A Doubly Stochastically Perturbed Algorithm for Linearly Constrained Bilevel Optimization

In this work, we develop analysis and algorithms for a class of (stochastic) bilevel optimization problems whose lower-level (LL) problem is strongly convex and linearly constrained. Most existing approaches for solving such problems rely on unrealistic assumptions or penalty function-based approximate reformulations that are not necessarily equivalent to the original problem. In this work, we develop a stochastic algorithm based on an implicit gradient approach, suitable for data-intensive applications. It is well-known that for the class of problems of interest, the implicit function is nonsmooth. To circumvent this difficulty, we apply a smoothing technique that involves adding small random (linear) perturbations to the LL objective and then taking the expectation of the implicit objective over these perturbations. This approach gives rise to a novel stochastic formulation that ensures the differentiability of the implicit function and leads to the design of a novel and efficient doubly stochastic algorithm. We show that the proposed algorithm converges to an $(\epsilon, \overline{\delta})$-Goldstein stationary point of the stochastic objective in $\widetilde{{O}}(\epsilon^{-4} \overline{\delta}^{-1})$ iterations. Moreover, under certain additional assumptions, we establish the same convergence guarantee for the algorithm to achieve a $(3\epsilon, \overline{\delta} + {O}(\epsilon))$-Goldstein stationary point of the original objective. Finally, we perform experiments on adversarial training (AT) tasks to showcase the utility of the proposed algorithm.