A Fully First-Order Method for Stochastic Bilevel Optimization

• URL: http://arxiv.org/abs/2301.10945v1
• Date: Thu, 26 Jan 2023 05:34:21 GMT
• Title: A Fully First-Order Method for Stochastic Bilevel Optimization
• Authors: Jeongyeol Kwon, Dohyun Kwon, Stephen Wright, Robert Nowak
• Abstract summary: We consider unconstrained bilevel optimization problems when only the first-order gradient oracles are available. We propose a Fully First-order Approximation (F2SA) method, and study its non-asymptotic convergence properties. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.
• Score: 8.663726907303303
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider stochastic unconstrained bilevel optimization problems when only the first-order gradient oracles are available. While numerous optimization methods have been proposed for tackling bilevel problems, existing methods either tend to require possibly expensive calculations regarding Hessians of lower-level objectives, or lack rigorous finite-time performance guarantees. In this work, we propose a Fully First-order Stochastic Approximation (F2SA) method, and study its non-asymptotic convergence properties. Specifically, we show that F2SA converges to an $\epsilon$-stationary solution of the bilevel problem after $\epsilon^{-7/2}, \epsilon^{-5/2}$, and $\epsilon^{-3/2}$ iterations (each iteration using $O(1)$ samples) when stochastic noises are in both level objectives, only in the upper-level objective, and not present (deterministic settings), respectively. We further show that if we employ momentum-assisted gradient estimators, the iteration complexities can be improved to $\epsilon^{-5/2}, \epsilon^{-4/2}$, and $\epsilon^{-3/2}$, respectively. We demonstrate even superior practical performance of the proposed method over existing second-order based approaches on MNIST data-hypercleaning experiments.

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