Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion

• URL: http://arxiv.org/abs/2302.03775v1
• Date: Tue, 7 Feb 2023 22:09:20 GMT
• Title: Optimal Stochastic Non-smooth Non-convex Optimization through Online-to-Non-convex Conversion
• Authors: Ashok Cutkosky, Harsh Mehta, Francesco Orabona
• Abstract summary: We present new algorithms for optimizing non-known, non-smooth objectives based on a novel analysis technique. For deterministic second-order smooth objectives, applying advanced optimistic online learning techniques enables a new $O(delta0.5) all$ to recover optimal or best-known results.
• Score: 56.92236659731376
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 techniques also recover all optimal or best-known results for finding $\epsilon$ stationary points of smooth or second-order smooth objectives in both stochastic and deterministic settings.

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