Controlled Langevin Dynamics for Sampling of Feedforward Neural Networks Trained with Minibatches

2026-03-16

AI Summary

This work addresses the computational intractability of exact sampling methods—such as Hybrid Monte Carlo—for large-scale neural networks, which stems from their reliance on full-batch gradient evaluations to sample from the Boltzmann distribution. The authors propose a controlled minibatch pseudo-Langevin dynamics approach that leverages the statistical properties of minibatch gradient noise, adjusting virtual mass and friction coefficients to dramatically improve computational efficiency while preserving sampling accuracy. Grounded in stochastic differential equations and equilibrium distribution theory, the method enables scalable sampling in high-dimensional parameter spaces. Experiments demonstrate that on networks with millions of parameters, the samples generated at moderate temperatures achieve generalization performance comparable to that of stochastic gradient descent, without requiring a validation set or early stopping.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Sampling the parameter space of artificial neural networks according to a Boltzmann distribution provides insight into the geometry of low-loss solutions and offers an alternative to conventional loss minimization for training. However, exact sampling methods such as hybrid Monte Carlo (hMC), while formally correct, become computationally prohibitive for realistic datasets because they require repeated evaluation of full-batch gradients. We introduce a pseudo-Langevin (pL) dynamics that enables efficient Boltzmann sampling of feed-forward neural networks trained with large datasets by using minibatches in a controlled manner. The method exploits the statistical properties of minibatch gradient noise and adjusts fictitious masses and friction coefficients to ensure that the induced stochastic process samples efficiently the desired equilibrium distribution. We validate numerically the approach by comparing its equilibrium statistics with those obtained from exact hMC sampling. Performance benchmarks demonstrate that, while hMC rapidly becomes inefficient as network size increases, the pL scheme maintains high computational diffusion and scales favorably to networks with over one million parameters. Finally, we show that sampling at intermediate temperatures yields optimal generalization performance, comparable to SGD, without requiring a validation set or early stopping procedure. These results establish controlled minibatch Langevin dynamics as a practical and scalable tool for exploring and exploiting the solution space of large neural networks.

Problem

• Boltzmann sampling
• neural networks
• minibatch gradients
• computational efficiency
• parameter space exploration

Innovation

• pseudo-Langevin dynamics
• minibatch sampling
• Boltzmann distribution
• scalable neural network sampling
• controlled Langevin dynamics