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Error bound condition has recently gained revived interest in optimization.It has been leveraged to derive faster convergence for many popular algorithms,including subgradient methods, proximal gradient method and acceleratedproximal gradient method. However, it is still unclear whether the Frank-Wolfe(FW) method can enjoy faster convergence under error bound condition. In thisshort note, we give an affirmative answer to this question. We show that the FWmethod (with a line search for the step size) for optimization over a stronglyconvex set is automatically adaptive to the error bound condition of theproblem. In particular, the iteration complexity of FW can be characterized by$O(\max(1/\epsilon^{1-\theta}, \log(1/\epsilon)))$ where $\theta\in[0,1]$ is aconstant that characterizes the error bound condition. Our results imply thatif the constrained set is characterized by a strongly convex function and theobjective function can achieve a smaller value outside the considered domain,then the FW method enjoys a fast rate of $O(1/t^2)$.

Frank-Wolfe Method is Automatically Adaptive to Error Bound Condition