Analyzing and Improving Greedy 2-Coordinate Updates for Equality-Constrained Optimization via Steepest Descent in the 1-Norm

• URL: http://arxiv.org/abs/2307.01169v1
• Date: Mon, 3 Jul 2023 17:27:18 GMT
• Title: Analyzing and Improving Greedy 2-Coordinate Updates for Equality-Constrained Optimization via Steepest Descent in the 1-Norm
• Authors: Amrutha Varshini Ramesh, Aaron Mishkin, Mark Schmidt, Yihan Zhou, Jonathan Wilder Lavington, Jennifer She
• Abstract summary: We exploit a connection between the greedy 2-coordinate update for this problem and equality-constrained steepest descent in the 1-norm. We then consider minimizing with both a summation constraint and bound constraints, as arises in the support vector machine dual problem.
• Score: 12.579063422860072
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We consider minimizing a smooth function subject to a summation constraint over its variables. By exploiting a connection between the greedy 2-coordinate update for this problem and equality-constrained steepest descent in the 1-norm, we give a convergence rate for greedy selection under a proximal Polyak-Lojasiewicz assumption that is faster than random selection and independent of the problem dimension $n$. We then consider minimizing with both a summation constraint and bound constraints, as arises in the support vector machine dual problem. Existing greedy rules for this setting either guarantee trivial progress only or require $O(n^2)$ time to compute. We show that bound- and summation-constrained steepest descent in the L1-norm guarantees more progress per iteration than previous rules and can be computed in only $O(n \log n)$ time.

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