Non-convex online learning via algorithmic equivalence

Part of Advances in Neural Information Processing Systems 35 (NeurIPS 2022) Main Conference Track

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Udaya Ghai, Zhou Lu, Elad Hazan


We study an algorithmic equivalence technique between non-convex gradient descent and convex mirror descent. We start by looking at a harder problem of regret minimization in online non-convex optimization. We show that under certain geometric and smoothness conditions, online gradient descent applied to non-convex functions is an approximation of online mirror descent applied to convex functions under reparameterization. In continuous time, the gradient flow with this reparameterization was shown to be \emph{exactly} equivalent to continuous-time mirror descent by Amid and Warmuth, but theory for the analogous discrete time algorithms is left as an open problem. We prove an $O(T^{\frac{2}{3}})$ regret bound for non-convex online gradient descent in this setting, answering this open problem. Our analysis is based on a new and simple algorithmic equivalence method.