In this paper, we propose approximate Frank-Wolfe (FW) algorithms to solve convex optimization problems over graph-structured support sets where the \textit{linear minimization oracle} (LMO) cannot be efficiently obtained in general. We first demonstrate that two popular approximation assumptions (\textit{additive} and \textit{multiplicative gap errors)}, are not valid for our problem, in that no cheap gap-approximate LMO oracle exists in general. Instead, a new \textit{approximate dual maximization oracle} (DMO) is proposed, which approximates the inner product rather than the gap. When the objective is -smooth, we prove that the standard FW method using a -approximate DMO converges as in general, and as over a -relaxation of the constraint set. Additionally, when the objective is -strongly convex and the solution is unique, a variant of FW converges to with the same per-iteration complexity. Our empirical results suggest that even these improved bounds are pessimistic, with significant improvement in recovering real-world images with graph-structured sparsity.
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