Non-convex online learning via algorithmic equivalence

TL;DR

This paper establishes an equivalence between non-convex gradient descent and convex mirror descent under reparameterization, providing new regret bounds for non-convex online learning.

Contribution

It introduces a novel algorithmic equivalence technique that connects non-convex gradient descent with convex mirror descent, and proves regret bounds in discrete time.

Findings

Proves $O(T^{2/3})$ regret bound for non-convex online gradient descent.

Establishes discrete-time equivalence between non-convex gradient descent and convex mirror descent.

Provides a new simple method for analyzing non-convex online learning algorithms.

Abstract

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 exactly equivalent to continuous-time mirror descent by Amid and Warmuth 2020, 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.