Homotopy Smoothing for Non-Smooth Problems with Lower Complexity than $O(1/\epsilon)$

TL;DR

This paper introduces the HOPS algorithm, which achieves lower iteration complexity than existing methods for non-smooth optimization problems by leveraging local sharpness and smoothing techniques.

Contribution

The paper presents a novel homotopy smoothing (HOPS) algorithm that reduces iteration complexity for non-smooth problems without strong convexity assumptions, outperforming previous methods.

Findings

HOPS achieves a complexity of O(1/ extepsilon^{1- heta}) for non-smooth problems.

HOPS demonstrates linear convergence on several well-known non-smooth problems.

Experimental results show HOPS outperforms Nesterov's smoothing and primal-dual methods.

Abstract

In this paper, we develop a novel {\bf ho}moto{\bf p}y {\bf s}moothing (HOPS) algorithm for solving a family of non-smooth problems that is composed of a non-smooth term with an explicit max-structure and a smooth term or a simple non-smooth term whose proximal mapping is easy to compute. The best known iteration complexity for solving such non-smooth optimization problems is $O(1/\epsilon)$ without any assumption on the strong convexity. In this work, we will show that the proposed HOPS achieved a lower iteration complexity of $\widetilde O(1/\epsilon^{1-\theta})$\footnote{$\widetilde O()$ suppresses a logarithmic factor.} with $\theta\in(0,1]$ capturing the local sharpness of the objective function around the optimal solutions. To the best of our knowledge, this is the lowest iteration complexity achieved so far for the considered non-smooth optimization problems without strong convexity assumption. The HOPS algorithm employs Nesterov's smoothing technique and Nesterov's accelerated gradient method and runs in stages, which gradually decreases the smoothing parameter in a stage-wise manner until it yields a sufficiently good approximation of the original function. We show that HOPS enjoys a linear convergence for many well-known non-smooth problems (e.g., empirical risk minimization with a piece-wise linear loss function and $\ell_1$ norm regularizer, finding a point in a polyhedron, cone programming, etc). Experimental results verify the effectiveness of HOPS in comparison with Nesterov's smoothing algorithm and the primal-dual style of first-order methods.