Fast Rates of ERM and Stochastic Approximation: Adaptive to Error Bound Conditions

TL;DR

This paper develops adaptive fast convergence rates for empirical risk minimization and stochastic approximation algorithms under error bound conditions, enhancing statistical learning efficiency.

Contribution

It introduces new fast and intermediate convergence rates for ERM and stochastic approximation under EBC, which are adaptive and do not require prior knowledge of EBC.

Findings

Convergence rates range from rac{1}{\u221a n} to rac{1}{n} depending on EBC.

ERM can achieve faster than rac{1}{n} in some cases.

Algorithms adapt automatically to EBC without prior information.

Abstract

Error bound conditions (EBC) are properties that characterize the growth of an objective function when a point is moved away from the optimal set. They have recently received increasing attention in the field of optimization for developing optimization algorithms with fast convergence. However, the studies of EBC in statistical learning are hitherto still limited. The main contributions of this paper are two-fold. First, we develop fast and intermediate rates of empirical risk minimization (ERM) under EBC for risk minimization with Lipschitz continuous, and smooth convex random functions. Second, we establish fast and intermediate rates of an efficient stochastic approximation (SA) algorithm for risk minimization with Lipschitz continuous random functions, which requires only one pass of $n$ samples and adapts to EBC. For both approaches, the convergence rates span a full spectrum between $\widetilde O(1/\sqrt{n})$ and $\widetilde O(1/n)$ depending on the power constant in EBC, and could be even faster than $O(1/n)$ in special cases for ERM. Moreover, these convergence rates are automatically adaptive without using any knowledge of EBC. Overall, this work not only strengthens the understanding of ERM for statistical learning but also brings new fast stochastic algorithms for solving a broad range of statistical learning problems.