Stochastic optimization with momentum: convergence, fluctuations, and traps avoidance

TL;DR

This paper analyzes a unified stochastic optimization framework, including popular methods like Adam and S-NAG, proving convergence, stability, and trap avoidance in non-convex settings through differential equation analysis.

Contribution

It introduces a comprehensive analysis of stochastic optimization algorithms as noisy discretizations of differential equations, establishing convergence and trap avoidance results.

Findings

Proves almost sure convergence to critical points in non-convex settings.

Establishes convergence rates via a Central Limit Theorem.

Shows non-convergence to saddle points and maxima under certain conditions.

Abstract

In this paper, a general stochastic optimization procedure is studied, unifying several variants of the stochastic gradient descent such as, among others, the stochastic heavy ball method, the Stochastic Nesterov Accelerated Gradient algorithm (S-NAG), and the widely used Adam algorithm. The algorithm is seen as a noisy Euler discretization of a non-autonomous ordinary differential equation, recently introduced by Belotto da Silva and Gazeau, which is analyzed in depth. Assuming that the objective function is non-convex and differentiable, the stability and the almost sure convergence of the iterates to the set of critical points are established. A noteworthy special case is the convergence proof of S-NAG in a non-convex setting. Under some assumptions, the convergence rate is provided under the form of a Central Limit Theorem. Finally, the non-convergence of the algorithm to undesired critical points, such as local maxima or saddle points, is established. Here, the main ingredient is a new avoidance of traps result for non-autonomous settings, which is of independent interest.