Non-Uniform Smoothness for Gradient Descent

• URL: http://arxiv.org/abs/2311.08615v1
• Date: Wed, 15 Nov 2023 00:44:08 GMT
• Title: Non-Uniform Smoothness for Gradient Descent
• Authors: Albert S. Berahas, Lindon Roberts, Fred Roosta
• Abstract summary: We introduce a local first-order smoothness oracle (LFSO) which generalizes the Lipschitz continuous gradient smoothness condition. We show that this oracle can encode all relevant problem information for tuning stepsizes for a suitably modified gradient descent method. 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.
• Score: 5.64297382055816
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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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