Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints

2024-02-28

International Conference on Artificial Intelligence and Statistics

AI Summary

We address black-box optimization where both the objective function and feasibility constraints are unknown. To this end, we propose a diffusion-based implicit constraint modeling framework. Our core insight reformulates constrained optimization as a sampling problem over the joint support of the data manifold and a Boltzmann distribution. This work introduces the first general-purpose constraint sampler based on diffusion models, featuring a novel two-stage guided diffusion with Langevin correction—adaptable to both differentiable and non-differentiable objectives—and an iterative importance sampling paradigm. Evaluated on synthetic benchmarks, six real-world black-box tasks, and multi-objective molecular design, our method achieves significant improvements or matches state-of-the-art performance, while markedly enhancing solution feasibility and practical utility.

Technology Category

• Search and Optimization: Sampling/Simulation-based Search
• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Addressing real-world optimization problems becomes particularly challenging when analytic objective functions or constraints are unavailable. While numerous studies have addressed the issue of unknown objectives, limited research has focused on scenarios where feasibility constraints are not given explicitly. Overlooking these constraints can lead to spurious solutions that are unrealistic in practice. To deal with such unknown constraints, we propose to perform optimization within the data manifold using diffusion models. To constrain the optimization process to the data manifold, we reformulate the original optimization problem as a sampling problem from the product of the Boltzmann distribution defined by the objective function and the data distribution learned by the diffusion model. Depending on the differentiability of the objective function, we propose two different sampling methods. For differentiable objectives, we propose a two-stage framework that begins with a guided diffusion process for warm-up, followed by a Langevin dynamics stage for further correction. For non-differentiable objectives, we propose an iterative importance sampling strategy using the diffusion model as the proposal distribution. Comprehensive experiments on a synthetic dataset, six real-world black-box optimization datasets, and a multi-objective molecule optimization dataset show that our method achieves better or comparable performance with previous state-of-the-art baselines.

Problem

• Optimizing with unknown feasibility constraints using diffusion models
• Reformulating optimization as constrained sampling from data manifold
• Handling both differentiable and non-differentiable objective functions

Innovation

• Uses diffusion models as constrained samplers
• Reformulates optimization as Boltzmann-data product sampling
• Proposes differentiable and non-differentiable sampling methods