On the Convergence Rates of Iterative Regularization Algorithms for Composite Bilevel Optimization

Abstract. This paper investigates iterative methods for solving bilevel optimization problems where both inner and outer functions have a composite structure. We establish novel theoretical results, including the first analysis that provides simultaneous convergence rates for the iteratively regularized proximal gradient (IRE-PG) method, a variant of Solodov’s algorithm. These rates for the inner and outer functions highlight the inherent trade-offs between their respective convergence behaviors. We further extend this analysis to an accelerated version of IRE-PG, proving faster convergence rates under specific settings. Additionally, we propose a new scheme for handling cases where these methods cannot be directly applied to the bilevel problem due to the difficulty of computing the associated proximal operator. This scheme offers surrogate functions to approximate the original problem and a framework to translate convergence rates between the surrogate and original functions. Our results show that the accelerated method’s advantage diminishes under this translation.