The Role of Memory in Stochastic Optimization
This work addresses a foundational issue in machine learning optimization, offering incremental improvements by providing a unified framework for memory analysis and more stable algorithms for practitioners.
The paper tackles the problem of how memory in stochastic optimization affects convergence, using stochastic differential equations to model arbitrary memory types and deriving a flexible discrete-time algorithm with better stability than classical momentum, eliminating the need for iterate averaging in convex stochastic settings.
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.