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

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.

Machine Learning (cs.LG)

arXiv:2602.23633 [cs.LG]

(or arXiv:2602.23633v2 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2602.23633 Focus to learn more arXiv-issued DOI via DataCite

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