Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints (2402.18012v3)

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.

• The paper introduces a novel method that leverages diffusion models as constrained samplers by integrating Boltzmann and data distribution densities.
• The approach uses a structured two-stage framework with guided diffusion warm-up followed by Langevin dynamics to hone in on feasible, optimized solutions.
• Experimental results demonstrate that the method outperforms existing baselines in multi-objective and real-world optimization tasks, including molecular design.

Diffusion Models as Constrained Samplers for Optimization with Unknown Constraints

Recent advancements in optimization, especially within complex real-world scenarios, often confront bounded constraints or scenarios where these constraints are unknown. Existing methods that handle unknown objective functions, categorized broadly under black-box optimization, have insufficiently addressed these challenges where feasibility constraints might not be articulated. When these constraints are disregarded, resulting solutions may lack viability when applied practically.

This paper proposes a novel approach leveraging diffusion models to address optimization problems on the data manifold, especially when faced with unknown constraints. By reframing optimization challenges as sampling problems from a product of probabilistic models, the authors have delineated a methodology that efficiently combines optimization objectives with the data space learned by a diffusion model.

Methodology

The approach is based on sampling from the product of two densities: the Boltzmann distribution, which is formed by the objective function, and the data distribution acquired via a diffusion model. Given that diffusion models have demonstrated a robust ability to learn intricate data distributions, utilizing them allows the practical constraints inherent in the data samples to be modeled effectively within the optimization process.

To enhance sampling efficiency, the authors proposed a structured two-stage framework. The initial stage involves a guided diffusion process that serves as a warm-up. This stage shifts the distribution focus towards feasible solutions, offering a preferable initial point for the subsequent step. Following this, Langevin dynamics refine solutions, pushing the exploration further and correcting initial samples. This two-stage approach ensures that the resultant samples respect learned constraints and minimize the objective within these bounds.

Experimental Results

Experiments validate the efficacy of this approach using synthetic data, six datasets from real-world black-box optimization scenarios, and a multi-objective optimization task within the field of molecular design. Notably, on tasks such as Superconductor, which involves optimizing material properties, the novel method surpassed existing baseline strategies in performance by significant margins.

The contribution also holds strong numerical results in multi-objective optimization tasks, where the model not only attains high scores in specific objectives but adeptly balances multiple objective constraints, showcasing higher validity rates and competitive optimization results compared to existing methods.

Theoretical Implications

From a theoretical perspective, the approach demonstrates that the guided diffusion stage can effectively limit the optimization to within feasible constraints, as diffusion models are calibrated to understand the underlying data manifolds—proven in image, video, and 3D space modeling. The proposed method further expands on diffusion models' capabilities, manifesting their potential beyond generative tasks and situating them as viable tools for complex constrained optimization problems.

Conclusions and Potential for Future Work

The research showcases a substantial leap in handling optimization under unknown constraints by adeptly employing diffusion models, providing a foundation for future developments in AI that explore real-world, conundrum-filled optimization landscapes. Practical applications range from drug design to material science, highlighting an attractive prospect for industries needing advanced, accurate optimization solutions under incomplete information about constraints.

Looking ahead, optimizing and enhancing manifold learning during the guided diffusion process could offer improvements, as would focusing on ensuring hard constraints directly within diffusion spaces. Additionally, extending this framework to cover derivative-free optimization represents a promising direction for further research.