@inproceedings{
cao2024an,
title={An Accelerated Gradient Method for Convex Smooth Simple Bilevel Optimization},
author={Jincheng Cao and Ruichen Jiang and Erfan Yazdandoost Hamedani and Aryan Mokhtari},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=aFOdln7jBV}
}
In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\\})$ iterations to find a solution that is $\epsilon_f$-suboptimal and $\epsilon_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th Hölderian error bound,  we show that our method achieves an iteration complexity of $\mathcal{O}(\max\\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.
An Accelerated Gradient Method for Convex Smooth Simple Bilevel Optimization
Jincheng Cao Ruichen Jiang Erfan Yazdandoost Hamedani Aryan Mokhtari