Higher-Order Accelerated Methods for Faster Non-Smooth Optimization

Abstract: We provide improved convergence rates for various \emph{non-smooth} optimization problems via higher-order accelerated methods. In the case of $\ell_\infty$ regression, we achieves an $O(\epsilon^{-4/5})$ iteration complexity, breaking the $O(\epsilon^{-1})$ barrier so far present for previous methods. We arrive at a similar rate for the problem of $\ell_1$-SVM, going beyond what is attainable by first-order methods with prox-oracle access for non-smooth non-strongly convex problems. We further show how to achieve even faster rates by introducing higher-order regularization. Our results rely on recent advances in near-optimal accelerated methods for higher-order smooth convex optimization. In particular, we extend Nesterov's smoothing technique to show that the standard softmax approximation is not only smooth in the usual sense, but also \emph{higher-order} smooth. With this observation in hand, we provide the first example of higher-order acceleration techniques yielding faster rates for \emph{non-smooth} optimization, to the best of our knowledge.