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A Single-Loop Gradient Algorithm for Pessimistic Bilevel Optimization via Smooth Approximation
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Proposes SiPBA, a fully first-order single-loop gradient method for pessimistic bilevel optimization based on a penalized and regularized smooth approximation of the value function.
Problem
Pessimistic bilevel optimization (PBO) has a non-smooth value function with a min-max-min structure, and existing methods need second-order information or double loops, so fully first-order single-loop gradient methods are lacking.
Approach
Approximate the PBO value function by a penalized (rho) and regularized (sigma, with coupling term <y,z>) minimax problem whose gradient is smooth; then run one projected gradient ascent-descent step per iteration on (y,z) to get an inexact gradient for x, with rho_k growing and sigma_k decaying.
Key results
- On the synthetic PBO example with n=100, SiPBA reaches relative error tolerance 1e-4 in all 10/10 valid runs with average time 1.03 s, versus 87.44 s for AdaProx-PD and 4.07 s for AdaProx-SG.
- On spam classification, SiPBA-Hinge achieves the best average accuracy/F1 of 85.5/84.7 when trained on TREC06, and SiPBA-CE achieves 75.7/74.6 when trained on TREC07 (best overall in those settings).
- The paper reports convergence guarantees for SiPBA of the form min over k of ||x^{k+1}-x^k||^2/alpha_k^2 = O(1/K^{1-2s}) under parameter conditions.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| use SiPBA instead of AdaProx-PD | Synthetic PBO, n=100 · Synthetic PBO, n=100 | Ave Time (s): 87.44 -> 1.03 | Helped |
| use SiPBA instead of AdaProx-PD | Synthetic PBO, n=100 · Synthetic PBO, n=100 | Min. relative error: 1.79\times10^{-7} -> 1.22 \times 10^{-6} | Hurt |
| use SiPBA-Hinge instead of SQP-Hinge | Spam classification trained on TREC06 · TREC06 train average | Accuracy: 85.1 -> 85.5 | Helped |
| use SiPBA-CE instead of SQP-CE | Spam classification trained on TREC07 · TREC07 train average | Accuracy: 76.9 -> 75.7 | Hurt |
| use SiPBA-CE instead of SQP-CE | Spam classification trained on EnronSpam · EnronSpam train average | Accuracy: 76.8 -> 77.4 | Helped |
| use SiPBA-Hinge instead of SQP-Hinge | Spam classification trained on LingSpam · LingSpam train average | Accuracy: 64.6 -> 75.0 | Helped |
| set alpha0 to 1 rather than 0.1 (ablation) | SiPBA · Synthetic PBO, n=100 | Time (s): 1.0$\pm0.1$ -> 0.1$\pm0.0$ | Helped |
| set alpha0 to 0.01 rather than 0.1 (ablation) | SiPBA · Synthetic PBO, n=100 | Time (s): 1.0$\pm0.1$ -> 14.5$\pm1.5$ | Hurt |
| set beta0 to 0.01 rather than 0.001 (ablation) | SiPBA · Synthetic PBO, n=100 | Time (s): 1.0$\pm0.1$ -> 0.5$\pm0.1$ | Helped |
| set beta0 to 0.0001 rather than 0.001 (ablation) | SiPBA · Synthetic PBO, n=100 | Time (s): 1.0$\pm0.1$ -> 16.4$\pm3.1$ | Hurt |
| vary p, q, or s individually from their reference values (ablation) | SiPBA · Synthetic PBO, n=100 | Time (s): p=0.01: 1.4$\pm0.3$; p=0.0001: 1.0$\pm0.1$; q=0.01: 1.2$\pm0.1$; q=0.0001: 1.1$\pm0.2$; s=0.3: 5.0$\pm0.2$; s=0.016: 0.8 | Mixed |
| set p=q=0.01 and s=0.16 rather than p=q=0.001 and s=0.1 (ablation) | SiPBA · Synthetic PBO, n=100 | Time (s): 1.0$\pm0.1$ -> 1.9$\pm0.3$ | Hurt |
Surprising findings
- SiPBA on the synthetic example is far faster than AdaProx-PD, which needed 87.44 s on average, despite AdaProx-PD having a lower minimum relative error.
Limitations
- Analysis is restricted to deterministic PBO problems; stochastic extension is left to future work.
- Convergence guarantees require compact X and specific diminishing parameter schedules (s < 1/2, 8p+8q <= s, small beta0/sigma0).
- Hyperparameter sensitivity: ablation shows runtime varies strongly with alpha0, beta0 and s.
Experimental setup
Datasets: TREC2006 TREC2007 EnronSpam LingSpam MNIST FashionMNIST synthetic linear hyper-representation data synthetic PBO example
Benchmarks: synthetic PBO example TREC06 TREC07 EnronSpam LingSpam linear hyper-representation synthetic MNIST FashionMNIST
Metrics: relative error runtime (s) valid runs accuracy F1 score test loss test accuracy
Baselines: AdaProx-PD AdaProx-SG Scholtes-C Scholtes-D SQP AID-FP AID-CG PZOBO scikit-learn SVC and Logistic Regression
Builds on: AdaProx Scholtes relaxation AID-CG AID-FP PZOBO
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