The Role of Memory in Stochastic Optimization

TL;DR

This paper uses stochastic differential equations to analyze how different memory mechanisms in gradient-based optimization algorithms affect convergence, stability, and performance in convex and nonconvex stochastic settings.

Contribution

It introduces a continuous-time model for arbitrary memory types, derives convergence guarantees, and proposes a flexible discretized algorithm with improved stability over classical momentum methods.

Findings

Memory choice significantly impacts convergence and stability.

The proposed algorithm outperforms classical momentum in stochastic convex optimization.

Long-term memory improves second-moment estimation in adaptive methods like Adam.

Abstract

The choice of how to retain information about past gradients dramatically affects the convergence properties of state-of-the-art stochastic optimization methods, such as Heavy-ball, Nesterov's momentum, RMSprop and Adam. Building on this observation, we use stochastic differential equations (SDEs) to explicitly study the role of memory in gradient-based algorithms. We first derive a general continuous-time model that can incorporate arbitrary types of memory, for both deterministic and stochastic settings. We provide convergence guarantees for this SDE for weakly-quasi-convex and quadratically growing functions. We then demonstrate how to discretize this SDE to get a flexible discrete-time algorithm that can implement a board spectrum of memories ranging from short- to long-term. Not only does this algorithm increase the degrees of freedom in algorithmic choice for practitioners but it also comes with better stability properties than classical momentum in the convex stochastic setting. In particular, no iterate averaging is needed for convergence. Interestingly, our analysis also provides a novel interpretation of Nesterov's momentum as stable gradient amplification and highlights a possible reason for its unstable behavior in the (convex) stochastic setting. Furthermore, we discuss the use of long term memory for second-moment estimation in adaptive methods, such as Adam and RMSprop. Finally, we provide an extensive experimental study of the effect of different types of memory in both convex and nonconvex settings.