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kkkFW: A Frank-Wolfe style algorithm with stronger subproblem oracles

29 June 2020
Lijun Ding
Jicong Fan
Madeleine Udell
ArXiv (abs)PDFHTML
Abstract

This paper proposes a new variant of Frank-Wolfe (FW), called kkkFW. Standard FW suffers from slow convergence: iterates often zig-zag as update directions oscillate around extreme points of the constraint set. The new variant, kkkFW, overcomes this problem by using two stronger subproblem oracles in each iteration. The first is a kkk linear optimization oracle (kkkLOO) that computes the kkk best update directions (rather than just one). The second is a kkk direction search (kkkDS) that minimizes the objective over a constraint set represented by the kkk best update directions and the previous iterate. When the problem solution admits a sparse representation, both oracles are easy to compute, and kkkFW converges quickly for smooth convex objectives and several interesting constraint sets: kkkFW achieves finite 4Lf3D4γδ2\frac{4L_f^3D^4}{\gamma\delta^2}γδ24Lf3​D4​ convergence on polytopes and group norm balls, and linear convergence on spectrahedra and nuclear norm balls. Numerical experiments validate the effectiveness of kkkFW and demonstrate an order-of-magnitude speedup over existing approaches.

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