Sharper Analysis of Single-Loop Methods for Bilevel Optimization

• URL: http://arxiv.org/abs/2607.10263v1
• Date: Sat, 11 Jul 2026 11:41:42 GMT
• Title: Sharper Analysis of Single-Loop Methods for Bilevel Optimization
• Abstract summary: We establish sharper convergence results for single-loop approximate implicit differentiation (AID) and iterative differentiation (ITD) methods.<n>We prove that the error is $mathcalO(2)$, exactly matching the known lower bound and improving upon the previous $mathcalO(3)$ guarantee.
• Score: 81.36947969465848
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Bilevel optimization underpins many machine learning applications, including hyperparameter optimization, meta-learning, neural architecture search, and reinforcement learning. While hypergradient-based methods have advanced significantly, a gap persists between theoretical guarantees and practical single-loop implementations required for efficiency. We bridge this gap by establishing sharper convergence results for single-loop approximate implicit differentiation (AID) and iterative differentiation (ITD) methods, leveraging our proposed analytical framework, decoupled norm analysis (DNA). For AID, we improve the convergence rate from $\mathcal{O}(κ^6/K)$ to $\mathcal{O}(κ^5/K)$, where $κ$ is the condition number of the inner-level problem. For ITD, we prove that the asymptotic error is $\mathcal{O}(κ^2)$, exactly matching the known lower bound and improving upon the previous $\mathcal{O}(κ^3)$ guarantee. Numerical experiments on synthetic and real tasks corroborate our theoretical findings.

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