Parameter-Agnostic Optimization under Relaxed Smoothness
Florian Hübler, Junchi Yang, Xiang Li, Niao He
Abstract
Tuning hyperparameters, such as the stepsize, presents a major challenge of training machine learning models. To address this challenge, numerous adaptive optimization algorithms have been developed that achieve near-optimal complexities, even when stepsizes are independent of problem-specific parameters, provided that the loss function is $L$-smooth. However, as the assumption is relaxed to the more realistic $(L_0, L_1)$-smoothness, all existing convergence results still necessitate tuning of the stepsize. In this study, we demonstrate that Normalized Stochastic Gradient Descent with Momentum (NSGD-M) can achieve a (nearly) rate-optimal complexity without prior knowledge of any problem parameter, though this comes at the cost of introducing an exponential term dependent on $L_1$ in the complexity. We further establish that this exponential term is inevitable to such schemes by introducing a theoretical framework of lower bounds tailored explicitly for parameter-agnostic algorithms. Interestingly, in deterministic settings, the exponential factor can be neutralized by employing Gradient Descent with a Backtracking Line Search. To the best of our knowledge, these findings represent the first parameter-agnostic convergence results under the generalized smoothness condition. Our empirical experiments further confirm our theoretical insights.
BibTeX
@InProceedings{pmlr-v238-hubler24a,
title = {Parameter-Agnostic Optimization under Relaxed Smoothness},
author = {H\"{u}bler, Florian and Yang, Junchi and Li, Xiang and He, Niao},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {4861--4869},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/hubler24a/hubler24a.pdf},
url = {https://proceedings.mlr.press/v238/hubler24a.html},
abstract = {Tuning hyperparameters, such as the stepsize, presents a major challenge of training machine learning models. To address this challenge, numerous adaptive optimization algorithms have been developed that achieve near-optimal complexities, even when stepsizes are independent of problem-specific parameters, provided that the loss function is $L$-smooth. However, as the assumption is relaxed to the more realistic $(L_0, L_1)$-smoothness, all existing convergence results still necessitate tuning of the stepsize. In this study, we demonstrate that Normalized Stochastic Gradient Descent with Momentum (NSGD-M) can achieve a (nearly) rate-optimal complexity without prior knowledge of any problem parameter, though this comes at the cost of introducing an exponential term dependent on $L_1$ in the complexity. We further establish that this exponential term is inevitable to such schemes by introducing a theoretical framework of lower bounds tailored explicitly for parameter-agnostic algorithms. Interestingly, in deterministic settings, the exponential factor can be neutralized by employing Gradient Descent with a Backtracking Line Search. To the best of our knowledge, these findings represent the first parameter-agnostic convergence results under the generalized smoothness condition. Our empirical experiments further confirm our theoretical insights.}
}