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Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion
Ashok Cutkosky · Harsh Mehta · Francesco Orabona

Wed Jul 26 02:00 PM -- 03:30 PM (PDT) @ Exhibit Hall 1 #623
We present new algorithms for optimizing non-smooth, non-convex stochastic objectives based on a novel analysis technique. This improves the current best-known complexity for finding a $(\delta,\epsilon)$-stationary point from $O(\epsilon^{-4}\delta^{-1})$ stochastic gradient queries to $O(\epsilon^{-3}\delta^{-1})$, which we also show to be optimal. Our primary technique is a reduction from non-smooth non-convex optimization to *online learning*, after which our results follow from standard regret bounds in online learning. For *deterministic and second-order smooth* objectives, applying more advanced optimistic online learning techniques enables a new complexity of $O(\epsilon^{-1.5}\delta^{-0.5})$. Our improved non-smooth analysis also immediately recovers all optimal or best-known results for finding $\epsilon$ stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

Author Information

Ashok Cutkosky (Boston University)
Harsh Mehta (Google Research)
Francesco Orabona (KAUST)
Francesco Orabona

Francesco Orabona is an Associate Professor at KAUST. His background covers both theoretical and practical aspects of machine learning and optimization. His current research interests lie in online learning, and more generally the problem of designing and analyzing adaptive and parameter-free learning algorithms. He received the PhD degree in Electrical Engineering at the University of Genoa in 2007. He is (co)author of more than 60 peer reviewed papers.

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