Stochastic Langevin Differential Inclusions with Applications to Machine Learning

TL;DR

This paper extends the theoretical understanding of Langevin-type stochastic differential inclusions, crucial for machine learning, by establishing foundational results including solution existence and asymptotic behavior under non-smooth potential functions.

Contribution

It introduces foundational results for Langevin-type stochastic differential inclusions with non-smooth potentials, relevant to machine learning applications.

Findings

Proves strong existence of solutions for Langevin-type stochastic differential inclusions.

Demonstrates asymptotic minimization of the free-energy functional.

Addresses non-Lipschitz drift scenarios in machine learning contexts.

Abstract

Stochastic differential equations of Langevin-diffusion form have received significant attention, thanks to their foundational role in both Bayesian sampling algorithms and optimization in machine learning. In the latter, they serve as a conceptual model of the stochastic gradient flow in training over-parameterized models. However, the literature typically assumes smoothness of the potential, whose gradient is the drift term. Nevertheless, there are many problems for which the potential function is not continuously differentiable, and hence the drift is not Lipschitz continuous everywhere. This is exemplified by robust losses and Rectified Linear Units in regression problems. In this paper, we show some foundational results regarding the flow and asymptotic properties of Langevin-type Stochastic Differential Inclusions under assumptions appropriate to the machine-learning settings. In particular, we show strong existence of the solution, as well as an asymptotic minimization of the canonical free-energy functional.