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Approximate Frank-Wolfe Algorithms over Graph-structured Support Sets

29 June 2021
Baojian Zhou
Yifan Sun
ArXiv (abs)PDFHTML
Abstract

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 LLL-smooth, we prove that the standard FW method using a δ\deltaδ-approximate DMO converges as O(L/δt+(1−δ)(δ−1+δ−2))\mathcal{O}(L / \delta t + (1-\delta)(\delta^{-1} + \delta^{-2}))O(L/δt+(1−δ)(δ−1+δ−2)) in general, and as O(L/(δ2(t+2)))\mathcal{O}(L/(\delta^2(t+2)))O(L/(δ2(t+2))) over a δ\deltaδ-relaxation of the constraint set. Additionally, when the objective is μ\muμ-strongly convex and the solution is unique, a variant of FW converges to O(L2log⁡(t)/(μδ6t2))\mathcal{O}(L^2\log(t)/(\mu \delta^6 t^2))O(L2log(t)/(μδ6t2)) 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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