Diff2SP: Diffusion Models for Correlated Scenario Generation in Stochastic Programming

Technical Summary of Diff2SP: Diffusion Models for Correlated Scenario Generation in Stochastic Programming

Problem Statement
Scenario generation is a foundational component of stochastic programming (SP), where a finite set of representative scenarios approximates underlying uncertainty to inform decision-making. Existing methodologies face significant trade-offs between fidelity, diversity, and computational cost. Classical sampling-based techniques (e.g., Monte Carlo) often require vast numbers of scenarios to capture rare, high-cost tail events and complex dependencies, rendering them computationally expensive. Conversely, supervised learning approaches require fixed input-output pairs and fail to generate diverse, realistic scenarios beyond predefined patterns. While unsupervised generative models like Generative Adversarial Networks (GANs) have been applied to scenario generation, they suffer from training instability, mode collapse, and a lack of explicit likelihood estimates, making it difficult to assess the representation of tail events or cross-variable dependencies. Crucially, most existing methods treat scenario generation and downstream decision-making as separate steps, generating scenarios to match historical data distributions rather than to optimize decision quality.

Methodology: The Diff2SP Framework
The authors propose Diff2SP, a diffusion-based generative framework that integrates downstream stochastic optimization objectives directly into the scenario generation process. The framework consists of three primary components:

1. Diffusion-Based Architecture with Transformers:

   • The core utilizes a Denoising Diffusion Probabilistic Model (DDPM). The forward process progressively corrupts data samples (e.g., multivariate time series of wind, solar, and load) with Gaussian noise over $K$ steps.
   • The reverse process learns to denoise, synthesizing new samples by iteratively removing noise.
   • Unlike standard DDPMs that often use U-Net (CNN-based) architectures, Diff2SP employs a Transformer-based architecture for the denoising network. This allows the model to capture long-range temporal dependencies and complex cross-variable correlations via self-attention mechanisms, which are critical for high-dimensional multivariate time series.
   • The model supports conditional generation, allowing it to adapt to exogenous contexts (e.g., time of day, weather variables) via an optional conditioning vector $c$.

2. Optimization-Guided Training:

   • Diff2SP introduces a unified training objective that combines three loss components:
     • Noise Prediction Loss: Standard DDPM training to estimate the added noise.
     • Reconstruction Loss: Encourages the recovery of temporally consistent samples.
     • Optimization Loss: A novel term that penalizes discrepancies in the optimal objective value of the downstream SP problem when using generated scenarios versus historical data.
   • To enable gradient-based training through the optimization loss, the SP problem is embedded as a differentiable optimization layer within the neural network. Gradients are propagated through the optimal solution using implicit differentiation of the optimality conditions (based on Fiacco and Ishizuka, 1990).
   • The total loss is a weighted sum: $L_k(\theta) = \lambda_{noise}L_{noise} + \lambda_{recon}L_{recon} + \lambda_{opt}L_{opt}$.

3. Theoretical Justification:

   • The paper establishes regret bounds linking distributional accuracy (measured by the 1-Wasserstein distance) to decision quality.
   • It derives sample complexity guarantees, showing that by directly minimizing an optimization-aware loss, Diff2SP achieves a decision regret rate of $O(N^{-1/2})$. In contrast, GANs, which minimize statistical divergence (e.g., Jensen-Shannon) indirectly, achieve a slower rate of $O(N^{-1/4})$.

Key Contributions

1. Framework Development: The authors develop a diffusion-based framework that embeds a stochastic optimization problem as a differentiable layer, coupling uncertainty modeling with downstream decision-making.
2. Theoretical Regret Guarantees: They establish regret bounds quantifying how errors in the learned uncertainty distribution translate into losses in decision quality, demonstrating that decision-aware training yields stronger theoretical guarantees than distribution-matching alone.
3. Sample Complexity Analysis: The paper derives sample complexity bounds for optimization-aware scenario generation, comparing them against GAN-based learning to show that directly incorporating decision objectives leads to faster convergence and better robustness.

Empirical Results
The framework was evaluated on both synthetic and real-world datasets:

• Synthetic Experiments: Using a synthetic multivariate Gaussian distribution, Diff2SP demonstrated superior sample efficiency in recovering correlation structures compared to GANs, particularly in low-sample regimes. In downstream quadratic programming tasks, Diff2SP consistently achieved lower optimization errors across varying numbers of generated scenarios.
• Real-World Case Study: Applied to a power system stochastic programming task involving wind, solar, and load time series across six regions in the MISO footprint. The model generated scenarios used to solve a DC Optimal Power Flow (OPF) problem.
  • Statistical Fidelity: Generated scenarios captured major peaks and valleys, though minor overestimations of extreme peaks were noted.
  • Decision Quality: An ablation study compared the full Diff2SP model against a vanilla diffusion model and a variant without the optimization loss. The full model, incorporating the optimization loss, achieved the lowest composite error across all evaluation metrics (objective deviation, mean/covariance deviation, and line-limit violations). This confirmed that purely statistical objectives fail to preserve decision-critical structures, whereas optimization-guided training ensures alignment with downstream constraints and costs.

Significance and Claims
The paper claims that Diff2SP addresses the persistent trade-off between statistical fidelity and decision relevance in scenario generation. By embedding the SP problem directly into the training loop, the framework generates scenarios that are not only statistically coherent but also "decision-aware." The authors emphasize that while diffusion is not strictly necessary for the theoretical regret bounds (any sufficiently regularized generator could theoretically achieve them), its stable, differentiable training objective makes it practically superior to adversarial models like GANs for incorporating high-variance, nonconvex decision-dependent losses. The work demonstrates that integrating optimization objectives into generative modeling leads to improved robustness and decision quality in complex, correlated uncertainty environments, specifically highlighting applications in modern energy systems with high renewable penetration.