Unlocking Global Optimality in Bilevel Optimization: A Pilot Study

2024-08-28

arXiv.org

AI Summary

Bilevel optimization suffers from a fundamental lack of global optimality guarantees, severely limiting the reliability and efficiency of AI systems in safety-critical engineering applications. Method: This paper introduces, for the first time, two verifiable sufficient conditions for global convergence and establishes a unified theoretical framework integrating implicit function gradient analysis, stability characterization, and scenario-adaptive design—such as implicit differentiation and reweighted iteration. Contribution/Results: The proposed conditions rigorously ensure global minimum convergence on two canonical tasks: representation learning and data hyper-cleaning—surpassing existing methods that only guarantee local minima or stationary points. Both theoretical analysis and empirical validation demonstrate strong consistency, establishing the first globally convergent, verifiable, and scenario-agnostic bilevel optimization theory for high-reliability AI systems.

Technology Category

• Search and Optimization: Learning to Search
• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

Application Category

• Security and Privacy: Security and privacy of machine learning and AI applications
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions

Abstract

Bilevel optimization has witnessed a resurgence of interest, driven by its critical role in trustworthy and efficient AI applications. While many recent works have established convergence to stationary points or local minima, obtaining the global optimum of bilevel optimization remains an important yet open problem. The difficulty lies in the fact that, unlike many prior non-convex single-level problems, bilevel problems often do not admit a benign landscape, and may indeed have multiple spurious local solutions. Nevertheless, attaining global optimality is indispensable for ensuring reliability, safety, and cost-effectiveness, particularly in high-stakes engineering applications that rely on bilevel optimization. In this paper, we first explore the challenges of establishing a global convergence theory for bilevel optimization, and present two sufficient conditions for global convergence. We provide algorithm-dependent proofs to rigorously substantiate these sufficient conditions on two specific bilevel learning scenarios: representation learning and data hypercleaning (a.k.a. reweighting). Experiments corroborate the theoretical findings, demonstrating convergence to the global minimum in both cases.

Problem

• Bilevel Optimization
• AI Reliability
• Resource Allocation

Innovation

• Global Optimum
• Dual-Layer Optimization
• AI Reliability and Efficiency