Near-Optimal Convex Simple Bilevel Optimization with a Bisection Method
Jiulin Wang, Xu Shi, Rujun Jiang
Abstract
This paper studies a class of simple bilevel optimization problems where we minimize a composite convex function at the upper-level subject to a composite convex lower-level problem. Existing methods either provide asymptotic guarantees for the upper-level objective or attain slow sublinear convergence rates. We propose a bisection algorithm to find a solution that is $\epsilon_f$-optimal for the upper-level objective and $\epsilon_g$-optimal for the lower-level objective. In each iteration, the binary search narrows the interval by assessing inequality system feasibility. Under mild conditions, the total operation complexity of our method is ${{\mathcal{O}}}\left(\max\{\sqrt{L_{f_1}/\epsilon_f},\sqrt{L_{g_1}/\epsilon_g}\} \right)$. Here, a unit operation can be a function evaluation, gradient evaluation, or the invocation of the proximal mapping, $L_{f_1}$ and $L_{g_1}$ are the Lipschitz constants of the upper- and lower-level objectives’ smooth components, and ${\mathcal{O}}$ hides logarithmic terms. Our approach achieves a near-optimal rate in unconstrained smooth or composite convex optimization when disregarding logarithmic terms. Numerical experiments demonstrate the effectiveness of our method.
BibTeX
@InProceedings{pmlr-v238-wang24d,
title = {Near-Optimal Convex Simple Bilevel Optimization with a Bisection Method},
author = {Wang, Jiulin and Shi, Xu and Jiang, Rujun},
booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics},
pages = {2008--2016},
year = {2024},
editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen},
volume = {238},
series = {Proceedings of Machine Learning Research},
month = {02--04 May},
publisher = {PMLR},
pdf = {https://proceedings.mlr.press/v238/wang24d/wang24d.pdf},
url = {https://proceedings.mlr.press/v238/wang24d.html},
abstract = {This paper studies a class of simple bilevel optimization problems where we minimize a composite convex function at the upper-level subject to a composite convex lower-level problem. Existing methods either provide asymptotic guarantees for the upper-level objective or attain slow sublinear convergence rates. We propose a bisection algorithm to find a solution that is $\epsilon_f$-optimal for the upper-level objective and $\epsilon_g$-optimal for the lower-level objective. In each iteration, the binary search narrows the interval by assessing inequality system feasibility. Under mild conditions, the total operation complexity of our method is ${{\mathcal{O}}}\left(\max\{\sqrt{L_{f_1}/\epsilon_f},\sqrt{L_{g_1}/\epsilon_g}\} \right)$. Here, a unit operation can be a function evaluation, gradient evaluation, or the invocation of the proximal mapping, $L_{f_1}$ and $L_{g_1}$ are the Lipschitz constants of the upper- and lower-level objectives’ smooth components, and ${\mathcal{O}}$ hides logarithmic terms. Our approach achieves a near-optimal rate in unconstrained smooth or composite convex optimization when disregarding logarithmic terms. Numerical experiments demonstrate the effectiveness of our method.}
}