BAGEL: Projection-Free Algorithm for Adversarially Constrained Online Convex Optimization

Projection-based algorithms for Constrained Online Convex Optimization (COCO) achieve optimal $\mathcal{O}(T^{1/2})$ regret guarantees but face scalability challenges due to the computational complexity of projections. To circumvent this, projection-free methods utilizing Linear Optimization Oracles (LOO) have been proposed, albeit typically achieving slower $\mathcal{O}(T^{3/4})$ regret rates. In this work, we examine whether the $\mathcal{O}(T^{1/2})$ rate can be recovered in the projection-free setting by strengthening the oracle assumption. We introduce BAGEL, an algorithm utilizing a Separation Oracle (SO) that achieves $\mathcal{O}(T^{1/2})$ regret and $\tilde{\mathcal{O}}(T^{1/2})$ cumulative constraint violation (CCV) for convex cost functions. Our analysis shows that by leveraging an infeasible projection via SO, we can match the time-horizon dependence of projection-based methods with $\tilde{\mathcal{O}}(T)$ oracle calls, provided dependence on the geometry of the action set. This establishes a specific regime where projection-free methods can attain the same convergence rates as projection-based counterparts.