Revisiting the Approximate Carathéodory Problem via the Frank-Wolfe Algorithm
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The approximate Carathéodory theorem states that given a polytope P, each point in P can be approximated within ϵ-accuracy in ℓ_p-norm as the convex combination of O(pD_p^2/ϵ^2) vertices, where p≥2 and D_p is the diameter of P in ℓ_p-norm. A solution satisfying these properties can be built using probabilistic arguments [Barman, 2015] or by applying mirror descent to the dual problem [Mirrokni et al., 2017]. We revisit the approximate Carathéodory problem by solving the primal problem via the Frank-Wolfe algorithm, providing a simplified analysis and leading to an efficient practical method. Sublinear to linear sparsity bounds are derived naturally using existing convergence results of the Frank-Wolfe algorithm in different scenarios.
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