Revisiting SGD with Increasingly Weighted Averaging: Optimization and Generalization Perspectives

TL;DR

This paper investigates how increasingly weighted averaging schemes in stochastic gradient descent influence both optimization and generalization errors across convex, strongly convex, and non-convex problems, revealing trade-offs and practical benefits.

Contribution

It provides a comprehensive analysis of increasingly weighted averaging in SGD for various objective types, addressing both optimization and generalization errors, which was previously under-explored.

Findings

Weighted averaging affects optimization and generalization errors differently.

Polynomially increased weighting can improve convergence and generalization.

Trade-offs exist between optimization speed and generalization performance.

Abstract

Stochastic gradient descent (SGD) has been widely studied in the literature from different angles, and is commonly employed for solving many big data machine learning problems. However, the averaging technique, which combines all iterative solutions into a single solution, is still under-explored. While some increasingly weighted averaging schemes have been considered in the literature, existing works are mostly restricted to strongly convex objective functions and the convergence of optimization error. It remains unclear how these averaging schemes affect the convergence of {\it both optimization error and generalization error} (two equally important components of testing error) for {\bf non-strongly convex objectives, including non-convex problems}. In this paper, we {\it fill the gap} by comprehensively analyzing the increasingly weighted averaging on convex, strongly convex and non-convex objective functions in terms of both optimization error and generalization error. In particular, we analyze a family of increasingly weighted averaging, where the weight for the solution at iteration $t$ is proportional to $t^{\alpha}$ ($\alpha > 0$). We show how $\alpha$ affects the optimization error and the generalization error, and exhibit the trade-off caused by $\alpha$. Experiments have demonstrated this trade-off and the effectiveness of polynomially increased weighted averaging compared with other averaging schemes for a wide range of problems including deep learning.