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Constrained Bi-Level Optimization: Proximal Lagrangian Value Function Approach and Hessian-free Algorithm
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Proposes LV-HBA, a single-loop, Hessian-free first-order algorithm for constrained bilevel optimization with lower-level constraints coupling upper and lower variables, based on a smooth proximal Lagrangian value function reformulation, with non-asymptotic convergence analysis for merely convex lower levels.
Problem
Constrained bilevel optimization where the lower-level problem has inequality constraints coupling upper- and lower-level variables; existing methods need Hessian computations and strong convexity of the lower-level objective.
Approach
Defines a smooth proximal Lagrangian value function (value of a strongly-convex-strongly-concave proximal min-max problem over the LL primal and multiplier), yielding a single-level reformulation with smooth constraints that can be solved by single-loop GDA-style updates using only first-order information.
Key results
- On SVM hyperparameter optimization with diabetes, LV-HBA reaches 75.07±1.6 accuracy in 6.65±1.3 s versus GAM's 74±1.4 in 33.06±2.2 s.
- On data hyper-cleaning with gisette, LV-HBA achieves 94.6±0.4 accuracy in 100 s versus GAM's 94.2±0.5 in 200 s.
- In federated loss tuning, LV-HBA reaches 90% test accuracy in 65 communication rounds versus 527 for E-AiPOD and 625 for FedNest.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use LV-HBA instead of GAM | Linear SVM hyperparameter optimization · diabetes | accuracy: 74 ± 1.4 -> 75.07 ± 1.6 | Helped |
| Use LV-HBA instead of GAM | Linear SVM hyperparameter optimization · fourclass | accuracy: 75.2 ± 1.6 -> 75.4 ± 1.2 | Helped |
| Use LV-HBA instead of GAM | Data hyper-cleaning · gisette | accuracy: 94.2 ± 0.5 -> 94.6 ± 0.4 | Helped |
| Use LV-HBA instead of GAM | Linear SVM hyperparameter optimization · diabetes | time (s): 33.06± 2.2 -> 6.65± 1.3 | Helped |
| Use LV-HBA instead of GAM | Linear SVM hyperparameter optimization · fourclass | time (s): 30± 2.2 -> 6± 1.8 | Helped |
| Use LV-HBA instead of GAM | Data hyper-cleaning · gisette | time (s): 200±11.5 -> 100±11.5 | Helped |
| Use LV-HBA instead of E-AiPOD or FedNest | Federated loss function tuning · Test accuracy 90 | communication rounds: 527 -> 65; 625 -> 65 | Helped |
| Use LV-HBA instead of E-AiPOD or FedNest | Federated loss function tuning · Test accuracy 92 | communication rounds: 1080 -> 149; 1575 -> 149 | Helped |
| Use LV-HBA instead of E-AiPOD or FedNest | Federated loss function tuning · Test accuracy 94 | communication rounds: 2256 -> 588; 3010 -> 588 | Helped |
Surprising findings
- AiPOD and E-AiPOD performed poorly on the merely convex synthetic problem, which the authors attribute to their need for strong convexity of the lower level.
Limitations
- Analysis is deterministic; stochastic variants are left to future work.
- Convergence requires step-size and proximal parameter conditions (e.g., gamma_1 < 1/rho_f) and a boundedness assumption on the penalized objective for the function-value bound.
Experimental setup
Datasets: diabetes fourclass gisette
Benchmarks: synthetic constrained BLO problems SVM hyperparameter optimization (diabetes, fourclass) data hyper-cleaning (gisette) federated loss function tuning
Metrics: accuracy running time communication rounds distance to optimal solution hyper-objective value
Baselines: AiPOD E-AiPOD GAM FedNest
Builds on: Moreau envelope value function AiPOD GAM
Closest papers by meaning
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