Parameter-Agnostic Optimization under Relaxed Smoothness

• URL: http://arxiv.org/abs/2311.03252v1
• Date: Mon, 6 Nov 2023 16:39:53 GMT
• Title: Parameter-Agnostic Optimization under Relaxed Smoothness
• Authors: Florian H\"ubler, Junchi Yang, Xiang Li, Niao He
• Abstract summary: We show that Normalized Gradient Descent with Momentum (NSGD-M) can achieve a rate-optimal complexity without prior knowledge of any problem parameter. In deterministic settings, the exponential factor can be neutralized by employing Gradient Descent with a Backtracking Line Search.
• Score: 25.608968462899316
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• 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.

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