First-Order Methods for Linearly Constrained Bilevel Optimization

• URL: http://arxiv.org/abs/2406.12771v1
• Date: Tue, 18 Jun 2024 16:41:21 GMT
• Title: First-Order Methods for Linearly Constrained Bilevel Optimization
• Authors: Guy Kornowski, Swati Padmanabhan, Kai Wang, Zhe Zhang, Suvrit Sra,
• Abstract summary: We present first-order linearly constrained optimization methods for high-level Hessian computations. For linear inequality constraints, we attain $(delta,epsilon)$-Goldstein stationarity in $widetildeO(ddelta-1 epsilon-3)$ gradient oracle calls.
• Score: 38.19659447295665
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Algorithms for bilevel optimization often encounter Hessian computations, which are prohibitive in high dimensions. While recent works offer first-order methods for unconstrained bilevel problems, the constrained setting remains relatively underexplored. We present first-order linearly constrained optimization methods with finite-time hypergradient stationarity guarantees. For linear equality constraints, we attain $\epsilon$-stationarity in $\widetilde{O}(\epsilon^{-2})$ gradient oracle calls, which is nearly-optimal. For linear inequality constraints, we attain $(\delta,\epsilon)$-Goldstein stationarity in $\widetilde{O}(d{\delta^{-1} \epsilon^{-3}})$ gradient oracle calls, where $d$ is the upper-level dimension. Finally, we obtain for the linear inequality setting dimension-free rates of $\widetilde{O}({\delta^{-1} \epsilon^{-4}})$ oracle complexity under the additional assumption of oracle access to the optimal dual variable. Along the way, we develop new nonsmooth nonconvex optimization methods with inexact oracles. We verify these guarantees with preliminary numerical experiments.

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