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This paper presents new projection-free algorithms for Online ConvexOptimization (OCO) over a convex domain $\mathcal{K} \subset \mathbb{R}^d$.Classical OCO algorithms (such as Online Gradient Descent) typically need toperform Euclidean projections onto the convex set $\cK$ to ensure feasibilityof their iterates. Alternative algorithms, such as those based on theFrank-Wolfe method, swap potentially-expensive Euclidean projections onto$\mathcal{K}$ for linear optimization over $\mathcal{K}$. However, suchalgorithms have a sub-optimal regret in OCO compared to projection-basedalgorithms. In this paper, we look at a third type of algorithms that outputapproximate Newton iterates using a self-concordant barrier for the set ofinterest. The use of a self-concordant barrier automatically ensuresfeasibility without the need for projections. However, the computation of theNewton iterates requires a matrix inverse, which can still be expensive. As ourmain contribution, we show how the stability of the Newton iterates can beleveraged to compute the inverse Hessian only a vanishing fraction of therounds, leading to a new efficient projection-free OCO algorithm with astate-of-the-art regret bound.

Projection-Free Online Convex Optimization via Efficient Newton Iterations