Frank-Wolfe algorithms (FW) are popular first-order methods to solve convex constrained optimization problems that rely on a linear minimization oracle instead of potentially expensive projection-like oracles. Many works have identified accelerated convergence rates under various structural assumptions on the optimization problem and for specific FW variants when using line search or short-step, requiring feedback from the objective function. Little is known about accelerated convergence regimes when utilizing open loop step-size rules, a.k.a. FW with pre-determined step-sizes, which are algorithmically extremely simple and stable. Not only is FW with open loop step-size rules not always subject to the same convergence rate lower bounds as FW with line search or short-step, but in some specific cases, such as kernel herding in infinite-dimensions, it is observed that FW with open loop step-size rules leads to faster convergence as opposed to FW with line search or short-step. We propose a partial answer to this open problem in kernel herding, characterize a general setting for which FW with open loop step-size rules converges non-asymptotically faster than with line search or short-step, and derive several accelerated convergence results for FW (and two of its variants) with open loop step-size rules. Finally, our numerical experiments highlight potential gaps in our current understanding of the FW method in general.
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