On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation

In this work, we study first-order algorithms for solving Bilevel Optimization (BO) where the objective functions are smooth but possibly nonconvex in both levels and the variables are restricted to closed convex sets. As a first step, we study the landscape of BO through the lens of penalty methods, in which the upper- and lower-level objectives are combined in a weighted sum with penalty parameter σ > 0. In particular, we establish a strong connection between the penalty function and the hyper-objective by explicitly characterizing the conditions under which the values and derivatives of the two must be O(σ)-close. A by-product of our analysis is the explicit formula for the gradient of hyper-objective when the lower-level problem has multiple solutions under minimal conditions, which could be of independent interest. Next, viewing the penalty formulation as O(σ)-approximation of the original BO, we propose first-order algorithms that find an ε-stationary solution by optimizing the penalty formulation with σ = O(ε). When the perturbed lower-level problem uniformly satisfies the small-error proximal error-bound (EB) condition, we propose a first-order algorithm that converges to an ε-stationary point of the penalty function, using in total O(ε-3) and O(ε-7) accesses to first-order (stochastic) gradient oracles when the oracle is deterministic and oracles are noisy, respectively. Under an additional assumption on stochastic oracles, we show that the algorithm can be implemented in a fully it single-loop manner, i.e., with O(1) samples per iteration, and achieves the improved oracle-complexity of O(ε-3) and O(ε-5), respectively.