Multilevel Monte-Carlo (MLMC) technique and establish its sample and computational complexities.

However, all the above results are limited to the deterministic setting.

Bilevel optimization problems involve two nested objectives, where an upper-level objective depends on a solution to a lower-level problem. When the latter is non-convex, multiple critical points may be present, leading to an ambiguous definition of the problem. In this paper, we introduce a key ingredient for resolving this ambiguity through the concept of a selection map which allows one to choose a particular solution to the lower-level problem. Using such maps, we define a class of hierarchical games between two agents that resolve the ambiguity in bilevel problems. This new class of games requires introducing new analytical tools in Morse theory to extend implicit differentiation, a technique used in bilevel optimization resulting from the implicit function theorem. In particular, we establish the validity of such a method even when the latter theorem is inapplicable due to degenerate critical points.Finally, we show that algorithms for solving bilevel problems based on unrolled optimization solve these games up to approximation errors due to finite computational power. A simple correction to these algorithms is then proposed for removing these errors.

Learning to Optimize (L2O) is a subfield of machine learning (ML) in which ML models are trained to solve parametric optimization problems. The general goal is to learn a fast approximator of solutions to constrained optimization problems, as a function of their defining parameters. Prior L2O methods focus almost entirely on single-level programs, in contrast to the bilevel programs, whose constraints are themselves expressed in terms of optimization subproblems. Bilevel programs have numerous important use cases but are notoriously difficult to solve, particularly under stringent time demands. This paper proposes a framework for learning to solve a broad class of challenging bilevel optimization problems, by leveraging modern techniques for differentiation through optimization problems. The framework is illustrated on an array of synthetic bilevel programs, as well as challenging control system co-design problems, showing how neural networks can be trained as efficient approximators of parametric bilevel optimization.

Bilevel optimization minimizes an objective function, defined by an upper-level problem whose feasible region is the solution of a lower-level problem. We study the oracle complexity of finding an $ε$-stationary point with first-order methods when the upper-level problem is nonconvex and the lower-level problem is strongly convex. Recent works (Ji et al., ICML 2021; Arbel and Mairal, ICLR 2022; Chen el al., JMLR 2025) achieve a $\tilde{\mathcal{O}}(κ^4 ε^{-2})$ upper bound that is near-optimal in $ε$. However, the optimal dependency on the condition number $κ$ is unknown. In this work, we establish a new $Ω(κ^2 ε^{-2})$ lower bound and $\tilde{\mathcal{O}}(κ^{7/2} ε^{-2})$ upper bound for this problem, establishing the first provable gap between bilevel problems and minimax problems in this setup. Our lower bounds can be extended to various settings, including high-order smooth functions, stochastic oracles, and convex hyper-objectives: (1) For second-order and arbitrarily smooth problems, we show $Ω(κ_y^{13/4} ε^{-12/7})$ and $Ω(κ^{17/10} ε^{-8/5})$ lower bounds, respectively. (2) For convex-strongly-convex problems, we improve the previously best lower bound (Ji and Liang, JMLR 2022) from $Ω(κ/\sqrtε)$ to $Ω(κ^{5/4} / \sqrtε)$. (3) For smooth stochastic problems, we show an $Ω(κ^4 ε^{-4})$ lower bound.

This class of problems is referred to as the "simple bilevel problem" [

Specifically, at each iteration, both the designer and the agents only move one step.