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

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 |nablaPhi(x)|2≤ε, the corrected proof establishes an oracle complexity of O(kappa14ε-2), equivalently an averaged stationarity rate of O(kappa7K-1/2). The result preserves the canonical O(ε-2) dependence on the target accuracy while giving an explicit characterization of the condition-number dependence for stochastic AID-based single-loop methods.