A New Randomized Primal-Dual Algorithm for Convex Optimization with Optimal Last Iterate Rates

• URL: http://arxiv.org/abs/2003.01322v3
• Date: Thu, 28 Oct 2021 14:32:18 GMT
• Title: A New Randomized Primal-Dual Algorithm for Convex Optimization with Optimal Last Iterate Rates
• Authors: Quoc Tran-Dinh and Deyi Liu
• Abstract summary: We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems. We prove that our algorithm achieves optimal convergence rates in two cases: general convexity and strong convexity. Our results show that the proposed method has encouraging performance on different experiments.
• Score: 16.54912614895861
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We develop a novel unified randomized block-coordinate primal-dual algorithm to solve a class of nonsmooth constrained convex optimization problems, which covers different existing variants and model settings from the literature. We prove that our algorithm achieves optimal $\mathcal{O}(n/k)$ and $\mathcal{O}(n^2/k^2)$ convergence rates (up to a constant factor) in two cases: general convexity and strong convexity, respectively, where $k$ is the iteration counter and n is the number of block-coordinates. Our convergence rates are obtained through three criteria: primal objective residual and primal feasibility violation, dual objective residual, and primal-dual expected gap. Moreover, our rates for the primal problem are on the last iterate sequence. Our dual convergence guarantee requires additionally a Lipschitz continuity assumption. We specify our algorithm to handle two important special cases, where our rates are still applied. Finally, we verify our algorithm on two well-studied numerical examples and compare it with two existing methods. Our results show that the proposed method has encouraging performance on different experiments.

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