@inproceedings{
mishkin2024directional,
title={Directional Smoothness and Gradient Methods: Convergence and Adaptivity},
author={Aaron Mishkin and Ahmed Khaled and Yuanhao Wang and Aaron Defazio and Robert M. Gower},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=m9WZrEXWl5}
}
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.
Directional Smoothness and Gradient Methods: Convergence and Adaptivity
Aaron Mishkin Ahmed Khaled Yuanhao Wang Aaron Defazio Robert M. Gower