Barzilai-Borwein Step Size for Stochastic Gradient Descent

TL;DR

This paper introduces a novel approach for automatically selecting step sizes in stochastic gradient descent methods using the Barzilai-Borwein technique, improving convergence and reducing manual tuning.

Contribution

It proposes the BB-based step size methods for SGD and SVRG, providing theoretical convergence guarantees and demonstrating practical effectiveness through experiments.

Findings

SVRG-BB converges linearly for strongly convex functions

The methods outperform or match well-tuned SGD variants

Automatic step size selection simplifies practical implementation

Abstract

One of the major issues in stochastic gradient descent (SGD) methods is how to choose an appropriate step size while running the algorithm. Since the traditional line search technique does not apply for stochastic optimization algorithms, the common practice in SGD is either to use a diminishing step size, or to tune a fixed step size by hand, which can be time consuming in practice. In this paper, we propose to use the Barzilai-Borwein (BB) method to automatically compute step sizes for SGD and its variant: stochastic variance reduced gradient (SVRG) method, which leads to two algorithms: SGD-BB and SVRG-BB. We prove that SVRG-BB converges linearly for strongly convex objective functions. As a by-product, we prove the linear convergence result of SVRG with Option I proposed in [10], whose convergence result is missing in the literature. Numerical experiments on standard data sets show that the performance of SGD-BB and SVRG-BB is comparable to and sometimes even better than SGD and SVRG with best-tuned step sizes, and is superior to some advanced SGD variants.