Fast Nonsmooth Regularized Risk Minimization with Continuation

TL;DR

This paper introduces a continuation algorithm for nonsmooth regularized risk minimization that achieves faster convergence rates and outperforms existing methods in classification and regression tasks.

Contribution

The paper presents a flexible continuation algorithm applicable to a broad class of nonsmooth problems with proven convergence rates and improved empirical performance.

Findings

Achieves $O(1/T^2)$ convergence on strongly convex problems.

Achieves $O(1/T)$ convergence on general convex problems.

Outperforms state-of-the-art methods in experiments.

Abstract

In regularized risk minimization, the associated optimization problem becomes particularly difficult when both the loss and regularizer are nonsmooth. Existing approaches either have slow or unclear convergence properties, are restricted to limited problem subclasses, or require careful setting of a smoothing parameter. In this paper, we propose a continuation algorithm that is applicable to a large class of nonsmooth regularized risk minimization problems, can be flexibly used with a number of existing solvers for the underlying smoothed subproblem, and with convergence results on the whole algorithm rather than just one of its subproblems. In particular, when accelerated solvers are used, the proposed algorithm achieves the fastest known rates of $O(1/T^2)$ on strongly convex problems, and $O(1/T)$ on general convex problems. Experiments on nonsmooth classification and regression tasks demonstrate that the proposed algorithm outperforms the state-of-the-art.