Revisiting SGD with Increasingly Weighted Averaging: Optimization and Generalization Perspectives
This work addresses a gap in understanding averaging schemes for non-strongly convex objectives, which is incremental but relevant for practitioners optimizing machine learning models.
The paper analyzes increasingly weighted averaging in SGD for convex, strongly convex, and non-convex objectives, showing how the weight parameter α affects both optimization and generalization errors, with experiments demonstrating trade-offs and effectiveness across various problems including deep learning.
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 $t^α$ ($α> 0$). We show how $α$ affects the optimization error and the generalization error, and exhibit the trade-off caused by $α$. 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.