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A Single-Loop First-Order Algorithm for Linearly Constrained Bilevel Optimization
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Proposes SFLCB, a single-loop, Hessian-free first-order GDA algorithm for bilevel optimization with strongly convex lower level and linear coupled constraints, using penalty and augmented Lagrangian reformulation, achieving O(ε^-3) complexity.
Problem
Bilevel optimization where the lower-level problem is strongly convex with coupled linear constraints Bx+Ay-b<=0; the hyper-objective may be non-smooth and Hessian computations are costly.
Approach
Reformulates the problem via a penalty function Phi_delta with slack variables and an augmented Lagrangian, proves closeness of values and derivatives to the hyper-objective, and runs single-loop GDA on the resulting minimax problem, avoiding Hessian computations and projections onto coupled constraint sets.
Key results
- Improves iteration complexity for linearly constrained BLO from O(ε^-3 log(1/ε)) (BLOCC) to O(ε^-3) with a single-loop algorithm.
- SFLCB converges significantly faster than GAM, LV-HBA, BLOCC and BiC-GAFFA on SVM hyperparameter optimization on the diabetes dataset (as claimed by the authors).
- On transportation network design, SFLCB significantly outperforms BLOCC on 3-node and 9-node networks.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use SFLCB instead of the prior double-loop method | Bilevel optimization with coupled linear constraints and full-row-rank A · Coupled linear constraints, A full row rank | iteration complexity: O(ε^{-3} log(ε^{-1})) -> O(ε^{-3}) | Helped |
| Use SFLCB instead of GAM, LV-HBA, BLOCC, and BiC-GAFFA | SVM hyperparameter optimization · Diabetes dataset | convergence speed | Helped |
| Use SFLCB instead of BLOCC | Transportation network design · Transportation network design, 3-node and 9-node networks | UL utility | Helped |
| Vary δ from 0.01 to 0.05, 0.1, or 0.5 (ablation) | SFLCB · 3-node transportation network | final performance: 0.01, 0.05, 0.1, 0.5 | Mixed |
| Increase δ to 1 (ablation) | SFLCB · 3-node transportation network | final performance: 1 | Hurt |
| Decrease δ to 0.01 (ablation) | SFLCB · 3-node transportation network | convergence speed: 0.01 | Hurt |
| Increase δ (ablation) | SFLCB · 3-node transportation network | initial convergence speed | Helped |
Limitations
- Analysis restricted to deterministic settings
- Linear constraint structure only; general coupled constraints not covered by the algorithm
- Full row rank of A needed for the O(ε^-3) rate in the coupled case; without it the rate is O(ε^-4)
Experimental setup
Datasets: diabetes dataset
Benchmarks: toy bilevel problem SVM hyperparameter optimization on diabetes transportation network design (3-node and 9-node)
Metrics: upper-level objective value UL utility
Baselines: GAM LV-HBA BLOCC BiC-GAFFA
Builds on: penalty-based bilevel reformulation BLOCC augmented Lagrangian method
Closest papers by meaning
- An Accelerated Gradient Method for Convex Smooth Simple Bilevel Optimization (NeurIPS 2024 · similarity 0.92)
- On Penalty Methods for Nonconvex Bilevel Optimization and First-Order Stochastic Approximation (ICLR 2024 · similarity 0.92)
- Double Momentum Method for Lower-Level Constrained Bilevel Optimization (ICML 2024 · similarity 0.92)
- Penalty-Based First-Order Methods for Bilevel Optimization with Minimax and Constrained Lower-Level Problems (NeurIPS 2026 · similarity 0.91)
- Penalty-based Methods for Simple Bilevel Optimization under Hölderian Error Bounds (NeurIPS 2024 · similarity 0.91)
- Functionally Constrained Algorithm Solves Convex Simple Bilevel Problem (NeurIPS 2024 · similarity 0.90)
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