Projection-Free Methods for Stochastic Simple Bilevel Optimization with Convex Lower-level Problem

In this paper, we study a class of stochastic bilevel optimization problems, also known as stochastic simple bilevel optimization, where we minimize a smooth stochastic objective function over the optimal solution set of another stochastic convex optimization problem. We introduce novel stochastic bilevel optimization methods that locally approximate the solution set of the lower-level problem via a stochastic cutting plane, and then run a conditional gradient update with variance reduction techniques to control the error induced by using stochastic gradients. For the case that the upper-level function is convex, our method requires Õ(max{1/ϵ²_f, 1/ϵ²_g}) stochastic oracle queries to obtain a solution that is ϵ_foptimal for the upper-level and ϵ_g-optimal for the lower-level. This guarantee improves the previous best-known complexity of O(max{1/ϵ⁴_f, 1/ϵ⁴_g}). Moreover, for the case that the upper-level function is non-convex, our method requires at most Õ(max{1/ϵ³_f, 1/ϵ³_g}) stochastic oracle queries to find an (ϵ_f, ϵ_g)-stationary point. In the finite-sum setting, we show that the number of stochastic oracle calls required by our method are Õ(√n/ϵ) and Õ(√n/ϵ²) for the convex and non-convex settings, respectively, where ϵ = min{ϵ_f, ϵ_g}.