On the Online Frank-Wolfe Algorithms for Convex and Non-convex Optimizations
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In this paper, the online variants of the classical Frank-Wolfe algorithm are considered. We consider minimizing the regret with a stochastic cost. The online algorithms only require simple iterative updates and a non-adaptive step size rule, in contrast to the hybrid schemes commonly considered in the literature. Several new results are derived for convex and non-convex losses. With a strongly convex stochastic cost and when the optimal solution lies in the interior of the constraint set or the constraint set is a polytope, the regret bound and anytime optimality are shown to be O( ^3 T / T ) and O( ^2 T / T), respectively, where T is the number of rounds played. These results are based on an improved analysis on the stochastic Frank-Wolfe algorithms. Moreover, the online algorithms are shown to converge even when the loss is non-convex, i.e., the algorithms find a stationary point to the time-varying/stochastic loss at a rate of O(√(1/T)). Numerical experiments on realistic data sets are presented to support our theoretical claims.
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