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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 anon-smooth term with an explicit max-structure and a smooth term or a simplenon-smooth term whose proximal mapping is easy to compute. The best knowniteration 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 alogarithmic factor.} with $\theta\in(0,1]$ capturing the local sharpness of theobjective function around the optimal solutions. To the best of our knowledge,this is the lowest iteration complexity achieved so far for the considerednon-smooth optimization problems without strong convexity assumption. The HOPSalgorithm employs Nesterov's smoothing technique and Nesterov's acceleratedgradient method and runs in stages, which gradually decreases the smoothingparameter in a stage-wise manner until it yields a sufficiently goodapproximation of the original function. We show that HOPS enjoys a linearconvergence for many well-known non-smooth problems (e.g., empirical riskminimization with a piece-wise linear loss function and $\ell_1$ normregularizer, finding a point in a polyhedron, cone programming, etc).Experimental results verify the effectiveness of HOPS in comparison withNesterov's smoothing algorithm and the primal-dual style of first-ordermethods.

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