On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation

• URL: http://arxiv.org/abs/2309.01753v2
• Date: Sun, 11 Feb 2024 08:10:01 GMT
• Title: On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation
• Authors: Jeongyeol Kwon, Dohyun Kwon, Stephen Wright, Robert Nowak
• Abstract summary: We show first-order algorithms for solving Bilevel Optimization problems. In particular, we show a strong connection between the penalty function and the hyper-objective. We show an improved oracle-complexity of $O(epsilon-3)$ and $O(epsilon-5)$, respectively.
• Score: 13.813242559935732
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter $\sigma > 0$. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be $O(\sigma)$-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as $O(\sigma)$-approximation of the original BO, we propose first-order algorithms that find an $\epsilon$-stationary solution by optimizing the penalty formulation with $\sigma = O(\epsilon)$. When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an $\epsilon$-stationary point of the penalty function, using in total $O(\epsilon^{-3})$ and $O(\epsilon^{-7})$ accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully {\it single-loop} manner, i.e., with $O(1)$ samples per iteration, and achieves the improved oracle-complexity of $O(\epsilon^{-3})$ and $O(\epsilon^{-5})$, respectively.

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