Sharp Oracle-Regret Tradeoffs for Projection-Free Online Convex Optimization

2026-09-23

AI Summary

This study addresses the trade-off in regret bounds for projection-free online convex optimization under constrained linear optimization oracle calls. Methodologically, it proposes a universal feasibility construction, fixed-body coupling, and an interleaved block-counting approximate gradient algorithm. The primary contributions lie in establishing dimension-free tight regret lower bounds with matching upper bounds, thereby revealing a sharp trade-off between the total oracle budget and per-round call limits. Furthermore, this work identifies critical smoothness thresholds and characterizes optimal convergence rates alongside the requisite budget sizes under specific constraints. Collectively, these results constitute a significant theoretical advance in projection-free online optimization.

Abstract

We characterize the regret attainable in online convex optimization when access to the feasible set is limited to an exact linear optimization oracle. The learner is given an inscribed ball and a diameter bound and must remain feasible on every consistent instance. For convex $G$-Lipschitz losses, diameter at most $D$, a total allowance of $Q$ oracle calls, and a strict limit of $B$ calls per round, the dimension-free minimax expected regret is $Θ(GD\max\{\sqrt T,T/(1+\min\{Q,BT\})^{1/4}\})$. The lower bound applies to arbitrary randomized learners. Universal feasibility first forces each action into the hull of the supplied ball and the preceding oracle replies. A fixed-body construction then couples fresh phase directions to a shared simplex, making useful replies costly repeatedly even though all losses have a common minimizer. A counted approximate-gradient method with interleaved blocks attains the matching rate. Total-budget and strict per-round guarantees follow as special cases, including the $T^{3/4}$ rate with one call per round and the quadratic total budget needed for $\sqrt T$ regret. For prescribed smoothness $β$, an analytic construction yields a curvature-dependent lower bound and identifies the threshold above which the general characterization remains sharp.

Problem

• Online Convex Optimization
• Projection-Free
• Oracle-Regret Tradeoffs
• Linear Optimization Oracle
• Minimax Regret

Innovation

• Projection-Free Online Convex Optimization
• Linear Optimization Oracle
• Minimax Regret
• Oracle-Regret Tradeoffs
• Approximate-Gradient Method