Projection-Free Online Convex Optimization via Efficient Newton Iterations

This paper presents new projection-free algorithms for Online Convex Optimization (OCO) over a convex domain K subset ℝd. Classical OCO algorithms (such as Online Gradient Descent) typically need to perform Euclidean projections onto the convex set cK to ensure feasibility of their iterates. Alternative algorithms, such as those based on the Frank-Wolfe method, swap potentially-expensive Euclidean projections onto K for linear optimization over K. However, such algorithms have a sub-optimal regret in OCO compared to projection-based algorithms. In this paper, we look at a third type of algorithms that output approximate Newton iterates using a self-concordant barrier for the set of interest. The use of a self-concordant barrier automatically ensures feasibility without the need for projections. However, the computation of the Newton iterates requires a matrix inverse, which can still be expensive. As our main contribution, we show how the stability of the Newton iterates can be leveraged to compute the inverse Hessian only a vanishing fraction of the rounds, leading to a new efficient projection-free OCO algorithm with a state-of-the-art regret bound.