On the Complexity of Finding Stationary Points in Nonconvex Simple Bilevel Optimization

TL;DR

This paper introduces a polynomial-time first-order algorithm for finding stationary points in nonconvex simple bilevel optimization problems, addressing a gap in understanding the complexity of such problems.

Contribution

It proposes a novel notion of stationarity and demonstrates the first complexity bounds for algorithms guaranteeing joint stationarity in general nonconvex bilevel problems.

Findings

The proposed method achieves polynomial complexity for reaching stationarity.

First complexity result for algorithms guaranteeing joint stationarity in nonconvex bilevel problems.

Effective solution up to stationarity for nonconvex simple bilevel optimization.

Abstract

In this paper, we study the problem of solving a simple bilevel optimization problem, where the upper-level objective is minimized over the solution set of the lower-level problem. We focus on the general setting in which both the upper- and lower-level objectives are smooth but potentially nonconvex. Due to the absence of additional structural assumptions for the lower-level objective-such as convexity or the Polyak-{\L}ojasiewicz (PL) condition-guaranteeing global optimality is generally intractable. Instead, we introduce a suitable notion of stationarity for this class of problems and aim to design a first-order algorithm that finds such stationary points in polynomial time. Intuitively, stationarity in this setting means the upper-level objective cannot be substantially improved locally without causing a larger deterioration in the lower-level objective. To this end, we show that a simple and implementable variant of the dynamic barrier gradient descent (DBGD) framework can effectively solve the considered nonconvex simple bilevel problems up to stationarity. Specifically, to reach an $(\epsilon_f, \epsilon_g)$-stationary point-where $\epsilon_f$ and $\epsilon_g$ denote the target stationarity accuracies for the upper- and lower-level objectives, respectively-the considered method achieves a complexity of $\mathcal{O}\left(\max\left(\epsilon_f^{-\frac{3+p}{1+p}}, \epsilon_g^{-\frac{3+p}{2}}\right)\right)$, where $p \geq 0$ is an arbitrary constant balancing the terms. To the best of our knowledge, this is the first complexity result for a discrete-time algorithm that guarantees joint stationarity for both levels in general nonconvex simple bilevel problems.