An Augmented Lagrangian Value Function Method for Lower-level Constrained Stochastic Bilevel Optimization

Abstract: Recently, lower-level constrained bilevel optimization has attracted increasing attention. However, existing methods mostly focus on either deterministic cases or problems with linear constraints. The main challenge in stochastic cases with general constraints is the bias and variance of the hyper-gradient, arising from the inexact solution of the lower-level problem. In this paper, we propose a novel stochastic augmented Lagrangian value function method for solving stochastic bilevel optimization problems with nonlinear lower-level constraints. Our approach reformulates the original bilevel problem using an augmented Lagrangian-based value function and then applies a penalized stochastic gradient method that carefully manages the noise from stochastic oracles. We establish an equivalence between the stochastic single-level reformulation and the original constrained bilevel problem and provide a non-asymptotic rate of convergence for the proposed method. The rate is further enhanced by employing variance reduction techniques. Extensive experiments on synthetic problems and real-world applications demonstrate the effectiveness of our approach.