Non-Uniform Smoothness for Gradient Descent

Albert S. Berahas; Lindon Roberts; Fred Roosta

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arXiv · 2311.08615

Non-Uniform Smoothness for Gradient Descent

Albert S. Berahas·Lindon Roberts·Fred Roosta

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Abstract

The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given problem. In this work we introduce a local first-order smoothness oracle (LFSO) which generalizes the Lipschitz continuous gradients smoothness condition and is applicable to any twice-differentiable function. We show that this oracle can encode all relevant problem information for tuning stepsizes for a suitably modified gradient descent method and give global and local convergence results. We also show that LFSOs in this modified first-order method can yield global linear convergence rates for non-strongly convex problems with extremely flat minima, and thus improve over the lower bound on rates achievable by general (accelerated) first-order methods.

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BibTeX RIS

Albert S. Berahas, Lindon Roberts, Fred Roosta. 2023-11-15. Non-Uniform Smoothness for Gradient Descent. https://arxiv.org/abs/2311.08615

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