Convergence and Stability of the Stochastic Proximal Point Algorithm with Momentum

TL;DR

This paper analyzes the convergence and stability of a stochastic proximal point algorithm with momentum, demonstrating improved convergence rates and broader stability conditions compared to stochastic gradient methods.

Contribution

It introduces and studies the stochastic proximal point algorithm with momentum, showing it achieves faster convergence and better stability properties.

Findings

SPPAM converges faster to a neighborhood than SPPA.

SPPAM has a more favorable dependence on problem constants than SGDM.

Wider range of hyperparameters ensures convergence for SPPAM.

Abstract

Stochastic gradient descent with momentum (SGDM) is the dominant algorithm in many optimization scenarios, including convex optimization instances and non-convex neural network training. Yet, in the stochastic setting, momentum interferes with gradient noise, often leading to specific step size and momentum choices in order to guarantee convergence, set aside acceleration. Proximal point methods, on the other hand, have gained much attention due to their numerical stability and elasticity against imperfect tuning. Their stochastic accelerated variants though have received limited attention: how momentum interacts with the stability of (stochastic) proximal point methods remains largely unstudied. To address this, we focus on the convergence and stability of the stochastic proximal point algorithm with momentum (SPPAM), and show that SPPAM allows a faster linear convergence to a neighborhood compared to the stochastic proximal point algorithm (SPPA) with a better contraction factor, under proper hyperparameter tuning. In terms of stability, we show that SPPAM depends on problem constants more favorably than SGDM, allowing a wider range of step size and momentum that lead to convergence.