Linear Convergence of a Frank-Wolfe Type Algorithm over Trace-Norm Balls

Part of Advances in Neural Information Processing Systems 30 (NIPS 2017)

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Authors

Zeyuan Allen-Zhu, Elad Hazan, Wei Hu, Yuanzhi Li

Abstract

<p>We propose a rank-k variant of the classical Frank-Wolfe algorithm to solve convex optimization over a trace-norm ball. Our algorithm replaces the top singular-vector computation (1-SVD) in Frank-Wolfe with a top-k singular-vector computation (k-SVD), which can be done by repeatedly applying 1-SVD k times. Alternatively, our algorithm can be viewed as a rank-k restricted version of projected gradient descent. We show that our algorithm has a linear convergence rate when the objective function is smooth and strongly convex, and the optimal solution has rank at most k. This improves the convergence rate and the total time complexity of the Frank-Wolfe method and its variants.</p>