Provable and Practical Online Learning Rate Adaptation with Hypergradient Descent
It addresses the problem of adaptive learning rate tuning for practitioners in optimization, offering provable and practical improvements over existing methods.
This paper provides the first rigorous convergence analysis of the hypergradient descent method (HDM), a heuristic for adaptive stepsize selection, and develops new adaptive gradient methods that achieve local superlinear convergence and robust performance, with HDM-HB often matching L-BFGS using less memory and cheaper iterations.
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.