A Stochastic Quasi-Newton Method for Non-convex Optimization with Non-uniform Smoothness

• URL: http://arxiv.org/abs/2403.15244v2
• Date: Thu, 26 Sep 2024 15:56:30 GMT
• Title: A Stochastic Quasi-Newton Method for Non-convex Optimization with Non-uniform Smoothness
• Authors: Zhenyu Sun, Ermin Wei,
• Abstract summary: We propose a fast quasi-Newton method when there exists non-uniformity in smoothness. Our algorithm can achieve the best-known $mathcalO(epsilon-3)$ sample complexity and enjoys convergence speedup. Our numerical experiments show that our proposed algorithm outperforms the state-of-the-art approaches.
• Score: 4.097563258332958
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Classical convergence analyses for optimization algorithms rely on the widely-adopted uniform smoothness assumption. However, recent experimental studies have demonstrated that many machine learning problems exhibit non-uniform smoothness, meaning the smoothness factor is a function of the model parameter instead of a universal constant. In particular, it has been observed that the smoothness grows with respect to the gradient norm along the training trajectory. Motivated by this phenomenon, the recently introduced $(L_0, L_1)$-smoothness is a more general notion, compared to traditional $L$-smoothness, that captures such positive relationship between smoothness and gradient norm. Under this type of non-uniform smoothness, existing literature has designed stochastic first-order algorithms by utilizing gradient clipping techniques to obtain the optimal $\mathcal{O}(\epsilon^{-3})$ sample complexity for finding an $\epsilon$-approximate first-order stationary solution. Nevertheless, the studies of quasi-Newton methods are still lacking. Considering higher accuracy and more robustness for quasi-Newton methods, in this paper we propose a fast stochastic quasi-Newton method when there exists non-uniformity in smoothness. Leveraging gradient clipping and variance reduction, our algorithm can achieve the best-known $\mathcal{O}(\epsilon^{-3})$ sample complexity and enjoys convergence speedup with simple hyperparameter tuning. Our numerical experiments show that our proposed algorithm outperforms the state-of-the-art approaches.

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