Local Optimality and Generalization Guarantees for the Langevin Algorithm via Empirical Metastability

TL;DR

This paper analyzes the path-wise behavior of the Langevin algorithm in non-convex ERM, revealing empirical metastability and providing insights into its convergence and escape times, with implications for generalization.

Contribution

It introduces a metastability framework for Langevin dynamics in non-convex optimization, connecting path-wise behavior with generalization guarantees and escape time scaling.

Findings

Langevin trajectories exhibit empirical metastability near local optima.

Escape times scale exponentially, aligning with Eyring-Kramers law.

Langevin algorithm can visit all optima without strong initialization conditions.

Abstract

We study the detailed path-wise behavior of the discrete-time Langevin algorithm for non-convex Empirical Risk Minimization (ERM) through the lens of metastability, adopting some techniques from Berglund and Gentz (2003. For a particular local optimum of the empirical risk, with an arbitrary initialization, we show that, with high probability, at least one of the following two events will occur: (1) the Langevin trajectory ends up somewhere outside the $\varepsilon$-neighborhood of this particular optimum within a short recurrence time; (2) it enters this $\varepsilon$-neighborhood by the recurrence time and stays there until a potentially exponentially long escape time. We call this phenomenon empirical metastability. This two-timescale characterization aligns nicely with the existing literature in the following two senses. First, the effective recurrence time (i.e., number of iterations multiplied by stepsize) is dimension-independent, and resembles the convergence time of continuous-time deterministic Gradient Descent (GD). However unlike GD, the Langevin algorithm does not require strong conditions on local initialization, and has the possibility of eventually visiting all optima. Second, the scaling of the escape time is consistent with the Eyring-Kramers law, which states that the Langevin scheme will eventually visit all local minima, but it will take an exponentially long time to transit among them. We apply this path-wise concentration result in the context of statistical learning to examine local notions of generalization and optimality.