, the first benchmark specifically designed for dLLMs, featuring realistic tasks that are trivial for humans and autoregressive LLMs yet exceptionally challenging for dLLMs under parallel decoding. Using ParallelBench, we systematically analyze both dLLMs and autoregressive LLMs, revealing that: (i) dLLMs under parallel decoding can suffer dramatic quality degradation in real-world scenarios, and (ii) current parallel decoding strategies struggle to adapt their degree of parallelism based on task difficulty, thus failing to achieve meaningful speedup without compromising quality. Our findings underscore the pressing need for innovative decoding methods that can overcome the current speed-quality trade-off. We release our benchmark to help accelerate the development of truly efficient dLLMs.

About 90% of the computing resources available to the LHCb experiment has been spent to produce simulated data samples for Run 2 of the Large Hadron Collider at CERN. The upgraded LHCb detector will be able to collect larger data samples, requiring many more simulated events to analyze the data to be collected in Run 3. Simulation is a key necessity of analysis to interpret signal, reject background and measure efficiencies. The needed simulation will far exceed the pledged resources, requiring an evolution in technologies and techniques to produce these simulated data samples. In this contribution, we discuss Lamarr, a Gaudi-based framework to speed-up the simulation production parameterizing both the detector response and the reconstruction algorithms of the LHCb experiment. Deep Generative Models powered by several algorithms and strategies are employed to effectively parameterize the high-level response of the single components of the LHCb detector, encoding within neural networks the experimental errors and uncertainties introduced in the detection and reconstruction phases. Where possible, models are trained directly on real data, statistically subtracting any background components by applying appropriate reweighing procedures. Embedding Lamarr in the general LHCb Gauss Simulation framework allows to combine its execution with any of the available generators in a seamless way. The resulting software package enables a simulation process independent of the detailed simulation used to date.

The auto-regressive decoding of Large Language Models (LLMs) results in significant overheads in their hardware performance. While recent research has investigated various speculative decoding techniques for multi-token generation, these efforts have primarily focused on improving processing speed such as throughput. Crucially, they often neglect other metrics essential for real-life deployments, such as memory consumption and training cost. To overcome these limitations, we propose a novel parallel prompt decoding that requires only 0.0002% trainable parameters, enabling efficient training on a single A100-40GB GPU in just 16 hours. Inspired by the human natural language generation process, PPD approximates outputs generated at future timesteps in parallel by using multiple prompt tokens. This approach partially recovers the missing conditional dependency information necessary for multi-token generation, resulting in up to a 28% higher acceptance rate for long-range predictions. Furthermore, we present a hardware-aware dynamic sparse tree technique that adaptively optimizes this decoding scheme to fully leverage the computational capacities on different GPUs. Through extensive experiments across LLMs ranging from MobileLlama to Vicuna-13B on a wide range of benchmarks, our approach demonstrates up to 2.49times speedup and maintains a minimal runtime memory overhead of just 0.0004%. More importantly, our parallel prompt decoding can serve as an orthogonal optimization for synergistic integration with existing speculative decoding, showing up to 1.22times further speed improvement. Our code is available at https://github.com/hmarkc/parallel-prompt-decoding.

ML-augmented algorithms utilize predictions to achieve performance beyond their worst-case bounds. Producing these predictions might be a costly operation -- this motivated Im et al. '22 to introduce the study of algorithms which use predictions parsimoniously. We design parsimonious algorithms for caching and MTS with action predictions, proposed by Antoniadis et al. '20, focusing on the parameters of consistency (performance with perfect predictions) and smoothness (dependence of their performance on the prediction error). Our algorithm for caching is 1-consistent, robust, and its smoothness deteriorates with the decreasing number of available predictions. We propose an algorithm for general MTS whose consistency and smoothness both scale linearly with the decreasing number of predictions. Without the restriction on the number of available predictions, both algorithms match the earlier guarantees achieved by Antoniadis et al. '20.

Sequence parallelism (SP), which divides the sequence dimension of input tensors across multiple computational devices, is becoming key to unlocking the long-context capabilities of generative AI models. This paper investigates the state-of-the-art SP approaches, i.e. DeepSpeed-Ulysses and Ring-Attention, and proposes a unified SP approach, which is more robust to transformer model architectures and network hardware topology. This paper compares the communication and memory cost of SP and existing parallelism, including data/tensor/zero/expert/pipeline parallelism, and discusses the best practices for designing hybrid 4D parallelism involving SP. We achieved 86% MFU on two 8xA800 nodes using SP for sequence length 208K for the LLAMA3-8B model. Our code is publicly available on https://github.com/feifeibear/long-context-attention.

In this study, we propose a simple method for fault-tolerant Strassen-like matrix multiplications. The proposed method is based on using two distinct Strassen-like algorithms instead of replicating a given one. We have realized that using two different algorithms, new check relations arise resulting in more local computations. These local computations are found using computer aided search. To improve performance, special parity (extra) sub-matrix multiplications (PSMMs) are generated (two of them) at the expense of increasing communication/computation cost of the system. Our preliminary results demonstrate that the proposed method outperforms a Strassen-like algorithm with two copies and secures a very close performance to three copy version using only 2 PSMMs, reducing the total number of compute nodes by around 24\% i.e., from 21 to 16.

The inference process in Large Language Models (LLMs) is often limited due to the absence of parallelism in the auto-regressive decoding process, resulting in most operations being restricted by the memory bandwidth of accelerators. While methods such as speculative decoding have been suggested to address this issue, their implementation is impeded by the challenges associated with acquiring and maintaining a separate draft model. In this paper, we present Medusa, an efficient method that augments LLM inference by adding extra decoding heads to predict multiple subsequent tokens in parallel. Using a tree-based attention mechanism, Medusa constructs multiple candidate continuations and verifies them simultaneously in each decoding step. By leveraging parallel processing, Medusa introduces only minimal overhead in terms of single-step latency while substantially reducing the number of decoding steps required. We present two levels of fine-tuning procedures for Medusa to meet the needs of different use cases: Medusa-1: Medusa is directly fine-tuned on top of a frozen backbone LLM, enabling lossless inference acceleration. Medusa-2: Medusa is fine-tuned together with the backbone LLM, enabling better prediction accuracy of Medusa heads and higher speedup but needing a special training recipe that preserves the backbone model's capabilities. Moreover, we propose several extensions that improve or expand the utility of Medusa, including a self-distillation to handle situations where no training data is available and a typical acceptance scheme to boost the acceptance rate while maintaining generation quality. We evaluate Medusa on models of various sizes and training procedures. Our experiments demonstrate that Medusa-1 can achieve over 2.2x speedup without compromising generation quality, while Medusa-2 further improves the speedup to 2.3-3.6x.

Inverse of an invertible convolution is an important operation that comes up in Normalizing Flows, Image Deblurring, etc. The naive algorithm for backpropagation of this operation using Gaussian elimination has running time O(n^3) where n is the number of pixels in the image. We give a fast parallel backpropagation algorithm with running time O(n) for a square image and provide a GPU implementation of the same. Inverse Convolutions are usually used in Normalizing Flows in the sampling pass, making them slow. We propose to use Inverse Convolutions in the forward (image to latent vector) pass of the Normalizing flow. Since the sampling pass is the inverse of the forward pass, it will use convolutions only, resulting in efficient sampling times. We use our parallel backpropagation algorithm for optimizing the inverse convolution layer resulting in fast training times also. We implement this approach in various Normalizing Flow backbones, resulting in our Inverse-Flow models. We benchmark Inverse-Flow on standard datasets and show significantly improved sampling times with similar bits per dimension compared to previous models.

The imperative need to scale computation across numerous nodes highlights the significance of efficient parallel computing, particularly in the realm of Message Passing Interface (MPI) integration. The challenging parallel programming task of generating MPI-based parallel programs has remained unexplored. This study first investigates the performance of state-of-the-art language models in generating MPI-based parallel programs. Findings reveal that widely used models such as GPT-3.5 and PolyCoder (specialized multi-lingual code models) exhibit notable performance degradation, when generating MPI-based programs compared to general-purpose programs. In contrast, domain-specific models such as MonoCoder, which are pretrained on MPI-related programming languages of C and C++, outperform larger models. Subsequently, we introduce a dedicated downstream task of MPI-based program generation by fine-tuning MonoCoder on HPCorpusMPI. We call the resulting model as MPIrigen. We propose an innovative preprocessing for completion only after observing the whole code, thus enabling better completion with a wider context. Comparative analysis against GPT-3.5 zero-shot performance, using a novel HPC-oriented evaluation method, demonstrates that MPIrigen excels in generating accurate MPI functions up to 0.8 accuracy in location and function predictions, and with more than 0.9 accuracy for argument predictions. The success of this tailored solution underscores the importance of domain-specific fine-tuning in optimizing language models for parallel computing code generation, paving the way for a new generation of automatic parallelization tools. The sources of this work are available at our GitHub MPIrigen repository: https://github.com/Scientific-Computing-Lab-NRCN/MPI-rigen

Despite advances in the programmable logic capabilities of modern trigger systems, a significant bottleneck remains in the amount of data to be transported from the detector to off-detector logic where trigger decisions are made. We demonstrate that a neural network autoencoder model can be implemented in a radiation tolerant ASIC to perform lossy data compression alleviating the data transmission problem while preserving critical information of the detector energy profile. For our application, we consider the high-granularity calorimeter from the CMS experiment at the CERN Large Hadron Collider. The advantage of the machine learning approach is in the flexibility and configurability of the algorithm. By changing the neural network weights, a unique data compression algorithm can be deployed for each sensor in different detector regions, and changing detector or collider conditions. To meet area, performance, and power constraints, we perform a quantization-aware training to create an optimized neural network hardware implementation. The design is achieved through the use of high-level synthesis tools and the hls4ml framework, and was processed through synthesis and physical layout flows based on a LP CMOS 65 nm technology node. The flow anticipates 200 Mrad of ionizing radiation to select gates, and reports a total area of 3.6 mm^2 and consumes 95 mW of power. The simulated energy consumption per inference is 2.4 nJ. This is the first radiation tolerant on-detector ASIC implementation of a neural network that has been designed for particle physics applications.

Feedforward computation, such as evaluating a neural network or sampling from an autoregressive model, is ubiquitous in machine learning. The sequential nature of feedforward computation, however, requires a strict order of execution and cannot be easily accelerated with parallel computing. To enable parallelization, we frame the task of feedforward computation as solving a system of nonlinear equations. We then propose to find the solution using a Jacobi or Gauss-Seidel fixed-point iteration method, as well as hybrid methods of both. Crucially, Jacobi updates operate independently on each equation and can be executed in parallel. Our method is guaranteed to give exactly the same values as the original feedforward computation with a reduced (or equal) number of parallelizable iterations, and hence reduced time given sufficient parallel computing power. Experimentally, we demonstrate the effectiveness of our approach in accelerating (i) backpropagation of RNNs, (ii) evaluation of DenseNets, and (iii) autoregressive sampling of MADE and PixelCNN++, with speedup factors between 2.1 and 26 under various settings.

We introduce Pion, a spectrum-preserving optimizer for large language model (LLM) training based on orthogonal equivalence transformation. Unlike additive optimizers such as Adam and Muon, Pion updates each weight matrix through left and right orthogonal transformations, preserving its singular values throughout training. This yields an optimization mechanism that modulates the geometry of weight matrices while keeping their spectral norm fixed. We derive the Pion update rule, systematically examine its design choices, and analyze its convergence behavior along with several key properties. Empirical results show that Pion offers a stable and competitive alternative to standard optimizers for both LLM pretraining and finetuning.

Expert parallelism has been introduced as a strategy to distribute the computational workload of sparsely-gated mixture-of-experts (MoE) models across multiple computing devices, facilitating the execution of these increasingly large-scale models. However, the All-to-All communication intrinsic to expert parallelism constitutes a significant overhead, diminishing the MoE models' efficiency. Current optimization approaches offer some relief, yet they are constrained by the sequential interdependence of communication and computation operations. To address this limitation, we present a novel shortcut-connected MoE architecture with overlapping parallel strategy, designated as ScMoE, which effectively decouples communication from its conventional sequence, allowing for a substantial overlap of 70% to 100% with computation. When compared with the prevalent top-2 MoE architecture, ScMoE demonstrates training speed improvements of 30% and 11%, and inference improvements of 40% and 15%, in our PCIe and NVLink hardware environments, respectively, where communication constitutes 60% and 15% of the total MoE time consumption. On the other hand, extensive experiments and theoretical analyses indicate that ScMoE not only achieves comparable but in some instances surpasses the model quality of existing approaches in vision and language tasks.

Rapid advancement of machine learning solutions has often coincided with the production of a test public data set. Such datasets reduce the largest barrier to entry for tackling a problem -- procuring data -- while also providing a benchmark to compare different solutions. Furthermore, large datasets have been used to train high-performing feature finders which are then used in new approaches to problems beyond that initially defined. In order to encourage the rapid development in the analysis of data collected using liquid argon time projection chambers, a class of particle detectors used in high energy physics experiments, we have produced the PILArNet, first 2D and 3D open dataset to be used for a couple of key analysis tasks. The initial dataset presented in this paper contains 300,000 samples simulated and recorded in three different volume sizes. The dataset is stored efficiently in sparse 2D and 3D matrix format with auxiliary information about simulated particles in the volume, and is made available for public research use. In this paper we describe the dataset, tasks, and the method used to procure the sample.

Scaling the size of language models to tens of billions of parameters has led to impressive performance on a wide range of tasks. At generation, these models are used auto-regressively, requiring a forward pass for each generated token, and thus reading the full set of parameters from memory. This memory access forms the primary bottleneck for generation and it worsens as the model size increases. Moreover, executing a forward pass for multiple tokens in parallel often takes nearly the same time as it does for just one token. These two observations lead to the development of speculative sampling, where a second smaller model is used to draft a few tokens, that are then validated or rejected using a single forward pass of the large model. Unfortunately, this method requires two models that share the same tokenizer and thus limits its adoption. As an alternative, we propose to use parallel decoding as a way to draft multiple tokens from a single model with no computational cost, nor the need for a second model. Our approach only requires an additional input token that marks the words that will be generated simultaneously. We show promising performance (up to 30% speed-up) while requiring only as few as O(d_{emb}) additional parameters.

Autoregressive decoding limits the efficiency of transformers for Machine Translation (MT). The community proposed specific network architectures and learning-based methods to solve this issue, which are expensive and require changes to the MT model, trading inference speed at the cost of the translation quality. In this paper, we propose to address the problem from the point of view of decoding algorithms, as a less explored but rather compelling direction. We propose to reframe the standard greedy autoregressive decoding of MT with a parallel formulation leveraging Jacobi and Gauss-Seidel fixed-point iteration methods for fast inference. This formulation allows to speed up existing models without training or modifications while retaining translation quality. We present three parallel decoding algorithms and test them on different languages and models showing how the parallelization introduces a speedup up to 38% w.r.t. the standard autoregressive decoding and nearly 2x when scaling the method on parallel resources. Finally, we introduce a decoding dependency graph visualizer (DDGviz) that let us see how the model has learned the conditional dependence between tokens and inspect the decoding procedure.

Tau leptons serve as an important tool for studying the production of Higgs and electroweak bosons, both within and beyond the Standard Model of particle physics. Accurate reconstruction and identification of hadronically decaying tau leptons is a crucial task for current and future high energy physics experiments. Given the advances in jet tagging, we demonstrate how tau lepton reconstruction can be decomposed into tau identification, kinematic reconstruction, and decay mode classification in a multi-task machine learning setup. Based on an electron-positron collision dataset with full detector simulation and reconstruction, we show that common jet tagging architectures can be effectively used for these sub-tasks. We achieve comparable momentum resolutions of 2-3% with all the tested models, while the precision of reconstructing individual decay modes is between 80-95%. We find ParticleTransformer to be the best-performing approach, significantly outperforming the heuristic baseline. This paper also serves as an introduction to a new publicly available Fuτure dataset for the development of tau reconstruction algorithms. This allows to further study the resilience of ML models to domain shifts and the efficient use of foundation models for such tasks.

Identifying and reconstructing hadronic τ decays (τ_{h}) is an important task at current and future high-energy physics experiments, as τ_{h} represent an important tool to analyze the production of Higgs and electroweak bosons as well as to search for physics beyond the Standard Model. The identification of τ_{h} can be viewed as a generalization and extension of jet-flavour tagging, which has in the recent years undergone significant progress due to the use of deep learning. Based on a granular simulation with realistic detector effects and a particle flow-based event reconstruction, we show in this paper that deep learning-based jet-flavour-tagging algorithms are powerful τ_{h} identifiers. Specifically, we show that jet-flavour-tagging algorithms such as LorentzNet and ParticleTransformer can be adapted in an end-to-end fashion for discriminating τ_{h} from quark and gluon jets. We find that the end-to-end transformer-based approach significantly outperforms contemporary state-of-the-art τ_{h} reconstruction and identification algorithms currently in use at the Large Hadron Collider.

Novel machine learning methods for tabular data generation are often developed on small datasets which do not match the scale required for scientific applications. We investigate a recent proposal to use XGBoost as the function approximator in diffusion and flow-matching models on tabular data, which proved to be extremely memory intensive, even on tiny datasets. In this work, we conduct a critical analysis of the existing implementation from an engineering perspective, and show that these limitations are not fundamental to the method; with better implementation it can be scaled to datasets 370x larger than previously used. Our efficient implementation also unlocks scaling models to much larger sizes which we show directly leads to improved performance on benchmark tasks. We also propose algorithmic improvements that can further benefit resource usage and model performance, including multi-output trees which are well-suited to generative modeling. Finally, we present results on large-scale scientific datasets derived from experimental particle physics as part of the Fast Calorimeter Simulation Challenge. Code is available at https://github.com/layer6ai-labs/calo-forest.

Geometry-aware optimization algorithms, such as Muon, have achieved remarkable success in training deep neural networks (DNNs). These methods leverage the underlying geometry of DNNs by selecting appropriate norms for different layers and updating parameters via norm-constrained linear minimization oracles (LMOs). However, even within a group of layers associated with the same norm, the local curvature can be heterogeneous across layers and vary dynamically over the course of training. For example, recent work shows that sharpness varies substantially across transformer layers and throughout training, yet standard geometry-aware optimizers impose fixed learning rates to layers within the same group, which may be inefficient for DNN training. In this paper, we introduce a noise-adaptive layerwise learning rate scheme on top of geometry-aware optimization algorithms and substantially accelerate DNN training compared to methods that use fixed learning rates within each group. Our method estimates gradient variance in the dual norm induced by the chosen LMO on the fly, and uses it to assign time-varying noise-adaptive layerwise learning rates within each group. We provide a theoretical analysis showing that our algorithm achieves a sharp convergence rate. Empirical results on transformer architectures such as LLaMA and GPT demonstrate that our approach achieves faster convergence than state-of-the-art optimizers.

Sparse matrices, more specifically SpGEMM kernels, are commonly found in a wide range of applications, spanning graph-based path-finding to machine learning algorithms (e.g., neural networks). A particular challenge in implementing SpGEMM kernels has been the pressure placed on DRAM memory. One approach to tackle this problem is to use an inner product method for the SpGEMM kernel implementation. While the inner product produces fewer intermediate results, it can end up saturating the memory bandwidth, given the high number of redundant fetches of the input matrix elements. Using an outer product-based SpGEMM kernel can reduce redundant fetches, but at the cost of increased overhead due to extra computation and memory accesses for producing/managing partial products. In this thesis, we introduce a novel SpGEMM kernel implementation based on the row-wise product approach. We leverage atomic instructions to merge intermediate partial products as they are generated. The use of atomic instructions eliminates the need to create partial product matrices. To evaluate our row-wise product approach, we map an optimized SpGEMM kernel to a custom accelerator designed to accelerate graph-based applications. The targeted accelerator is an experimental system named PIUMA, being developed by Intel. PIUMA provides several attractive features, including fast context switching, user-configurable caches, globally addressable memory, non-coherent caches, and asynchronous pipelines. We tailor our SpGEMM kernel to exploit many of the features of the PIUMA fabric. This thesis compares our SpGEMM implementation against prior solutions, all mapped to the PIUMA framework. We briefly describe some of the PIUMA architecture features and then delve into the details of our optimized SpGEMM kernel. Our SpGEMM kernel can achieve 9.4x speedup as compared to competing approaches.

This paper presents a novel approach, named the Group Marching Tree (GMT*) algorithm, to planning on GPUs at rates amenable to application within control loops, allowing planning in real-world settings via repeated computation of near-optimal plans. GMT*, like the Fast Marching Tree (FMT) algorithm, explores the state space with a "lazy" dynamic programming recursion on a set of samples to grow a tree of near-optimal paths. GMT*, however, alters the approach of FMT with approximate dynamic programming by expanding, in parallel, the group of all active samples with cost below an increasing threshold, rather than only the minimum cost sample. This group approximation enables low-level parallelism over the sample set and removes the need for sequential data structures, while the "lazy" collision checking limits thread divergence---all contributing to a very efficient GPU implementation. While this approach incurs some suboptimality, we prove that GMT* remains asymptotically optimal up to a constant multiplicative factor. We show solutions for complex planning problems under differential constraints can be found in ~10 ms on a desktop GPU and ~30 ms on an embedded GPU, representing a significant speed up over the state of the art, with only small losses in performance. Finally, we present a scenario demonstrating the efficacy of planning within the control loop (~100 Hz) towards operating in dynamic, uncertain settings.

Speculative decoding (SD) has emerged as a promising technique to accelerate LLM inference by employing a small draft model to propose draft tokens in advance, and validating them in parallel with the large target model. However, the existing SD methods still remain constrained by their serialized execution, which causes the mutual waiting bubbles between the draft and target models. To address this challenge, we draw inspiration from branch prediction in modern processors and propose a novel framework SpecBranch to unlock branch parallelism in SD. Specifically, we first take an in-depth analysis of the potential of branch parallelism in SD, and recognize that the key challenge lies in the trade-offs between parallelization and token rollback. Based on the analysis, we introduce parallel speculative branches to preemptively hedge against likely rejections. Meanwhile, to enhance parallelism, we jointly orchestrate adaptive draft lengths with a hybrid combination of the implicit draft model confidence and explicit reusing of target model features. Extensive experiments across various models and benchmarks show that SpecBranch achieves over 1.8times sim 4.5times speedups against the auto-regressive decoding and reduces rollback tokens by 50\% for poorly aligned models, while maintaining an identical sampling distribution.

It is commonly believed that scaling language models should commit a significant space or time cost, by increasing the parameters (parameter scaling) or output tokens (inference-time scaling). We introduce the third and more inference-efficient scaling paradigm: increasing the model's parallel computation during both training and inference time. We apply P diverse and learnable transformations to the input, execute forward passes of the model in parallel, and dynamically aggregate the P outputs. This method, namely parallel scaling (ParScale), scales parallel computation by reusing existing parameters and can be applied to any model structure, optimization procedure, data, or task. We theoretically propose a new scaling law and validate it through large-scale pre-training, which shows that a model with P parallel streams is similar to scaling the parameters by O(log P) while showing superior inference efficiency. For example, ParScale can use up to 22times less memory increase and 6times less latency increase compared to parameter scaling that achieves the same performance improvement. It can also recycle an off-the-shelf pre-trained model into a parallelly scaled one by post-training on a small amount of tokens, further reducing the training budget. The new scaling law we discovered potentially facilitates the deployment of more powerful models in low-resource scenarios, and provides an alternative perspective for the role of computation in machine learning.

Diffusion Large Language Models (dLLMs) have emerged as a promising paradigm for parallel token generation, with block-wise variants garnering significant research interest. Despite their potential, existing dLLMs typically suffer from a rigid accuracy-parallelism trade-off: increasing the number of tokens per forward (TPF) via aggressive parallel decoding often leads to performance degradation and increased generation instability. We identify that this limitation stems from the model's inability to navigate high-parallelism regimes where approximation errors and local corruptions accumulate, ultimately undermining the reliability of parallel generation. To address this, we propose LightningRL, a post-training framework designed to directly optimize the speed-quality Pareto frontier of pre-trained dLLMs. Instead of forcing uniform parallelization, our approach leverages reinforcement learning to identify and reinforce high-parallelism trajectories that maintain generation accuracy. Built upon the Group Relative Policy Optimization (GRPO) framework, LightningRL introduces several enhancements tailored for dLLMs: (1) stabilized training via per-reward decoupled normalization; (2) token-level negative log-likelihood (NLL) regularization on correct trajectories to anchor model performance; and (3) a dynamic sampling strategy with TPF-aware filtering to enhance training efficiency. Experimental results across mathematical and coding benchmarks demonstrate that LightningRL consistently advances the Pareto frontier, achieving competitive task accuracy while significantly increasing parallelism, reaching an average TPF of 7.32 (with a peak of 11.10 on the MBPP dataset). Our code is available at https://github.com/SJTU-DENG-Lab/LightningRL.

Pipeline parallelism is one of the key components for large-scale distributed training, yet its efficiency suffers from pipeline bubbles which were deemed inevitable. In this work, we introduce a scheduling strategy that, to our knowledge, is the first to successfully achieve zero pipeline bubbles under synchronous training semantics. The key idea behind this improvement is to split the backward computation into two parts, one that computes gradient for the input and another that computes for the parameters. Based on this idea, we handcraft novel pipeline schedules that significantly outperform the baseline methods. We further develop an algorithm that automatically finds an optimal schedule based on specific model configuration and memory limit. Additionally, to truly achieve zero bubble, we introduce a novel technique to bypass synchronizations during the optimizer step. Experimental evaluations show that our method outperforms the 1F1B schedule up to 23% in throughput under a similar memory limit. This number can be further pushed to 31% when the memory constraint is relaxed. We believe our results mark a major step forward in harnessing the true potential of pipeline parallelism. We open sourced our implementation based on the popular Megatron-LM repository on https://github.com/sail-sg/zero-bubble-pipeline-parallelism.

Recently, diffusion models have achieved significant advances in vision, text, and robotics. However, they still face slow generation speeds due to sequential denoising processes. To address this, a parallel sampling method based on Picard iteration was introduced, effectively reducing sequential steps while ensuring exact convergence to the original output. Nonetheless, Picard iteration does not guarantee faster convergence, which can still result in slow generation in practice. In this work, we propose a new parallelization scheme, the Picard Consistency Model (PCM), which significantly reduces the number of generation steps in Picard iteration. Inspired by the consistency model, PCM is directly trained to predict the fixed-point solution, or the final output, at any stage of the convergence trajectory. Additionally, we introduce a new concept called model switching, which addresses PCM's limitations and ensures exact convergence. Extensive experiments demonstrate that PCM achieves up to a 2.71x speedup over sequential sampling and a 1.77x speedup over Picard iteration across various tasks, including image generation and robotic control.

Large language models (LLMs) such as GPT-3, OPT, and LLaMA have demonstrated remarkable accuracy in a wide range of tasks. However, training these models can incur significant expenses, often requiring tens of thousands of GPUs for months of continuous operation. Typically, this training is carried out in specialized GPU clusters equipped with homogeneous high-speed Remote Direct Memory Access (RDMA) network interface cards (NICs). The acquisition and maintenance of such dedicated clusters is challenging. Current LLM training frameworks, like Megatron-LM and Megatron-DeepSpeed, focus primarily on optimizing training within homogeneous cluster settings. In this paper, we introduce Holmes, a training framework for LLMs that employs thoughtfully crafted data and model parallelism strategies over the heterogeneous NIC environment. Our primary technical contribution lies in a novel scheduling method that intelligently allocates distinct computational tasklets in LLM training to specific groups of GPU devices based on the characteristics of their connected NICs. Furthermore, our proposed framework, utilizing pipeline parallel techniques, demonstrates scalability to multiple GPU clusters, even in scenarios without high-speed interconnects between nodes in distinct clusters. We conducted comprehensive experiments that involved various scenarios in the heterogeneous NIC environment. In most cases, our framework achieves performance levels close to those achievable with homogeneous RDMA-capable networks (InfiniBand or RoCE), significantly exceeding training efficiency within the pure Ethernet environment. Additionally, we verified that our framework outperforms other mainstream LLM frameworks under heterogeneous NIC environment in terms of training efficiency and can be seamlessly integrated with them.

This paper presents a comprehensive comparison of three dominant parallel programming models in High Performance Computing (HPC): Message Passing Interface (MPI), Open Multi-Processing (OpenMP), and Compute Unified Device Architecture (CUDA). Selecting optimal programming approaches for modern heterogeneous HPC architectures has become increasingly critical. We systematically analyze these models across multiple dimensions: architectural foundations, performance characteristics, domain-specific suitability, programming complexity, and recent advancements. We examine each model's strengths, weaknesses, and optimization techniques. Our investigation demonstrates that MPI excels in distributed memory environments with near-linear scalability for communication-intensive applications, but faces communication overhead challenges. OpenMP provides strong performance and usability in shared-memory systems and loop-centric tasks, though it is limited by shared memory contention. CUDA offers substantial performance gains for data-parallel GPU workloads, but is restricted to NVIDIA GPUs and requires specialized expertise. Performance evaluations across scientific simulations, machine learning, and data analytics reveal that hybrid approaches combining two or more models often yield optimal results in heterogeneous environments. The paper also discusses implementation challenges, optimization best practices, and emerging trends such as performance portability frameworks, task-based programming, and the convergence of HPC and Big Data. This research helps developers and researchers make informed decisions when selecting programming models for modern HPC applications, emphasizing that the best choice depends on application requirements, hardware, and development constraints.

Gradient signals in LLM training are highly anisotropic: recurrent linguistic structure concentrates energy into a small set of dominant spectral directions, while context specific information resides in a long tail. We show that this spike tail separation persists throughout training, with the spike occupying only about 1.5% of directions yet dominating optimizer statistics. This dominance suppresses tail learning by contracting tail updates through second moment normalization and tightening the globally stable learning rate bound. Motivated by this analysis, we propose Spectra, a spike aware optimizer that suppresses the dominant low rank spike subspace without amplifying the noise sensitive spectral tail. Spectra tracks the spike subspace via cached, warm started power iteration and applies low rank spectral shaping with negligible overhead and substantially reduced optimizer state memory. On LLaMA3 8B trained on 50B tokens, Spectra reaches the same target loss 30% faster than AdamW, reduces per step end to end overhead by 0.7%, cuts optimizer state memory by 49.25%, and improves average downstream accuracy by 1.62%. Compared to Muon, Spectra is 5.1x faster in optimizer processing time, achieves a lower final loss, and improves average accuracy by 0.66%.