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arXiv · 2602.23633

On the Convergence of Single-Loop Stochastic Bilevel Optimization with Approximate Implicit Differentiation

Yubo Zhou·Luo Luo·Guang Dai·Haishan Ye
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Abstract

Stochastic Bilevel Optimization has emerged as a fundamental framework for meta-learning and hyperparameter optimization. Despite the practical prevalence of single-loop algorithms, their theoretical understanding in the stochastic regime remains less developed than that of multi-loop methods. In this paper, we provide a refined convergence analysis of the Single-loop Stochastic Approximate Implicit Differentiation (SSAID) algorithm. Under the squared-gradient stationarity criterion $\|\nabla\Phi(x)\|^2\le\epsilon$, the corrected proof establishes an oracle complexity of $\mathcal{O}(\kappa^{14}\epsilon^{-2})$, equivalently an averaged stationarity rate of $\mathcal{O}(\kappa^7K^{-1/2})$. The result preserves the canonical $\mathcal{O}(\epsilon^{-2})$ dependence on the target accuracy while giving an explicit characterization of the condition-number dependence for stochastic AID-based single-loop methods.

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Yubo Zhou, Luo Luo, Guang Dai, Haishan Ye. 2026-02-27. On the Convergence of Single-Loop Stochastic Bilevel Optimization with Approximate Implicit Differentiation. https://arxiv.org/abs/2602.23633

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