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Provable and Practical Online Learning Rate Adaptation with Hypergradient Descent

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arxiv 2502.11229 v2 pith:Y7WPACKD submitted 2025-02-16 math.OC cs.LG

classification math.OCcs.LG
keywords adaptiveanalysisconvergencemethodsdescentdevelopefficientfirst-order
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This paper investigates the convergence properties of the hypergradient descent method (HDM), a 25-year-old heuristic originally proposed for adaptive stepsize selection in stochastic first-order methods. We provide the first rigorous convergence analysis of HDM using the online learning framework of [Gao24] and apply this analysis to develop new state-of-the-art adaptive gradient methods with empirical and theoretical support. Notably, HDM automatically identifies the optimal stepsize for the local optimization landscape and achieves local superlinear convergence. Our analysis explains the instability of HDM reported in the literature and proposes efficient strategies to address it. We also develop two HDM variants with heavy-ball and Nesterov momentum. Experiments on deterministic convex problems show HDM with heavy-ball momentum (HDM-HB) exhibits robust performance and significantly outperforms other adaptive first-order methods. Moreover, HDM-HB often matches the performance of L-BFGS, an efficient and practical quasi-Newton method, using less memory and cheaper iterations.

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Cited by 4 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Gradient Methods with Online Scaling Part I. Theoretical Foundations

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    Online preconditioning for a GPU LP solver cuts iteration counts by roughly 10-30% on Netlib and MIPLIB benchmarks, with the learning rate tuned per instance.

  3. Online Learning-guided Learning Rate Adaptation via Gradient Alignment

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    GALA adapts the learning rate through online learning on gradient alignment, achieves a data-adaptive convergence rate for normalized SGD, and shows robust empirical performance across initial learning rates.

  4. Accelerating Optimization via Differentiable Stopping Time

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    A discrete stopping-time sensitivity defined from an ODE discretization approximates the continuous hitting-time gradient with O(h) error, enabling gradient-based optimization of iteration counts.

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