Gradient Descent Averaging and Primal-dual Averaging for Strongly Convex Optimization

TL;DR

This paper introduces gradient descent averaging and primal-dual averaging algorithms that achieve optimal convergence rates in strongly convex optimization, supported by theoretical proofs and empirical validation on SVMs and deep learning models.

Contribution

It develops GDA and SC-PDA algorithms that attain optimal convergence rates for strongly convex problems, filling gaps in existing convergence analysis.

Findings

GDA achieves optimal convergence in output averaging.

SC-PDA attains optimal individual convergence.

Experiments confirm theoretical results and effectiveness.

Abstract

Averaging scheme has attracted extensive attention in deep learning as well as traditional machine learning. It achieves theoretically optimal convergence and also improves the empirical model performance. However, there is still a lack of sufficient convergence analysis for strongly convex optimization. Typically, the convergence about the last iterate of gradient descent methods, which is referred to as individual convergence, fails to attain its optimality due to the existence of logarithmic factor. In order to remove this factor, we first develop gradient descent averaging (GDA), which is a general projection-based dual averaging algorithm in the strongly convex setting. We further present primal-dual averaging for strongly convex cases (SC-PDA), where primal and dual averaging schemes are simultaneously utilized. We prove that GDA yields the optimal convergence rate in terms of output averaging, while SC-PDA derives the optimal individual convergence. Several experiments on SVMs and deep learning models validate the correctness of theoretical analysis and effectiveness of algorithms.