Higher-Order Accelerated Methods for Faster Non-Smooth Optimization
This work provides faster optimization methods for machine learning and data analysis tasks involving non-smooth functions, representing an incremental advance by extending higher-order techniques to non-smooth settings.
The paper tackles non-smooth optimization problems, such as ℓ∞ regression and ℓ1-SVM, by developing higher-order accelerated methods that achieve improved convergence rates, including an O(ε^{-4/5}) iteration complexity for ℓ∞ regression, breaking the previous O(ε^{-1}) barrier.
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 $O(ε^{-4/5})$ iteration complexity, breaking the $O(ε^{-1})$ barrier so far present for previous methods. We arrive at a similar rate for the problem of $\ell_1$-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.