SPABA: A Single-Loop and Probabilistic Stochastic Bilevel Algorithm Achieving Optimal Sample Complexity

• URL: http://arxiv.org/abs/2405.18777v1
• Date: Wed, 29 May 2024 05:36:03 GMT
• Title: SPABA: A Single-Loop and Probabilistic Stochastic Bilevel Algorithm Achieving Optimal Sample Complexity
• Authors: Tianshu Chu, Dachuan Xu, Wei Yao, Jin Zhang,
• Abstract summary: We show that there is no gap in complexity analysis between bilevel and single-level optimization when implementing SPABA. We propose several other bi-loop or nested bi-level algorithms to improve the state of complexity analysis.
• Score: 5.046146134689904
• License: http://creativecommons.org/licenses/by-nc-sa/4.0/
• Abstract: While stochastic bilevel optimization methods have been extensively studied for addressing large-scale nested optimization problems in machine learning, it remains an open question whether the optimal complexity bounds for solving bilevel optimization are the same as those in single-level optimization. Our main result resolves this question: SPABA, an adaptation of the PAGE method for nonconvex optimization in (Li et al., 2021) to the bilevel setting, can achieve optimal sample complexity in both the finite-sum and expectation settings. We show the optimality of SPABA by proving that there is no gap in complexity analysis between stochastic bilevel and single-level optimization when implementing PAGE. Notably, as indicated by the results of (Dagr\'eou et al., 2022), there might exist a gap in complexity analysis when implementing other stochastic gradient estimators, like SGD and SAGA. In addition to SPABA, we propose several other single-loop stochastic bilevel algorithms, that either match or improve the state-of-the-art sample complexity results, leveraging our convergence rate and complexity analysis. Numerical experiments demonstrate the superior practical performance of the proposed methods.

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