Higher-Order Accelerated Methods for Faster Non-Smooth Optimization
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We provide improved convergence rates for various non-smooth optimization problems via higher-order accelerated methods. In the case of ℓ_∞ regression, we achieves an O(ϵ^-4/5) iteration complexity, breaking the O(ϵ^-1) barrier so far present for previous methods. We arrive at a similar rate for the problem of ℓ_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 higher-order smooth. With this observation in hand, we provide the first example of higher-order acceleration techniques yielding faster rates for non-smooth optimization, to the best of our knowledge.
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