Daily Papers

Understanding the properties of neural networks trained via stochastic gradient descent (SGD) is at the heart of the theory of deep learning. In this work, we take a mean-field view, and consider a two-layer ReLU network trained via SGD for a univariate regularized regression problem. Our main result is that SGD is biased towards a simple solution: at convergence, the ReLU network implements a piecewise linear map of the inputs, and the number of "knot" points - i.e., points where the tangent of the ReLU network estimator changes - between two consecutive training inputs is at most three. In particular, as the number of neurons of the network grows, the SGD dynamics is captured by the solution of a gradient flow and, at convergence, the distribution of the weights approaches the unique minimizer of a related free energy, which has a Gibbs form. Our key technical contribution consists in the analysis of the estimator resulting from this minimizer: we show that its second derivative vanishes everywhere, except at some specific locations which represent the "knot" points. We also provide empirical evidence that knots at locations distinct from the data points might occur, as predicted by our theory.

The pre-training cost of large language models (LLMs) is prohibitive. One cutting-edge approach to reduce the cost is zero-shot weight transfer, also known as model growth for some cases, which magically transfers the weights trained in a small model to a large model. However, there are still some theoretical mysteries behind the weight transfer. In this paper, inspired by prior applications of mean field theory to neural network dynamics, we introduce a mean field ansatz to provide a theoretical explanation for weight transfer. Specifically, we propose the row-column (RC) ansatz under the mean field point of view, which describes the measure structure of the weights in the neural network (NN) and admits a close measure dynamic. Thus, the weights of different sizes NN admit a common distribution under proper assumptions, and weight transfer methods can be viewed as sampling methods. We empirically validate the RC ansatz by exploring simple MLP examples and LLMs such as GPT-3 and Llama-3.1. We show the mean-field point of view is adequate under suitable assumptions which can provide theoretical support for zero-shot weight transfer.

Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from a theoretical point of view. The mathematical description of learning problems involves very large collections of interacting random variables, difficult to handle analytically as well as numerically. This complexity is precisely the object of study of statistical physics. Its mission, originally pointed towards natural systems, is to understand how macroscopic behaviors arise from microscopic laws. Mean-field methods are one type of approximation strategy developed in this view. We review a selection of classical mean-field methods and recent progress relevant for inference in neural networks. In particular, we remind the principles of derivations of high-temperature expansions, the replica method and message passing algorithms, highlighting their equivalences and complementarities. We also provide references for past and current directions of research on neural networks relying on mean-field methods.

In this paper we provide a novel analytical perspective on the theoretical understanding of gradient-based learning algorithms by interpreting consensus-based optimization (CBO), a recently proposed multi-particle derivative-free optimization method, as a stochastic relaxation of gradient descent. Remarkably, we observe that through communication of the particles, CBO exhibits a stochastic gradient descent (SGD)-like behavior despite solely relying on evaluations of the objective function. The fundamental value of such link between CBO and SGD lies in the fact that CBO is provably globally convergent to global minimizers for ample classes of nonsmooth and nonconvex objective functions, hence, on the one side, offering a novel explanation for the success of stochastic relaxations of gradient descent. On the other side, contrary to the conventional wisdom for which zero-order methods ought to be inefficient or not to possess generalization abilities, our results unveil an intrinsic gradient descent nature of such heuristics. This viewpoint furthermore complements previous insights into the working principles of CBO, which describe the dynamics in the mean-field limit through a nonlinear nonlocal partial differential equation that allows to alleviate complexities of the nonconvex function landscape. Our proofs leverage a completely nonsmooth analysis, which combines a novel quantitative version of the Laplace principle (log-sum-exp trick) and the minimizing movement scheme (proximal iteration). In doing so, we furnish useful and precise insights that explain how stochastic perturbations of gradient descent overcome energy barriers and reach deep levels of nonconvex functions. Instructive numerical illustrations support the provided theoretical insights.

In this paper, we explore the application of mean field theory, a technique from statistical physics, to deep metric learning and address the high training complexity commonly associated with conventional metric learning loss functions. By adapting mean field theory for deep metric learning, we develop an approach to design classification-based loss functions from pair-based ones, which can be considered complementary to the proxy-based approach. Applying the mean field theory to two pair-based loss functions, we derive two new loss functions, MeanFieldContrastive and MeanFieldClassWiseMultiSimilarity losses, with reduced training complexity. We extensively evaluate these derived loss functions on three image-retrieval datasets and demonstrate that our loss functions outperform baseline methods in two out of the three datasets.

We study in detail a one-dimensional lattice model of a continuum, conserved field (mass) that is transferred deterministically between neighbouring random sites. The model falls in a wider class of lattice models capturing the joint effect of random advection and diffusion and encompassing as specific cases, some models studied in the literature, like the Kang-Redner, Kipnis-Marchioro-Presutti, Takayasu-Taguchi, etc. The motivation for our setup comes from a straightforward interpretation as advection of particles in one-dimensional turbulence, but it is also related to a problem of synchronization of dynamical systems driven by common noise. For finite lattices, we study both the coalescence of an initially spread field (interpreted as roughening), and the statistical steady-state properties. We distinguish two main size-dependent regimes, depending on the strength of the diffusion term and on the lattice size. Using numerical simulations and mean-field approach, we study the statistics of the field. For weak diffusion, we unveil a characteristic hierarchical structure of the field. We also connect the model and the iterated function systems concept.

MeanFlow (MF) has recently been established as a framework for one-step generative modeling. However, its ``fastforward'' nature introduces key challenges in both the training objective and the guidance mechanism. First, the original MF's training target depends not only on the underlying ground-truth fields but also on the network itself. To address this issue, we recast the objective as a loss on the instantaneous velocity v, re-parameterized by a network that predicts the average velocity u. Our reformulation yields a more standard regression problem and improves the training stability. Second, the original MF fixes the classifier-free guidance scale during training, which sacrifices flexibility. We tackle this issue by formulating guidance as explicit conditioning variables, thereby retaining flexibility at test time. The diverse conditions are processed through in-context conditioning, which reduces model size and benefits performance. Overall, our improved MeanFlow (iMF) method, trained entirely from scratch, achieves 1.72 FID with a single function evaluation (1-NFE) on ImageNet 256times256. iMF substantially outperforms prior methods of this kind and closes the gap with multi-step methods while using no distillation. We hope our work will further advance fastforward generative modeling as a stand-alone paradigm.

Mean-field games have been used as a theoretical tool to obtain an approximate Nash equilibrium for symmetric and anonymous N-player games. However, limiting applicability, existing theoretical results assume variations of a "population generative model", which allows arbitrary modifications of the population distribution by the learning algorithm. Moreover, learning algorithms typically work on abstract simulators with population instead of the N-player game. Instead, we show that N agents running policy mirror ascent converge to the Nash equilibrium of the regularized game within mathcal{O}(varepsilon^{-2}) samples from a single sample trajectory without a population generative model, up to a standard O(1{N}) error due to the mean field. Taking a divergent approach from the literature, instead of working with the best-response map we first show that a policy mirror ascent map can be used to construct a contractive operator having the Nash equilibrium as its fixed point. We analyze single-path TD learning for N-agent games, proving sample complexity guarantees by only using a sample path from the N-agent simulator without a population generative model. Furthermore, we demonstrate that our methodology allows for independent learning by N agents with finite sample guarantees.

Coordinating large populations of interacting agents is a central challenge in multi-agent reinforcement learning (MARL), where the size of the joint state-action space scales exponentially with the number of agents. Mean-field methods alleviate this burden by aggregating agent interactions, but these approaches assume homogeneous interactions. Recent graphon-based frameworks capture heterogeneity, but are computationally expensive as the number of agents grows. Therefore, we introduce GMFS, a Graphon Mean-Field Subsampling framework for scalable cooperative MARL with heterogeneous agent interactions. By subsampling κ agents according to interaction strength, we approximate the graphon-weighted mean-field and learn a policy with sample complexity poly(κ) and optimality gap O(1/sqrtκ). We verify our theory with numerical simulations in robotic coordination, showing that GMFS achieves near-optimal performance.

The cost of hyperparameter tuning in deep learning has been rising with model sizes, prompting practitioners to find new tuning methods using a proxy of smaller networks. One such proposal uses muP parameterized networks, where the optimal hyperparameters for small width networks transfer to networks with arbitrarily large width. However, in this scheme, hyperparameters do not transfer across depths. As a remedy, we study residual networks with a residual branch scale of 1/text{depth} in combination with the muP parameterization. We provide experiments demonstrating that residual architectures including convolutional ResNets and Vision Transformers trained with this parameterization exhibit transfer of optimal hyperparameters across width and depth on CIFAR-10 and ImageNet. Furthermore, our empirical findings are supported and motivated by theory. Using recent developments in the dynamical mean field theory (DMFT) description of neural network learning dynamics, we show that this parameterization of ResNets admits a well-defined feature learning joint infinite-width and infinite-depth limit and show convergence of finite-size network dynamics towards this limit.

In this paper, we extend mean-field Langevin dynamics to minimax optimization over probability distributions for the first time with symmetric and provably convergent updates. We propose mean-field Langevin averaged gradient (MFL-AG), a single-loop algorithm that implements gradient descent ascent in the distribution spaces with a novel weighted averaging, and establish average-iterate convergence to the mixed Nash equilibrium. We also study both time and particle discretization regimes and prove a new uniform-in-time propagation of chaos result which accounts for the dependency of the particle interactions on all previous distributions. Furthermore, we propose mean-field Langevin anchored best response (MFL-ABR), a symmetric double-loop algorithm based on best response dynamics with linear last-iterate convergence. Finally, we study applications to zero-sum Markov games and conduct simulations demonstrating long-term optimality.

One-step generative modeling seeks to generate high-quality data samples in a single function evaluation, significantly improving efficiency over traditional diffusion or flow-based models. In this work, we introduce Modular MeanFlow (MMF), a flexible and theoretically grounded approach for learning time-averaged velocity fields. Our method derives a family of loss functions based on a differential identity linking instantaneous and average velocities, and incorporates a gradient modulation mechanism that enables stable training without sacrificing expressiveness. We further propose a curriculum-style warmup schedule to smoothly transition from coarse supervision to fully differentiable training. The MMF formulation unifies and generalizes existing consistency-based and flow-matching methods, while avoiding expensive higher-order derivatives. Empirical results across image synthesis and trajectory modeling tasks demonstrate that MMF achieves competitive sample quality, robust convergence, and strong generalization, particularly under low-data or out-of-distribution settings.

This work builds on the theoretical frameworks presented in "Liquidity pools as mean field games: A new framework" and "Liquidity pools as mean field games with transaction costs" by the same author, where the strategic interactions among traders in a constant-product market-making protocol were modelled using mean field games (MFG), first without transaction costs and then incorporating them. Here we present the formulation of a more complete model that integrates three types of agents: traders, liquidity providers (LPs), and arbitrageurs. While we do not establish existence results for this general model, the formulation identifies the main technical difficulties and lays the groundwork for future work. The LP acts as a dominating player in the sense of 'Mean field games with a dominating player' by Bensoussan, Chau, and Yam: its strategy influences the mean field distribution of the traders, and the equilibrium is sought as a solution to the coupled system of three problems that constitute a Major-Minor game. The arbitrageurs operate by solving the optimization problem presented in "An analysis of uniswap markets" by Angeris et al., and their impact on the LP is captured through the loss-versus-rebalancing of "Automated market making and loss-versus-rebalancing" by Milionis et al. The material in this article should be read as an open research proposal rather than a collection of closed results.

MeanFlow (MF) is a diffusion-motivated generative model that enables efficient few-step generation by learning long jumps directly from noise to data. In practice, it is often used as a latent MF by leveraging the pre-trained Stable Diffusion variational autoencoder (SD-VAE) for high-dimensional data modeling. However, MF training remains computationally demanding and is often unstable. During inference, the SD-VAE decoder dominates the generation cost, and MF depends on complex guidance hyperparameters for class-conditional generation. In this work, we develop an efficient training and sampling scheme for MF in the latent space of a Representation Autoencoder (RAE), where a pre-trained vision encoder (e.g., DINO) provides semantically rich latents paired with a lightweight decoder. We observe that naive MF training in the RAE latent space suffers from severe gradient explosion. To stabilize and accelerate training, we adopt Consistency Mid-Training for trajectory-aware initialization and use a two-stage scheme: distillation from a pre-trained flow matching teacher to speed convergence and reduce variance, followed by an optional bootstrapping stage with a one-point velocity estimator to further reduce deviation from the oracle mean flow. This design removes the need for guidance, simplifies training configurations, and reduces computation in both training and sampling. Empirically, our method achieves a 1-step FID of 2.03, outperforming vanilla MF's 3.43, while reducing sampling GFLOPS by 38% and total training cost by 83% on ImageNet 256. We further scale our approach to ImageNet 512, achieving a competitive 1-step FID of 3.23 with the lowest GFLOPS among all baselines. Code is available at https://github.com/sony/mf-rae.

In this paper, we introduce a new class of score-based generative models (SGMs) designed to handle high-cardinality data distributions by leveraging concepts from mean-field theory. We present mean-field chaos diffusion models (MF-CDMs), which address the curse of dimensionality inherent in high-cardinality data by utilizing the propagation of chaos property of interacting particles. By treating high-cardinality data as a large stochastic system of interacting particles, we develop a novel score-matching method for infinite-dimensional chaotic particle systems and propose an approximation scheme that employs a subdivision strategy for efficient training. Our theoretical and empirical results demonstrate the scalability and effectiveness of MF-CDMs for managing large high-cardinality data structures, such as 3D point clouds.

Modern diffusion/flow-based models for image generation typically exhibit two core characteristics: (i) using multi-step sampling, and (ii) operating in a latent space. Recent advances have made encouraging progress on each aspect individually, paving the way toward one-step diffusion/flow without latents. In this work, we take a further step towards this goal and propose "pixel MeanFlow" (pMF). Our core guideline is to formulate the network output space and the loss space separately. The network target is designed to be on a presumed low-dimensional image manifold (i.e., x-prediction), while the loss is defined via MeanFlow in the velocity space. We introduce a simple transformation between the image manifold and the average velocity field. In experiments, pMF achieves strong results for one-step latent-free generation on ImageNet at 256x256 resolution (2.22 FID) and 512x512 resolution (2.48 FID), filling a key missing piece in this regime. We hope that our study will further advance the boundaries of diffusion/flow-based generative models.

Learning the behavior of large agent populations is an important task for numerous research areas. Although the field of multi-agent reinforcement learning (MARL) has made significant progress towards solving these systems, solutions for many agents often remain computationally infeasible and lack theoretical guarantees. Mean Field Games (MFGs) address both of these issues and can be extended to Graphon MFGs (GMFGs) to include network structures between agents. Despite their merits, the real world applicability of GMFGs is limited by the fact that graphons only capture dense graphs. Since most empirically observed networks show some degree of sparsity, such as power law graphs, the GMFG framework is insufficient for capturing these network topologies. Thus, we introduce the novel concept of Graphex MFGs (GXMFGs) which builds on the graph theoretical concept of graphexes. Graphexes are the limiting objects to sparse graph sequences that also have other desirable features such as the small world property. Learning equilibria in these games is challenging due to the rich and sparse structure of the underlying graphs. To tackle these challenges, we design a new learning algorithm tailored to the GXMFG setup. This hybrid graphex learning approach leverages that the system mainly consists of a highly connected core and a sparse periphery. After defining the system and providing a theoretical analysis, we state our learning approach and demonstrate its learning capabilities on both synthetic graphs and real-world networks. This comparison shows that our GXMFG learning algorithm successfully extends MFGs to a highly relevant class of hard, realistic learning problems that are not accurately addressed by current MARL and MFG methods.

Bayesian optimisation is a popular approach for optimising expensive black-box functions. The next location to be evaluated is selected via maximising an acquisition function that balances exploitation and exploration. Gaussian processes, the surrogate models of choice in Bayesian optimisation, are often used with a constant prior mean function equal to the arithmetic mean of the observed function values. We show that the rate of convergence can depend sensitively on the choice of mean function. We empirically investigate 8 mean functions (constant functions equal to the arithmetic mean, minimum, median and maximum of the observed function evaluations, linear, quadratic polynomials, random forests and RBF networks), using 10 synthetic test problems and two real-world problems, and using the Expected Improvement and Upper Confidence Bound acquisition functions. We find that for design dimensions ge5 using a constant mean function equal to the worst observed quality value is consistently the best choice on the synthetic problems considered. We argue that this worst-observed-quality function promotes exploitation leading to more rapid convergence. However, for the real-world tasks the more complex mean functions capable of modelling the fitness landscape may be effective, although there is no clearly optimum choice.

Scaling Diffusion Transformers (DiTs) to hundreds of layers introduces a structural vulnerability: networks can enter a silent, mean-dominated collapse state that homogenizes token representations and suppresses centered variation. Through mechanistic auditing, we isolate the trigger event of this collapse as Mean Mode Screaming (MMS). MMS can occur even when training appears stable, with a mean-coherent backward shock on residual writers that opens deep residual branches and drives the network into a mean-dominated state. We show this behavior is driven by an exact decomposition of these gradients into mean-coherent and centered components, compounded by the structural suppression of attention-logit gradients through the null space of the Softmax Jacobian once values homogenize. To address this, we propose Mean-Variance Split (MV-Split) Residuals, which combine a separately gained centered residual update with a leaky trunk-mean replacement. On a 400-layer single-stream DiT, MV-Split prevents the divergent collapse that crashes the un-stabilized baseline; it tracks close to the baseline's pre-crash trajectory while remaining substantially better than token-isotropic gating methods such as LayerScale across the full schedule. Finally, we present a 1000-layer DiT as a scale-validation run at boundary scales, establishing that the architecture remains stably trainable at extreme depth.

We propose a principled and effective framework for one-step generative modeling. We introduce the notion of average velocity to characterize flow fields, in contrast to instantaneous velocity modeled by Flow Matching methods. A well-defined identity between average and instantaneous velocities is derived and used to guide neural network training. Our method, termed the MeanFlow model, is self-contained and requires no pre-training, distillation, or curriculum learning. MeanFlow demonstrates strong empirical performance: it achieves an FID of 3.43 with a single function evaluation (1-NFE) on ImageNet 256x256 trained from scratch, significantly outperforming previous state-of-the-art one-step diffusion/flow models. Our study substantially narrows the gap between one-step diffusion/flow models and their multi-step predecessors, and we hope it will motivate future research to revisit the foundations of these powerful models.

Recent frontier large language models predominantly rely on Mixture-of-Experts (MoE) architectures. Despite empirical progress, there is still no principled understanding of how hyperparameters should scale with network width N, expert width N_e, number of experts M, sparsity K, and depth L to ensure both stability and optimal performance at scale. We take a principled step toward resolving this gap by analyzing three different scaling regimes: (I) co-scaling Nasymp N_e, (II) co-scaling Nasymp Masymp K, and (III) full proportional scaling of N, N_e, M, and K. For each regime, we develop a novel Dynamical Mean Field Theory (DMFT) description of the limiting training dynamics of MoEs that provides a formal foundation for our analysis. Within this framework, we derive the unique parameterization for SGD and Adam satisfying all maximal-update (μ) desiderata. We then show that the resulting μP prescription does not reliably induce monotonic improvement with scale or robust learning-rate transfer. We trace these pathologies to scale-dependent observables in the aggregation dynamics, which motivates a refined set of desiderata that we term maximal scale stability. Guided by this principle, we derive a Maximally Scale-Stable Parameterization (MSSP) for both SGD and Adam in all three scaling regimes, and characterize the corresponding limiting dynamics - qualitatively distinct from the μP limit - through a separate DMFT analysis. Experiments verify that MSSP robustly recovers learning rate transfer and monotonic improvement with scale across regimes. Combined with existing depth-scaling theory, these results provide a complete scaling prescription for MoE architectures as a function of width, depth, expert width, and number of experts.

Deep Boltzmann machines (DBMs), one of the first ``deep'' learning methods ever studied, are multi-layered probabilistic models governed by a pairwise energy function that describes the likelihood of all variables/nodes in the network. In practice, DBMs are often constrained, i.e., via the restricted Boltzmann machine (RBM) architecture (which does not permit intra-layer connections), in order to allow for more efficient inference. In this work, we revisit the generic DBM approach, and ask the question: are there other possible restrictions to their design that would enable efficient (approximate) inference? In particular, we develop a new class of restricted model, the monotone DBM, which allows for arbitrary self-connection in each layer, but restricts the weights in a manner that guarantees the existence and global uniqueness of a mean-field fixed point. To do this, we leverage tools from the recently-proposed monotone Deep Equilibrium model and show that a particular choice of activation results in a fixed-point iteration that gives a variational mean-field solution. While this approach is still largely conceptual, it is the first architecture that allows for efficient approximate inference in fully-general weight structures for DBMs. We apply this approach to simple deep convolutional Boltzmann architectures and demonstrate that it allows for tasks such as the joint completion and classification of images, within a single deep probabilistic setting, while avoiding the pitfalls of mean-field inference in traditional RBMs.

Mixture-of-Experts (MoE) layers have emerged as an important tool in scaling up modern neural networks by decoupling total trainable parameters from activated parameters in the forward pass for each token. However, sparse MoEs add complexity to training due to (i) new trainable parameters (router weights) that, like all other parameter groups, require hyperparameter (HP) tuning; (ii) new architecture scale dimensions (number of and size of experts) that must be chosen and potentially taken large. To make HP selection cheap and reliable, we propose a new parameterization for transformer models with MoE layers when scaling model width, depth, number of experts, and expert (hidden) size. Our parameterization is justified by a novel dynamical mean-field theory (DMFT) analysis. When varying different model dimensions trained at a fixed token budget, we find empirically that our parameterization enables reliable HP transfer across models from 51M to over 2B total parameters. We further take HPs identified from sweeping small models on a short token horizon to train larger models on longer horizons and report performant model behaviors.

We study mean field portfolio games under Epstein-Zin preferences, which naturally encompass the classical time-additive power utility as a special case. In a general non-Markovian framework, we establish a uniqueness result by proving a one-to-one correspondence between Nash equilibria and the solutions to a class of BSDEs. A key ingredient in our approach is a necessary stochastic maximum principle tailored to Epstein-Zin utility and a nonlinear transformation. In the deterministic setting, we further derive an explicit closed-form solution for the equilibrium investment and consumption policies.

Generative models have shown great potential in trajectory planning. Recent studies demonstrate that anchor-guided generative models are effective in modeling the uncertainty of driving behaviors and improving overall performance. However, these methods rely on discrete anchor vocabularies that must sufficiently cover the trajectory distribution during testing to ensure robustness, inducing an inherent trade-off between vocabulary size and model performance. To overcome this limitation, we propose MeanFuser, an end-to-end autonomous driving method that enhances both efficiency and robustness through three key designs. (1) We introduce Gaussian Mixture Noise (GMN) to guide generative sampling, enabling a continuous representation of the trajectory space and eliminating the dependency on discrete anchor vocabularies. (2) We adapt ``MeanFlow Identity" to end-to-end planning, which models the mean velocity field between GMN and trajectory distribution instead of the instantaneous velocity field used in vanilla flow matching methods, effectively eliminating numerical errors from ODE solvers and significantly accelerating inference. (3) We design a lightweight Adaptive Reconstruction Module (ARM) that enables the model to implicitly select from all sampled proposals or reconstruct a new trajectory when none is satisfactory via attention weights.Experiments on the NAVSIM closed-loop benchmark demonstrate that MeanFuser achieves outstanding performance without the supervision of the PDM Score and exceptional inference efficiency, offering a robust and efficient solution for end-to-end autonomous driving. Our code and model are available at https://github.com/wjl2244/MeanFuser.

We study a fully connected Hopfield-type associative memory network with online activity-dependent synaptic plasticity, where neural states and synaptic couplings coevolve during retrieval. Using the generating-functional formalism, we derive a dynamical mean-field theory (DMFT) in the large-system limit with extensively many stored random patterns, and show that the many-body dynamics reduces to an effective single-site stochastic process with colored Gaussian crosstalk noise and delayed feedback terms. Numerical solutions of the DMFT equations agree well with direct simulations. We find that moderate plasticity enlarges the basin of attraction and increases the maximum retrievable memory load by generating a positive delayed feedback that stabilizes retrieval against crosstalk noise. However, excessively strong plasticity causes the network to imprint the imperfect initial cue itself, leading to spurious attractors and degraded retrieval performance. Consequently, an optimal plasticity strength emerges from the trade-off between memory stabilization and premature cue imprinting. These results extend the DMFT description of associative memory to networks with coevolving neural and synaptic dynamics.

Modern deep neural networks (DNNs) represent a formidable challenge for theorists: according to the commonly accepted probabilistic framework that describes their performance, these architectures should overfit due to the huge number of parameters to train, but in practice they do not. Here we employ results from replica mean field theory to compute the generalisation gap of machine learning models with quenched features, in the teacher-student scenario and for regression problems with quadratic loss function. Notably, this framework includes the case of DNNs where the last layer is optimised given a specific realisation of the remaining weights. We show how these results -- combined with ideas from statistical learning theory -- provide a stringent asymptotic upper bound on the generalisation gap of fully trained DNN as a function of the size of the dataset P. In particular, in the limit of large P and N_{out} (where N_out is the size of the last layer) and N_out ll P, the generalisation gap approaches zero faster than 2 N_out/P, for any choice of both architecture and teacher function. Notably, this result greatly improves existing bounds from statistical learning theory. We test our predictions on a broad range of architectures, from toy fully-connected neural networks with few hidden layers to state-of-the-art deep convolutional neural networks.

The forward pass of a Transformer can be seen as an interacting particle system on the unit sphere: time plays the role of layers, particles that of token embeddings, and the unit sphere idealizes layer normalization. In some weight settings the system can even be seen as a gradient flow for an explicit energy, and one can make sense of the infinite context length (mean-field) limit thanks to Wasserstein gradient flows. In this paper we study the effect of the perceptron block in this setting, and show that critical points are generically atomic and localized on subsets of the sphere.

Integrating first-order logic constraints (FOLCs) with neural networks is a crucial but challenging problem since it involves modeling intricate correlations to satisfy the constraints. This paper proposes a novel neural layer, LogicMP, whose layers perform mean-field variational inference over an MLN. It can be plugged into any off-the-shelf neural network to encode FOLCs while retaining modularity and efficiency. By exploiting the structure and symmetries in MLNs, we theoretically demonstrate that our well-designed, efficient mean-field iterations effectively mitigate the difficulty of MLN inference, reducing the inference from sequential calculation to a series of parallel tensor operations. Empirical results in three kinds of tasks over graphs, images, and text show that LogicMP outperforms advanced competitors in both performance and efficiency.

We revisit the flat-sky approximation for evaluating the angular power spectra of projected random fields by retaining information about the correlations along the line of sight. With broad, overlapping radial window functions, these line-of-sight correlations are suppressed and are ignored in the Limber approximation. However, retaining the correlations is important for narrow window functions or unequal-time spectra but introduces significant computational difficulties due to the highly oscillatory nature of the integrands involved. We deal with the integral over line-of-sight wave-modes in the flat-sky approximation analytically, using the FFTlog expansion of the 3D power spectrum. This results in an efficient computational method, which is a substantial improvement compared to any full-sky approaches. We apply our results to galaxy clustering (with and without redshift-space distortions), CMB lensing and galaxy lensing observables. For clustering, we find excellent agreement with the full-sky results on large (percent-level agreement) and intermediate or small (subpercent agreement) scales, dramatically out-performing the Limber approximation for both wide and narrow window functions, and in equal- and unequal-time cases. In the case of lensing, we show on the full sky that the angular power spectrum of the convergence can be very well approximated by projecting the 3D Laplacian (rather than the correct angular Laplacian) of the gravitational potential, even on large scales. Combining this approximation with our flat-sky techniques provides an efficient and accurate evaluation of the CMB lensing angular power spectrum on all scales.

We propose that the grokking phenomenon, where the train loss of a neural network decreases much earlier than its test loss, can arise due to a neural network transitioning from lazy training dynamics to a rich, feature learning regime. To illustrate this mechanism, we study the simple setting of vanilla gradient descent on a polynomial regression problem with a two layer neural network which exhibits grokking without regularization in a way that cannot be explained by existing theories. We identify sufficient statistics for the test loss of such a network, and tracking these over training reveals that grokking arises in this setting when the network first attempts to fit a kernel regression solution with its initial features, followed by late-time feature learning where a generalizing solution is identified after train loss is already low. We provide an asymptotic theoretical description of the grokking dynamics in this model using dynamical mean field theory (DMFT) for high dimensional data. We find that the key determinants of grokking are the rate of feature learning -- which can be controlled precisely by parameters that scale the network output -- and the alignment of the initial features with the target function y(x). We argue this delayed generalization arises when (1) the top eigenvectors of the initial neural tangent kernel and the task labels y(x) are misaligned, but (2) the dataset size is large enough so that it is possible for the network to generalize eventually, but not so large that train loss perfectly tracks test loss at all epochs, and (3) the network begins training in the lazy regime so does not learn features immediately. We conclude with evidence that this transition from lazy (linear model) to rich training (feature learning) can control grokking in more general settings, like on MNIST, one-layer Transformers, and student-teacher networks.

We propose a new method called the N-particle underdamped Langevin algorithm for optimizing a special class of non-linear functionals defined over the space of probability measures. Examples of problems with this formulation include training mean-field neural networks, maximum mean discrepancy minimization and kernel Stein discrepancy minimization. Our algorithm is based on a novel spacetime discretization of the mean-field underdamped Langevin dynamics, for which we provide a new, fast mixing guarantee. In addition, we demonstrate that our algorithm converges globally in total variation distance, bridging the theoretical gap between the dynamics and its practical implementation.

Minkowski functionals quantify the morphology of smooth random fields. They are widely used to probe statistical properties of cosmological fields. Analytic formulae for ensemble expectations of Minkowski functionals are well known for Gaussian and mildly non-Gaussian fields. In this paper we extend the formulae to composite fields which are sums of two fields and explicitly derive the expressions for the sum of uncorrelated mildly non-Gaussian and Gaussian fields. These formulae are applicable to observed data which is usually a sum of the true signal and one or more secondary fields that can be either noise, or some residual contaminating signal. Our formulae provide explicit quantification of the effect of the secondary field on the morphology and statistical nature of the true signal. As examples, we apply the formulae to determine how the presence of Gaussian noise can bias the morphological properties and statistical nature of Gaussian and non-Gaussian CMB temperature maps.

Finding the mixed Nash equilibria (MNE) of a two-player zero sum continuous game is an important and challenging problem in machine learning. A canonical algorithm to finding the MNE is the noisy gradient descent ascent method which in the infinite particle limit gives rise to the {\em Mean-Field Gradient Descent Ascent} (GDA) dynamics on the space of probability measures. In this paper, we first study the convergence of a two-scale Mean-Field GDA dynamics for finding the MNE of the entropy-regularized objective. More precisely we show that for each finite temperature (or regularization parameter), the two-scale Mean-Field GDA with a suitable {\em finite} scale ratio converges exponentially to the unique MNE without assuming the convexity or concavity of the interaction potential. The key ingredient of our proof lies in the construction of new Lyapunov functions that dissipate exponentially along the Mean-Field GDA. We further study the simulated annealing of the Mean-Field GDA dynamics. We show that with a temperature schedule that decays logarithmically in time the annealed Mean-Field GDA converges to the MNE of the original unregularized objective.

Recent reinforcement learning (RL) methods have achieved success in various domains. However, multi-agent RL (MARL) remains a challenge in terms of decentralization, partial observability and scalability to many agents. Meanwhile, collective behavior requires resolution of the aforementioned challenges, and remains of importance to many state-of-the-art applications such as active matter physics, self-organizing systems, opinion dynamics, and biological or robotic swarms. Here, MARL via mean field control (MFC) offers a potential solution to scalability, but fails to consider decentralized and partially observable systems. In this paper, we enable decentralized behavior of agents under partial information by proposing novel models for decentralized partially observable MFC (Dec-POMFC), a broad class of problems with permutation-invariant agents allowing for reduction to tractable single-agent Markov decision processes (MDP) with single-agent RL solution. We provide rigorous theoretical results, including a dynamic programming principle, together with optimality guarantees for Dec-POMFC solutions applied to finite swarms of interest. Algorithmically, we propose Dec-POMFC-based policy gradient methods for MARL via centralized training and decentralized execution, together with policy gradient approximation guarantees. In addition, we improve upon state-of-the-art histogram-based MFC by kernel methods, which is of separate interest also for fully observable MFC. We evaluate numerically on representative collective behavior tasks such as adapted Kuramoto and Vicsek swarming models, being on par with state-of-the-art MARL. Overall, our framework takes a step towards RL-based engineering of artificial collective behavior via MFC.

Few-step generation has been a long-standing goal, with recent one-step generation methods exemplified by MeanFlow achieving remarkable results. Existing research on MeanFlow primarily focuses on class-to-image generation. However, an intuitive yet unexplored direction is to extend the condition from fixed class labels to flexible text inputs, enabling richer content creation. Compared to the limited class labels, text conditions pose greater challenges to the model's understanding capability, necessitating the effective integration of powerful text encoders into the MeanFlow framework. Surprisingly, although incorporating text conditions appears straightforward, we find that integrating powerful LLM-based text encoders using conventional training strategies results in unsatisfactory performance. To uncover the underlying cause, we conduct detailed analyses and reveal that, due to the extremely limited number of refinement steps in the MeanFlow generation, such as only one step, the text feature representations are required to possess sufficiently high discriminability. This also explains why discrete and easily distinguishable class features perform well within the MeanFlow framework. Guided by these insights, we leverage a powerful LLM-based text encoder validated to possess the required semantic properties and adapt the MeanFlow generation process to this framework, resulting in efficient text-conditioned synthesis for the first time. Furthermore, we validate our approach on the widely used diffusion model, demonstrating significant generation performance improvements. We hope this work provides a general and practical reference for future research on text-conditioned MeanFlow generation. The code is available at https://github.com/AMAP-ML/EMF.

Grid edge resources refer to distributed energy resources (DERs) located on the consumer side of the electrical grid, controlled by consumers rather than utility companies. Integrating DERs with real-time electricity pricing can better align distributed supply with system demand, improving grid efficiency and reliability. However, DER owners, known as prosumers, often lack the expertise and resources to directly participate in wholesale energy markets, limiting their ability to fully realize the economic potential of their assets. Meanwhile, as DER adoption grows, the number of prosumers participating in the energy system is expected to increase significantly, creating additional challenges in coordination and market participation. To address these challenges, we propose a mean-field game framework that enables prosumers to autonomously learn optimal decision policies based on dynamic market prices and their variable solar generation. Our framework is designed to accommodate heterogeneous agents and demonstrates the existence of a mean-field equilibrium (MFE) in a wholesale energy market with many prosumers. Additionally, we introduce an algorithm that automates prosumers' resource control, facilitating real-time decision-making for energy storage management. Numerical experiments suggest that our approach converges towards an MFE and effectively reduces peak loads and price volatility, especially during periods of external demand or supply shocks. This study highlights the potential of a fully decentralized approach to integrating DERs into wholesale markets while improving market efficiency.

Effective macroeconomic policies play a crucial role in promoting economic growth and social stability. This paper models the optimal macroeconomic policy problem based on the Stackelberg Mean Field Game (SMFG), where the government acts as the leader in policy-making, and large-scale households dynamically respond as followers. This modeling method captures the asymmetric dynamic game between the government and large-scale households, and interpretably evaluates the effects of macroeconomic policies based on microfoundations, which is difficult for existing methods to achieve. We also propose a solution for SMFGs, incorporating pre-training on real data and a model-free Stackelberg mean-field reinforcement learning (SMFRL) algorithm, which operates independently of prior environmental knowledge and transitions. Our experimental results showcase the superiority of the SMFG method over other economic policies in terms of performance, efficiency-equity tradeoff, and SMFG assumption analysis. This paper significantly contributes to the domain of AI for economics by providing a powerful tool for modeling and solving optimal macroeconomic policies.

MeanFlow has recently emerged as a powerful framework for few-step generative modeling trained from scratch, but its success is not yet fully understood. In this work, we show that the MeanFlow objective naturally decomposes into two parts: trajectory flow matching and trajectory consistency. Through gradient analysis, we find that these terms are strongly negatively correlated, causing optimization conflict and slow convergence. Motivated by these insights, we introduce alpha-Flow, a broad family of objectives that unifies trajectory flow matching, Shortcut Model, and MeanFlow under one formulation. By adopting a curriculum strategy that smoothly anneals from trajectory flow matching to MeanFlow, alpha-Flow disentangles the conflicting objectives, and achieves better convergence. When trained from scratch on class-conditional ImageNet-1K 256x256 with vanilla DiT backbones, alpha-Flow consistently outperforms MeanFlow across scales and settings. Our largest alpha-Flow-XL/2+ model achieves new state-of-the-art results using vanilla DiT backbones, with FID scores of 2.58 (1-NFE) and 2.15 (2-NFE).

We develop a dynamical mean-field theory for random recurrent networks with low-rank structure and firing-rate-driven adaptation. When the random connectivity is strong enough to generate chaos, increasing adaptation strength drives the network through four regimes: a static coherent state, noise-sustained oscillations that progress from regular to irregular, stochastic switching between symmetric wells, and a global limit cycle. The theory identifies two instability mechanisms, chaos onset from the random connectivity and a Hopf bifurcation of the coherent mode, and shows how adaptation shapes both through the frequency-dependent single-neuron transfer function. A reduced three-dimensional model captures the bifurcation structure of the full network. Above the chaos threshold, coherent population-level oscillations coexist with heterogeneous firing rates and network-generated stochasticity at the single-neuron level. The interaction of adaptation with random and low-rank connectivity produces a rich oscillatory repertoire, including waxing-and-waning rhythmic episodes, persistent state switching, and slow Up-Down alternations, dynamics that have been observed during wakefulness, sleep, and anesthesia.

We characterise the dynamics of neuronal activity, in terms of field theory, using neural units placed on a 2D-lattice modelling the cortical surface. The electrical activity of neuronal units was analysed with the aim of deriving a neural field model with a simple functional form that still able to predict or reproduce empirical findings. Each neural unit was modelled using a neural mass and the accompanying field theory was derived in the continuum limit. The field theory comprised coupled (real) Klein-Gordon fields, where predictions of the model fall within the range of experimental findings. These predictions included the frequency spectrum of electric activity measured from the cortex, which was derived using an equipartition of energy over eigenfunctions of the neural fields. Moreover, the neural field model was invariant, within a set of parameters, to the dynamical system used to model each neuronal mass. Specifically, topologically equivalent dynamical systems resulted in the same neural field model when connected in a lattice; indicating that the fields derived could be read as a canonical cortical field theory. We specifically investigated non-dispersive fields that provide a structure for the coding (or representation) of afferent information. Further elaboration of the ensuing neural field theory, including the effect of dispersive forces, could be of importance in the understanding of the cortical processing of information.

We describe a novel method for the application of Convolutional Neural Networks (CNNs) to fields defined on the sphere, using the HEALPix tessellation scheme. Specifically, We have developed a pixel-based approach to implement convolutional layers on the spherical surface, similarly to what is commonly done for CNNs in Euclidian space. The algorithm is fully integrable with existing libraries for NNs (e.g., PyTorch or TensorFlow). We present two applications: (i) recognition of handwritten digits projected on the sphere; (ii) estimation of cosmological parameter from Cosmic Microwave Background (CMB) simulated maps. We have built a simple NN architecture, consisting in four convolutional+pooling layers, and have used it for all the applications explored herein. For what concerns the handwritten digits, our CNN reaches an accuracy of about 95%, comparable with other existing spherical CNNs. For CMB applications, we have tested the CNN on the estimation of a "mock" parameter, defining the angular scale at which the power spectrum of a Gaussian field projected on the sphere peaks. We have estimated this parameter directly from maps, in several cases: temperature and polarization, presence of noise and partial sky coverage. In all the cases, the NN performances are comparable with those from standard spectrum-based bayesian methods. We demonstrate, for the first time, the capability of CNNs to extract information from polarization fields and to distinguish between E and B-modes. Lastly, we have applied our CNN to the estimation of the Thomson scattering optical depth at reionization (tau) from simulated CMB maps. Even without any specific optimization of the NN architecture, we reach an accuracy comparable with standard bayesian methods. This work represents a first step towards the exploitation of NNs in CMB parameter estimation and demonstrates the feasibility of our approach.

This article considers the popular MCMC method of unadjusted Langevin Monte Carlo (LMC) and provides a non-asymptotic analysis of its sampling error in 2-Wasserstein distance. The proof is based on a refinement of mean-square analysis in Li et al. (2019), and this refined framework automates the analysis of a large class of sampling algorithms based on discretizations of contractive SDEs. Using this framework, we establish an O(d/epsilon) mixing time bound for LMC, without warm start, under the common log-smooth and log-strongly-convex conditions, plus a growth condition on the 3rd-order derivative of the potential of target measures. This bound improves the best previously known O(d/epsilon) result and is optimal (in terms of order) in both dimension d and accuracy tolerance epsilon for target measures satisfying the aforementioned assumptions. Our theoretical analysis is further validated by numerical experiments.

Neural ordinary differential equations (neural ODEs) have rapidly gained prominence as a powerful and unifying framework for conceptualizing artificial neural networks, elegantly connecting the continuous-time modeling of dynamical systems with the discrete, data-driven paradigm of modern deep learning. Beyond their practical advantages they offer fresh theoretical insights into the training and generalization properties of neural networks. The distinctive feature of this framework is its dual dynamical nature: inference dynamics, which govern the ODE evolution during forward computation, and training dynamics, which control the optimization of model parameters. This makes neural ODEs a particularly well-suited theoretical framework for studying a large variety of settings such as multi-layer neural networks (ResNets for example), autoregressive models (with next-token generation dynamics), generative models, and recurrent neural networks in theoretical neuroscience. In this work, we introduce a theoretically grounded class of models for studying neural ODEs trained via online stochastic gradient descent. We solve the training dynamics of these models via dynamical mean field theory and derive learning curves in the high-dimensional limit.