Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization

• URL: http://arxiv.org/abs/2212.05088v1
• Date: Fri, 9 Dec 2022 19:17:39 GMT
• Title: Cyclic Block Coordinate Descent With Variance Reduction for Composite Nonconvex Optimization
• Authors: Xufeng Cai, Chaobing Song, Stephen J. Wright, Jelena Diakonikolas
• Abstract summary: We propose methods for solving problems coordinate non-asymptotic gradient norm guarantees. Our results demonstrate the efficacy of the proposed cyclic-reduced scheme in training deep neural nets.
• Score: 26.218670461973705
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Nonconvex optimization is central in solving many machine learning problems, in which block-wise structure is commonly encountered. In this work, we propose cyclic block coordinate methods for nonconvex optimization problems with non-asymptotic gradient norm guarantees. Our convergence analysis is based on a gradient Lipschitz condition with respect to a Mahalanobis norm, inspired by a recent progress on cyclic block coordinate methods. In deterministic settings, our convergence guarantee matches the guarantee of (full-gradient) gradient descent, but with the gradient Lipschitz constant being defined w.r.t.~the Mahalanobis norm. In stochastic settings, we use recursive variance reduction to decrease the per-iteration cost and match the arithmetic operation complexity of current optimal stochastic full-gradient methods, with a unified analysis for both finite-sum and infinite-sum cases. We further prove the faster, linear convergence of our methods when a Polyak-{\L}ojasiewicz (P{\L}) condition holds for the objective function. To the best of our knowledge, our work is the first to provide variance-reduced convergence guarantees for a cyclic block coordinate method. Our experimental results demonstrate the efficacy of the proposed variance-reduced cyclic scheme in training deep neural nets.

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