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A Gradient Guided Diffusion Framework for Chance Constrained Programming
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Proposes GGDOpt, a gradient-guided diffusion framework that solves nonconvex chance-constrained programs from samples alone by reformulating CCP as sampling from a product of a data distribution and a Boltzmann distribution.
Problem
Chance constrained programming (CCP) with nonconvex feasible sets and unknown uncertainty distributions, where only samples are available, and existing convex approximation or SAA methods are restrictive or costly.
Approach
Treat CCP as sampling from the product of an unknown feasible-data distribution and a Boltzmann distribution of the objective; train a classifier-free-guided conditional diffusion model on feasible points generated by solving deterministic restricted problems, then run reverse SDE sampling with closed-form first- and second-order gradient guidance that needs no backpropagation through the network.
Key results
- On the linear CCP with n=8, GGDOpt second-order guidance reaches mean objective -0.6491 with std 0.0056 versus SOC_CVX -0.6586 and best SAA baseline SAA_PDCA -0.6389 (std 0.0314).
- GGDOpt runtime is about 0.05 s versus 0.63 s for SAA_PDCA and 0.32 s for SOC_CVX on the n=8 linear problem, roughly 80% overhead reduction.
- On robust waveform design (Nt=16, K=3), second-order GGDOpt achieves function value 0.1237-0.1260 with runtime 0.068-0.079 s, versus 1.29-1.47 s for convex approximation baselines.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| add first-order gradient guidance (ablation) | GGDOpt · Linear chance constrained problem (n=8) | FvalMean: 0.3481 -> -0.6483 | Helped |
| add first-order gradient guidance (ablation) | GGDOpt · Linear chance constrained problem (n=8) | FvalStd: 0.5486 -> 0.0051 | Helped |
| replace first-order with second-order gradient guidance (ablation) | GGDOpt · Linear chance constrained problem (n=8) | FvalMean: -0.6483 -> -0.6491 | Helped |
| replace first-order with second-order gradient guidance (ablation) | GGDOpt · Linear chance constrained problem (n=8) | FvalStd: 0.0051 -> 0.0056 | Mixed |
| increase inverse temperature beta from 0.1 to 10 (ablation) | GGDOpt Second-order · Linear chance constrained problem (n=8) | FvalMean: -0.6341 -> -0.6585 | Helped |
| increase inverse temperature beta from 0.1 to 10 (ablation) | GGDOpt Second-order · Linear chance constrained problem (n=8) | FvalStd: 5.6726e-3 -> 2.2329e-08 | Helped |
| use fixed variance sigma=0.1 instead of Tweedie's covariance (ablation) | GGDOpt Second-order · Linear chance constrained problem (n=8, rho=0.1) | time: 1.0984 -> 0.0493 | Helped |
| use fixed variance sigma=0.1 instead of Tweedie's covariance (ablation) | GGDOpt Second-order · Linear chance constrained problem (n=8, rho=0.1) | fval: -0.6571 -> -0.6545 | Hurt |
| use second-order guidance instead of Look-Ahead Guidance | GGDOpt · Linear chance constrained problem (rho=0.1) | fval: -0.5783 -> -0.6491 | Helped |
Surprising findings
- Without guidance, GGDOpt sampling performs very poorly (mean -objective 0.3481, std 0.5486), showing guidance is essential.
- On the waveform design task, GGDOpt's feasibility probabilities sometimes fall below the target (e.g., 0.81-0.86 vs 0.95 required), so constraint satisfaction is not always guaranteed.
Limitations
- Theoretical acceleration from second-order guidance is not established; second-order guidance requires Hessian inversion and fixed variance is a heuristic.
- Experiments cover only two problem types (synthetic linear CCP and waveform design).
- Reported chance-constraint violations in waveform design can exceed the target probability, and guarantees rely on mild assumptions and score-error bound delta.
Experimental setup
Datasets: synthetic Gaussian chance-constrained samples CVX-generated restricted-problem solutions for waveform design
Benchmarks: linear chance constrained problem (synthetic) robust waveform design (wireless)
Metrics: objective function value mean/std/median runtime violation probability function value
Baselines: SOC_CVX SAA_CVaR SAA_MIP SAA_PDCA SAA_SNSCO Sphere Bounding Bernstein-type Inequality Look-Ahead Guidance
Builds on: classifier-free guidance score-based SDE diffusion Tweedie's formula DDIM look-ahead guidance
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