Achieving ${O}(\epsilon^{-1.5})$ Complexity in Hessian/Jacobian-free Stochastic Bilevel Optimization

• URL: http://arxiv.org/abs/2312.03807v2
• Date: Wed, 20 Dec 2023 15:21:56 GMT
• Title: Achieving ${O}(\epsilon^{-1.5})$ Complexity in Hessian/Jacobian-free Stochastic Bilevel Optimization
• Authors: Yifan Yang, Peiyao Xiao, Kaiyi Ji
• Abstract summary: We show how to achieve an $O(epsilon1.5)$ sample complexity for non-strongly-accurate stationary point gradient bilevel optimization. As far as we know, this is the first Hessian/Jacobian-free method with an $O(epsilon1.5)$ sample complexity for non-strongly-accurate stationary point gradient optimization.
• Score: 21.661676550609876
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this paper, we revisit the bilevel optimization problem, in which the upper-level objective function is generally nonconvex and the lower-level objective function is strongly convex. Although this type of problem has been studied extensively, it still remains an open question how to achieve an ${O}(\epsilon^{-1.5})$ sample complexity in Hessian/Jacobian-free stochastic bilevel optimization without any second-order derivative computation. To fill this gap, we propose a novel Hessian/Jacobian-free bilevel optimizer named FdeHBO, which features a simple fully single-loop structure, a projection-aided finite-difference Hessian/Jacobian-vector approximation, and momentum-based updates. Theoretically, we show that FdeHBO requires ${O}(\epsilon^{-1.5})$ iterations (each using ${O}(1)$ samples and only first-order gradient information) to find an $\epsilon$-accurate stationary point. As far as we know, this is the first Hessian/Jacobian-free method with an ${O}(\epsilon^{-1.5})$ sample complexity for nonconvex-strongly-convex stochastic bilevel optimization.

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