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arXiv · 2403.04081

Directional Smoothness and Gradient Methods: Convergence and Adaptivity

Aaron Mishkin·Ahmed Khaled·Yuanhao Wang·Aaron Defazio·Robert M. Gower
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Abstract

We develop new sub-optimality bounds for gradient descent (GD) that depend on the conditioning of the objective along the path of optimization rather than on global, worst-case constants. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. Minimizing these upper-bounds requires solving implicit equations to obtain a sequence of strongly adapted step-sizes; we show that these equations are straightforward to solve for convex quadratics and lead to new guarantees for two classical step-sizes. For general functions, we prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness. Experiments on logistic regression show our convergence guarantees are tighter than the classical theory based on $L$-smoothness.

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Aaron Mishkin, Ahmed Khaled, Yuanhao Wang, Aaron Defazio, Robert M. Gower. 2024-03-06. Directional Smoothness and Gradient Methods: Convergence and Adaptivity. https://arxiv.org/abs/2403.04081

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