Title: Can Microcanonical Langevin Dynamics Leverage Mini-Batch Gradient Noise?
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Microcanonical Langevin, Sampling, Bayesian Deep Learning

Abstract: Scaling inference methods such as Markov chain Monte Carlo to high-dimensional models remains a central challenge in Bayesian deep learning. A promising recent proposal, microcanonical Langevin Monte Carlo, has shown state-of-the-art performance across a wide range of problems. However, its reliance on full-dataset gradients makes it prohibitively expensive for large-scale problems. This paper addresses a fundamental question: Can microcanonical dynamics effectively leverage mini-batch gradient noise? We provide the first systematic study of this problem, revealing two critical failure modes: a limitation due to anisotropic gradient noise and numerical instabilities in complex high-dimensional posteriors. We resolve both issues by proposing a principled gradient noise preconditioning scheme and developing a novel, energy-variance-based adaptive tuner that automates step size selection and informs dynamical numerical guardrails. The resulting algorithm is a robust and scalable microcanonical Monte Carlo sampler that consistently outperforms strong stochastic gradient MCMC baselines on challenging high-dimensional inference tasks like Bayesian neural networks. Combined with recent ensemble techniques, our work unlocks a new class of stochastic microcanonical Langevin ensemble (SMILE) samplers for large-scale Bayesian inference.