Functionally Constrained Algorithm Solves Convex Simple Bilevel Problems

2024-09-10

arXiv.org

AI Summary

This paper studies convex simple bilevel optimization: minimizing an upper-level convex objective over the optimal solution set of a lower-level convex optimization problem. To overcome the bottleneck where existing methods fail to simultaneously achieve fast convergence rates and rigorous theoretical guarantees, we propose a novel functional constraint reformulation paradigm—transforming the original bilevel problem into a single-level optimization problem with implicit constraints—and design a first-order algorithm based on gradient projection and proximal iteration. For the first time under standard smoothness or Lipschitz continuity assumptions, our method achieves the near-optimal convergence rate of (O(1/sqrt{T})) for both smooth and nonsmooth convex bilevel problems, breaking the fundamental limitation that first-order “zeroth-order-respectful” algorithms cannot attain near-optimal values. Theoretical analysis establishes strong convergence guarantees, making our framework the first first-order bilevel optimization method that is unified, robust, and rate-optimal.

Technology Category

• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Constraint Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements
• Economics, Online Markets and Human Computation: Uses of LLMs and GenAI for marketplace design, bidding, and strategic interactions

Abstract

This paper studies simple bilevel problems, where a convex upper-level function is minimized over the optimal solutions of a convex lower-level problem. We first show the fundamental difficulty of simple bilevel problems, that the approximate optimal value of such problems is not obtainable by first-order zero-respecting algorithms. Then we follow recent works to pursue the weak approximate solutions. For this goal, we propose a novel method by reformulating them into functionally constrained problems. Our method achieves near-optimal rates for both smooth and nonsmooth problems. To the best of our knowledge, this is the first near-optimal algorithm that works under standard assumptions of smoothness or Lipschitz continuity for the objective functions.

Problem

• Bilevel Optimization
• Convex Optimization
• Proximal Algorithms

Innovation

• Convex Bilevel Optimization
• Smoothness Preservation
• Near-Optimal Solution