On the Complexity of First-Order Methods in Stochastic Bilevel Optimization

We consider the problem of finding stationary points in Bilevel optimization when the lower-level problem is unconstrained and strongly convex. The problem has been extensively studied in recent years; the main technical challenge is to keep track of lower-level solutions y^*(x) in response to the changes in the upper-level variables x. Subsequently, all existing approaches tie their analyses to a genie algorithm that knows lower-level solutions and, therefore, need not query any points far from them. We consider a dual question to such approaches: suppose we have an oracle, which we call y^*-aware, that returns an O(ε)-estimate of the lower-level solution, in addition to first-order gradient estimators it locally unbiased within the Theta(ε)-ball around y^*(x). We study the complexity of finding stationary points with such an y^*-aware oracle: we propose a simple first-order method that converges to an ε stationary point using O(ε-6), O(ε-4) access to first-order y^*-aware oracles. Our upper bounds also apply to standard unbiased first-order oracles, improving the best-known complexity of first-order methods by O(ε) with minimal assumptions. We then provide the matching Omega(ε-6), Omega(ε-4) lower bounds without and with an additional smoothness assumption on y^*-aware oracles, respectively. Our results imply that any approach that simulates an algorithm with an y^*-aware oracle must suffer the same lower bounds.