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In this paper, we develop new efficient projection-free algorithms for OnlineConvex Optimization (OCO). Online Gradient Descent (OGD) is an example of aclassical OCO algorithm that guarantees the optimal $O(\sqrt{T})$ regret bound.However, OGD and other projection-based OCO algorithms need to perform aEuclidean projection onto the feasible set $\mathcal{C}\subset \mathbb{R}^d$whenever their iterates step outside $\mathcal{C}$. For various sets ofinterests, this projection step can be computationally costly, especially whenthe ambient dimension is large. This has motivated the development ofprojection-free OCO algorithms that swap Euclidean projections for often muchcheaper operations such as Linear Optimization (LO). However, state-of-the-artLO-based algorithms only achieve a suboptimal $O(T^{3/4})$ regret for generalOCO. In this paper, we leverage recent results in parameter-free OnlineLearning, and develop an OCO algorithm that makes two calls to an LO Oracle perround and achieves the near-optimal $\widetilde{O}(\sqrt{T})$ regret wheneverthe feasible set is strongly convex. We also present an algorithm for generalconvex sets that makes $\widetilde O(d)$ expected number of calls to an LOOracle per round and guarantees a $\widetilde O(T^{2/3})$ regret, improving onthe previous best $O(T^{3/4})$. We achieve the latter by approximating anyconvex set $\mathcal{C}$ by a strongly convex one, where LO can be performedusing $\widetilde {O}(d)$ expected number of calls to an LO Oracle for$\mathcal{C}$.

Exploiting the Curvature of Feasible Sets for Faster Projection-Free Online Learning