On Learning Rates and Schrödinger Operators
This work addresses the fundamental problem of understanding learning rate tuning in SGD for researchers and practitioners in machine learning, offering theoretical insights but being incremental as it builds on existing SDE-based analyses.
The paper presents a theoretical analysis of learning rates in stochastic gradient descent (SGD) using a learning-rate-dependent stochastic differential equation, establishing a linear convergence rate and deriving an explicit optimal rate via the Witten-Laplacian spectrum. It reveals that the convergence rate decreases to zero as the learning rate tends to zero for nonconvex functions but stays constant for convex ones, providing a mathematical interpretation for learning rate decay in nonconvex optimization.
The learning rate is perhaps the single most important parameter in the training of neural networks and, more broadly, in stochastic (nonconvex) optimization. Accordingly, there are numerous effective, but poorly understood, techniques for tuning the learning rate, including learning rate decay, which starts with a large initial learning rate that is gradually decreased. In this paper, we present a general theoretical analysis of the effect of the learning rate in stochastic gradient descent (SGD). Our analysis is based on the use of a learning-rate-dependent stochastic differential equation (lr-dependent SDE) that serves as a surrogate for SGD. For a broad class of objective functions, we establish a linear rate of convergence for this continuous-time formulation of SGD, highlighting the fundamental importance of the learning rate in SGD, and contrasting to gradient descent and stochastic gradient Langevin dynamics. Moreover, we obtain an explicit expression for the optimal linear rate by analyzing the spectrum of the Witten-Laplacian, a special case of the Schrödinger operator associated with the lr-dependent SDE. Strikingly, this expression clearly reveals the dependence of the linear convergence rate on the learning rate -- the linear rate decreases rapidly to zero as the learning rate tends to zero for a broad class of nonconvex functions, whereas it stays constant for strongly convex functions. Based on this sharp distinction between nonconvex and convex problems, we provide a mathematical interpretation of the benefits of using learning rate decay for nonconvex optimization.