Non-convex online learning via algorithmic equivalence
This provides a theoretical foundation for online learning with non-convex functions, addressing a gap in discrete-time algorithms for researchers in optimization and machine learning.
The paper tackles the open problem of regret minimization in online non-convex optimization by proving an O(T^{2/3}) regret bound for non-convex online gradient descent, using an algorithmic equivalence technique to relate it to convex mirror descent.
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.