New neural networks learn mappings between probability measures and functions.
problem Learning mappings between Wasserstein space of probability measures and function spaces.
method Two types of neural networks: bin density and cylindrical approximation, are proposed and supported by universal approximation theorems.
result Accuracy and efficiency of mean-field neural networks in generalization error with various test distributions.
Global convergence proved for three-layer neural networks in mean field regime.
problem Optimization efficiency of multilayer neural networks in the mean field regime.
method Developed a rigorous framework for mean field limit of three-layer networks using stochastic gradient descent and neuronal embedding.
result Global convergence guarantee for unregularized feedforward three-layer networks in the mean field regime.
This work shows linear convergence for two-layer neural networks in mean-field regime.
problem Optimizing two-layer neural networks in the mean-field regime.
method Mean-field analysis and continuous-time noisy gradient descent.
result Establishes linear convergence rate for two-layer neural networks.
New algorithm solves mean-field control problems using actor-critic learning with moment neural networks.
problem Solving mean-field control problems in continuous time reinforcement learning.
method Gradient-based policy and value function learning with moment neural networks on the Wasserstein space.
result Effective solution for diverse mean-field control problems, including multi-dimensional and nonlinear settings.
Two-layer neural networks learn efficiently using kernel methods in mean-field analysis.
problem Feature learning ability of two-layer neural networks in the mean-field regime.
method Mean-field analysis through kernel methods, focusing on dynamics of the first layer's kernel.
result Two-layer neural networks can learn a union of multiple reproducing kernel Hilbert spaces more efficiently than kernel methods.
Paper adds Fisher Information to mean field optimization for faster convergence.
problem Mean field optimization in neural networks training.
method Developed energy-dissipation method and gradient flow on probability space.
result Marginal distributions converge exponentially to minimizer.
Improved particle approximation for mean-field neural networks.
problem Particle approximation error for mean-field neural networks.
method Improved particle approximation error by leveraging the problem structure in risk minimization.
result Established an LSI-constant-free particle approximation error concerning the objective gap.
Recent studies have suggested that the cognitive process of the human brain is realized as probabilistic inference and can be further modeled by probabilistic graphical models like Markov random fields. Nevertheless, it remains unclear how probabilistic inference can be implemented by a network of spiking neurons in th…
The paper extends mean field results to three-layer neural networks using SGD.
problem Understanding the dynamics of training three-layer neural networks with SGD.
method Extending mean field results from two-layer networks to three-layer networks with two hidden layers, using non-linear partial differential equations.
result The distributions of weights in the two hidden layers are independent.
Elman-type RNNs converge to globally optimal solutions in the mean-field regime.
problem Optimizing feature learning in wide RNNs.
method Analysis of gradient descent dynamics and mean-field limits.
result Fixed points of infinite-width dynamics are globally optimal.
Study bounds graph neural networks' over-parameterized error.
problem Understanding graph neural networks' performance in over-parameterized regimes.
method Developed mean-field regime bounds for graph convolutional and message passing neural networks.
result Established upper bounds with a convergence rate of O(1/n) for generalization error. Deep Bayesian neural nets can use simpler weight approximations without sacrificing performance.
problem The need for complex weight posterior approximations in deep Bayesian neural networks.
method Theoretical and empirical analysis of mean-field variational inference in deep networks.
result Mean-field variational weight posteriors in deep networks can induce similar function-space distributions as complex approximations in shallower networks.
New framework analyzes deep neural networks using feature probabilities.
problem Degenerate situation in over-parameterized DNNs.
method Mean-field framework representing DNNs by feature probabilities and functions.
result Global convergence proof for over-parameterized Res-Net training.
Study shows polynomial-width neural networks can closely approximate infinite-width networks in polynomial time.
problem Approximating dynamics of polynomial-width neural networks with infinite-width networks.
method Bounding approximation gap through a differential equation governed by mean-field dynamics, considering local Hessian.
result Polynomially many neurons are sufficient to closely approximate mean-field dynamics.
Conservative SPDEs emerge from fluctuating SGD dynamics in neural networks.
problem Understanding the convergence of stochastic gradient descent to SPDEs.
method Mean-field analysis and central limit theorem for SPDEs.
result Optimal convergence rates for SPDEs derived from SGD.
A new algorithm for learning shallow neural networks with infinite width.
problem Learning shallow over-parameterized neural networks.
method Sinkhorn proximal algorithm approximating mean field learning dynamics.
result The algorithm performs gradient descent of the free energy associated with the risk functional.
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 i…
APAC-Net solves high-dimensional stochastic MFGs using neural networks.
problem High-dimensional stochastic mean-field games.
method Alternating population and control neural networks, parameterizing value and density functions.
result Solves up to 100-dimensional MFG problems.
The paper solves complex control problems using neural networks.
problem Solving McKean-Vlasov control problems.
method Mean-field neural networks and algorithms based on dynamic programming and stochastic maximum principle.
result Extensive numerical results show the accuracy of the proposed algorithms.
Softmax policy gradient achieves global optimality in wide neural networks with entropy regularization.
problem Optimizing softmax policies with neural networks in the mean-field regime.
method Modeling neural networks as Wasserstein gradient flows and proving global optimality of fixed points.
result Global optimality of softmax policy gradient in wide single hidden layer neural networks with entropy regularization.
Framework captures neural network learning in large-width limit.
problem Understanding learning dynamics in large neural networks.
method Developed a rigorous framework for multilayer neural networks in mean field limit.
result Global convergence guarantees for various network architectures and initializations.
PDA method optimizes neural networks with global convergence rate analysis.
problem Quantitative convergence rate for neural network optimization in mean field regime.
method Particle dual averaging (PDA) method, combining Langevin algorithm and outer loop optimization.
result Established quantitative global convergence for two-layer mean field neural networks.
Deep Galerkin Method estimates value function for mean-field control problem.
problem Optimal control of agents with average welfare as the objective.
method Apply DGM to estimate value function and distribution evolution.
result Neural network approximations converge to analytical solution.
Bayesian neural networks learn efficiently at infinite width, matching polynomial-width performance.
problem Understanding the inductive bias of infinite-width neural networks.
method Analyzing the reduced entropy and using subsampling techniques.
result The Bayesian mean-field learner generalizes exactly on polynomially-bounded targets.
Transformers approximate mean-field dynamics of indistinguishable particles.
problem Approximating the dynamics of indistinguishable particles in complex systems.
method Using transformers to model the mean-field dynamics of interacting particle systems.
result Theoretical bounds on the distance between true and transformer-obtained mean-field dynamics.
A new neural network approach for diffusion on networks.
problem Inference and estimation of diffusion on network structures.
method Neural mean-field dynamics derived from Mori-Zwanzig formalism, approximated by learnable time convolution operators.
result Significantly outperforms existing approaches in accuracy and efficiency.
New algorithm recovers sparse measures in polynomial time.
problem Recovering sparse measures from Fourier moments.
method Polynomial-time recovery method inspired by mean-field theory.
result Improves upon convex relaxation methods in specific parameter regime.
Study shows how neural networks generalize with minimal training data.
problem Understanding how neural networks generalize with limited data.
method Mean-field analysis of KL-regularized empirical risk minimization.
result Generalization error rate is O(1/n) for large n. The paper analyzes the dynamics of a simple neural network using a mean-field approach.
problem Understanding the training dynamics of neural networks, especially in classification tasks.
method Developed an analytic theory using a mean-field limit for a simple neural network.
result Explicitly solved the dynamics of a linearly separable dataset with a linear hinge loss.
Gradient descent struggles with high-dimensional data fitting.
problem Gradient descent struggles with high-dimensional data fitting.
method Gradient descent training of a two-layer neural network on empirical or population risk.
result Gradient descent training may not decrease population risk faster than t−4/(d−2) under mean field scaling. Study shows policy gradient convergence for entropy-regularized MDPs with neural nets in mean-field regime.
problem Global convergence of policy gradient for entropy-regularized MDPs with neural network approximation.
method Softmax policy with neural network approximation in mean-field regime, gradient flow in 2-Wasserstein metric, exponential convergence under sufficient regularization.
result Gradient flow converges exponentially fast to the unique stationary solution under sufficient regularization.
Improved sampling from mean-field stationary distributions.
problem Sampling from the stationary distribution of mean-field SDEs.
method Decoupling the problem into two aspects: approximation of mean-field SDE and sampling from finite-particle distribution.
result Improved guarantees in various settings, including optimizing neural networks.
Paper analyzes SHB method for neural networks, proving stability, connectivity, and global convergence.
problem Theoretical understanding of SHB method for neural networks.
method Mean-field analysis of SHB dynamics related to a partial differential equation.
result SHB method converges to global optimum and exhibits stability and connectivity.
Global convergence of multilayer neural networks proven for any depth.
problem Global convergence of multilayer neural networks in the mean field regime.
method Mean field limit framework, neuronal embedding, bidirectional diversity condition.
result Global convergence for multilayer networks of any depths, including correlated initializations.
Mean field theory explains gradient backpropagation in deep dropout networks.
problem Understanding gradient backpropagation in deep dropout networks.
method Applied mean field theory to dropout networks, considering realistic training conditions.
result Gradient backpropagation length is limited by depth scales, not just independence assumption.
Study shows benefits of transfer learning with neural networks.
problem Understanding generalization errors in transfer learning.
method Mean-field analysis applied to α-ERM and fine-tuning. result Established conditions for generalization error and convergence rates.
Improved PoC for MFLD reduces approximation error and provides model ensemble guarantees.
problem Quantifying optimization complexity in mean-field Langevin dynamics.
method Refined defective log-Sobolev inequality for neural network training.
result Improved PoC result with reduced approximation error and theoretical model ensemble guarantees.
Gradient descent dynamics in wide neural networks are analyzed using a dynamical CLT.
problem Understanding the fluctuations in wide shallow neural networks trained via gradient descent.
method Dynamical Central Limit Theorem (CLT) applied to neural network dynamics.
result Asymptotic fluctuations remain bounded in mean square throughout training.
Study analyzes adversarial training dynamics without data distribution assumptions.
problem Understanding training dynamics of adversarial training without data distribution assumptions.
method Mean field theory approach to analyze adversarial training in random deep neural networks.
result Upper bounds of adversarial loss derived empirically and theoretically.
Wide BNNs with odd activations fail to approximate data under mean-field inference.
problem Theoretical limitations of mean-field variational inference in wide, deep Bayesian neural networks.
method Analysis of mean-field variational inference in fully-connected BNNs with odd activation functions and Gaussian likelihood.
result The optimal mean-field variational posterior predictive distribution converges to the prior predictive distribution as network width increases.
Compact parameterization improves Bayesian neural network performance.
problem Improving performance of Bayesian neural networks using variational methods.
method Restricting variational distribution to a k-tied Normal distribution with low-rank factorization.
result Compact parameterization improves signal-to-noise ratio and convergence speed.
Gradient descent converges to minimum Bayes risk for two-layer ReLU networks in mean field regime.
problem Training two-layer ReLU networks using gradient descent in the mean field regime.
method Describes a condition for convergence to minimum Bayes risk, extending previous results to ReLU-activated networks.
result The condition for convergence does not depend on initialization and concerns weak convergence of network realization.
New GM layers improve neural network performance.
problem Improving neural network performance.
method Employing Gaussian mixture models and Wasserstein gradient flows.
result GM layers achieve comparable performance to two-layer networks.
The paper analyzes the mean field Langevin dynamics and its convergence rate.
problem The convergence property of the mean field Langevin dynamics in the context of neural networks.
method The analysis uses a proximal Gibbs distribution and techniques from convex optimization.
result A concise convergence rate analysis of the mean field Langevin dynamics in both continuous and discrete time settings.
The study analyzes a three-layer neural network's training dynamics using a functional-space mean-field theory.
problem Understanding the training dynamics of partially-trained three-layer neural networks.
method Generalized mean-field theory to functional spaces, proving convergence and feature learning.
result The training loss of the model decays to zero at a linear rate in the L2 regression setting. Bayesian neural networks ignore data in infinite units limit.
problem Pathological behavior of posterior in over-parameterized networks.
method Mean-field variational inference in infinite hidden units limit.
result Posterior mean converges to zero, ignoring data.
Study on neural networks' performance under different normalizations as N grows.
problem Characterizing neural networks' performance under various normalizations.
method Developed an asymptotic expansion to analyze statistical output of shallow neural networks.
result No bias-variance trade-off exists to leading order in N, and variance decreases as normalization approaches mean field.
Nonnegative Boltzmann machines (NNBMs) are recurrent probabilistic neural network models that can describe multi-modal nonnegative data. NNBMs form rectified Gaussian distributions that appear in biological neural network models, positive matrix factorization, nonnegative matrix factorization, and so on. In this paper,…