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 $\Vert \nabla \Phi (x){\Vert }^2\le \epsilon$, the corrected proof establishes an oracle complexity of $\mathcal{O}({\kappa }^{14}{\epsilon }^{-2})$, equivalently an averaged stationarity rate of $\mathcal{O}({\kappa }^7{K}^{-1/2})$. 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.