Towards Gradient Free and Projection Free Stochastic Optimization
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This paper focuses on the problem of constrainedstochastic optimization. A zeroth order Frank-Wolfe algorithm is proposed, which in addition to the projection-free nature of the vanilla Frank-Wolfe algorithm makes it gradient free. Under convexity and smoothness assumption, we show that the proposed algorithm converges to the optimal objective function at a rate O(1/T^1/3), where T denotes the iteration count. In particular, the primal sub-optimality gap is shown to have a dimension dependence of O(d^1/3), which is the best known dimension dependence among all zeroth order optimization algorithms with one directional derivative per iteration. For non-convex functions, we obtain the Frank-Wolfe gap to be O(d^1/3T^-1/4). Experiments on black-box optimization setups demonstrate the efficacy of the proposed algorithm.
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