Gradient Methods with Online Scaling

• URL: http://arxiv.org/abs/2411.01803v2
• Date: Tue, 05 Nov 2024 21:16:44 GMT
• Title: Gradient Methods with Online Scaling
• Authors: Wenzhi Gao, Ya-Chi Chu, Yinyu Ye, Madeleine Udell,
• Abstract summary: We introduce a framework to accelerate the convergence of gradient-based methods with online learning. We show for the first time that the widely-used hypergradient descent improves on the convergence of gradient descent.
• Score: 19.218484733179356
• License:
• Abstract: We introduce a framework to accelerate the convergence of gradient-based methods with online learning. The framework learns to scale the gradient at each iteration through an online learning algorithm and provably accelerates gradient-based methods asymptotically. In contrast with previous literature, where convergence is established based on worst-case analysis, our framework provides a strong convergence guarantee with respect to the optimal scaling matrix for the iteration trajectory. For smooth strongly convex optimization, our results provide an $O(\kappa^\star \log(1/\varepsilon)$) complexity result, where $\kappa^\star$ is the condition number achievable by the optimal preconditioner, improving on the previous $O(\sqrt{n}\kappa^\star \log(1/\varepsilon))$ result. In particular, a variant of our method achieves superlinear convergence on convex quadratics. For smooth convex optimization, we show for the first time that the widely-used hypergradient descent heuristic improves on the convergence of gradient descent.

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