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On PI Controllers for Updating Lagrange Multipliers in Constrained Optimization
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Proposes νPI, a PI controller with an EMA on the error signal for updating Lagrange multipliers, shown to generalize Polyak/Nesterov momentum and to damp GDA oscillations in constrained optimization.
Problem
Gradient descent-ascent on Lagrangian min-max formulations of constrained neural network training exhibits oscillations and overshoot in the Lagrange multiplier dynamics, and existing momentum/adaptive methods do not reliably fix this.
Approach
Update Lagrange multipliers with a PI-style controller whose proportional term acts on an exponential moving average of constraint violations (νPI); this generalizes UnifiedMomentum/Polyak/Nesterov and optimistic gradient, and provably damps multiplier oscillations.
Key results
- On CIFAR-10 layer-wise sparsity (30% density), νPI (κp=8.0) reaches 95.1% train / 86.2% test accuracy with violations in [-1.7, 0.1], versus GA at 92.0% / 84.1% with violations down to -27.8.
- In the fairness experiment, νPI with ν=0.99 reaches 92.4% training accuracy versus 89% for GA and reduces maximum violation faster.
- On hard-margin SVM, νPI is the only evaluated method that achieves zero constraint violation and converges to the optimal multiplier, while baselines oscillate.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use νPI with ν=0.99 instead of gradient ascent for multiplier updates | Adult statistical-parity classifier · Adult | training accuracy: 89% -> 92.4% | Helped |
| Use νPI with ν=0.99 instead of gradient ascent for multiplier updates | Adult statistical-parity classifier · Adult | maximum constraint violation | Helped |
| Use νPI instead of gradient ascent for multiplier updates | Hard-margin SVM · Iris binary classification | constraint violation | Helped |
| Increase κp in νPI (ablation) | νPI for sparsity-constrained ResNet-18 · CIFAR-10 global sparsity | overshoot | Mixed |
| Use νPI instead of gradient ascent and momentum updates | Sparsity-constrained ResNet-18 · CIFAR-10 layer-wise sparsity, 30% density target | train accuracy: 92.0 -> 95.1 | Helped |
| Use νPI instead of gradient ascent and momentum updates | Sparsity-constrained ResNet-18 · CIFAR-10 layer-wise sparsity, 30% density target | test accuracy: 84.1 -> 86.2 | Helped |
| Use νPI instead of gradient ascent and momentum updates | Sparsity-constrained ResNet-18 · CIFAR-10 layer-wise sparsity, 30% density target | violation range: 22.0 -> 1.8 | Helped |
| Replace ν=0 with ν=0.99 to filter noise in the constraint signal (ablation) | νPI for Adult fairness · Adult | multiplier stability | Helped |
| Use ν=0 without EMA filtering in νPI (ablation) | νPI for Adult fairness · Adult | multiplier stability | Hurt |
| Use positive Polyak momentum instead of νPI for multiplier updates | Sparsity-constrained ResNet-18 · CIFAR-10 sparsity | overshoot | Hurt |
| Increase κp in νPI (ablation) | νPI multiplier controller · Constrained optimization | damping of oscillations | Helped |
Surprising findings
- Positive Polyak momentum on Lagrange multipliers induces counter-intuitive updates that increase multipliers when feasibility improves, which may cause oscillations.
- Gradient ascent with dual restarts, a strong baseline, achieves comparable sparsity constraint satisfaction but with slightly infeasible solutions.
Limitations
- Theory and experiments restricted to linear dual players and primal updates treated as black-box; non-linear players left as future work.
- Fairness experiments show a final maximum violation around 1.7% that the authors attribute to stochastic primal updates.
- Tuning of kappa_p and kappa_i still requires search, though heuristics are suggested.
Experimental setup
Datasets: Iris Adult CIFAR-10
Benchmarks: Iris hard-margin SVM Adult statistical parity fairness CIFAR-10 sparsity
Metrics: training accuracy test accuracy maximum constraint violation multiplier convergence overshoot
Baselines: Gradient ascent Polyak momentum Nesterov momentum Adam GA with dual restarts
Builds on: PID controller for constrained RL (Stooke et al.) UnifiedMomentum dual restarts Cooper
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