Online Convex Optimization with a Separation Oracle
Zakaria Mhammedi
TL;DR
A new projection-free algorithm for Online Convex Optimization (OCO) with a state-of-the-art regret guarantee among separation-based algorithms and a state-of-the-art convergence rate for constrained stochastic convex optimization.
Abstract
In this paper, we introduce a new projection-free algorithm for Online Convex Optimization (OCO) with a state-of-the-art regret guarantee among separation-based algorithms. Existing projection-free methods based on the classical Frank-Wolfe algorithm achieve a suboptimal regret bound of $O(T^{3/4})$, while more recent separation-based approaches guarantee a regret bound of $O(κ\sqrt{T})$, where $κ$ denotes the asphericity of the feasible set, defined as the ratio of the radii of the containing and contained balls. However, for ill-conditioned sets, $κ$ can be arbitrarily large, potentially leading to poor performance. Our algorithm achieves a regret bound of $\widetilde{O}(\sqrt{dT} + κd)$, while requiring only $\widetilde{O}(1)$ calls to a separation oracle per round. Crucially, the main term in the bound, $\widetilde{O}(\sqrt{d T})$, is independent of $κ$, addressing the limitations of previous methods. Additionally, as a by-product of our analysis, we recover the $O(κ\sqrt{T})$ regret bound of existing OCO algorithms with a more straightforward analysis and improve the regret bound for projection-free online exp-concave optimization. Finally, for constrained stochastic convex optimization, we achieve a state-of-the-art convergence rate of $\widetilde{O}(σ/\sqrt{T} + κd/T)$, where $σ$ represents the noise in the stochastic gradients, while requiring only $\widetilde{O}(1)$ calls to a separation oracle per iteration.