An optimal control perspective on diffusion-based generative modeling (2211.01364v3)

Abstract: We establish a connection between stochastic optimal control and generative models based on stochastic differential equations (SDEs), such as recently developed diffusion probabilistic models. In particular, we derive a Hamilton-Jacobi-BeLLMan equation that governs the evolution of the log-densities of the underlying SDE marginals. This perspective allows to transfer methods from optimal control theory to generative modeling. First, we show that the evidence lower bound is a direct consequence of the well-known verification theorem from control theory. Further, we can formulate diffusion-based generative modeling as a minimization of the Kullback-Leibler divergence between suitable measures in path space. Finally, we develop a novel diffusion-based method for sampling from unnormalized densities -- a problem frequently occurring in statistics and computational sciences. We demonstrate that our time-reversed diffusion sampler (DIS) can outperform other diffusion-based sampling approaches on multiple numerical examples.

• The paper establishes a new link between diffusion models and optimal control by deriving an HJB equation for the evolution of SDE log-densities.
• It reframes the ELBO and introduces the Time-Reversed Diffusion Sampler, which efficiently samples from unnormalized densities.
• Numerical experiments demonstrate the method’s robustness on high-dimensional, multimodal distributions, paving the way for future research.

An Optimal Control Perspective on Diffusion-Based Generative Modeling

The paper "An Optimal Control Perspective on Diffusion-Based Generative Modeling" by Julius Berner, Lorenz Richter, and Karen Ullrich establishes a novel connection between stochastic optimal control and diffusion-based generative models. This work critically examines generative models based on stochastic differential equations (SDEs) and formulates them through an optimal control lens.

Key Contributions and Findings

The authors present several significant contributions:

1. PDE and Optimal Control Perspectives: The paper identifies a Hamilton-Jacobi-BeLLMan (HJB) equation governing the evolution of the log-densities of SDE marginals in diffusion models. This connection not only enhances theoretical understanding but also offers practical algorithms for numerical approximation, such as neural PDE solvers.
2. Evidence Lower Bound (ELBO): The work connects the ELBO commonly used in diffusion-based generative models to standard principles in control theory. This derivation underscores the ELBO as a practical consequence of control objectives.
3. Path Space Interpretation: By considering measures on path space, the authors provide a new perspective on diffusion models. This view facilitates understanding through Kullback-Leibler divergences between path measures, leading to improved loss functions and algorithms.
4. Novel Sampling Method: A significant contribution of the paper is a novel diffusion-based method for sampling from unnormalized densities. This is specifically beneficial for applications in Bayesian statistics, computational sciences, and related fields, where the target distribution is known only up to a normalization constant.

Methodological Innovations

The authors propose using time-reversed diffusion processes and stochastic control techniques to bridge the gap between generative models and optimal control. By leveraging the reversibility of diffusion processes, they define a control problem aimed at matching distributions across forward and reverse processes, simplifying the problem to one of divergence minimization. The introduction of a new method, the Time-Reversed Diffusion Sampler (DIS), demonstrates practical superiority over existing diffusion-based sampling approaches through numerous numerical experiments.

Numerical Results and Impact

The paper presents compelling numerical results showing that the DIS outperforms previous state-of-the-art diffusion samplers. The experiments span high-dimensional, multimodal distributions, highlighting the algorithm’s robustness and effectiveness. The work proposes a transformative view that not only applies to generative tasks but also opens doors for future work in sampling and density estimation under the optimal control framework.

Implications and Future Directions

The theoretical implications of this paper suggest promising future research directions, especially in the intersection of control theory, statistics, and machine learning. Potential advancements include using alternative divergences on path space, improved numerical schemes from control, and extensions to Schrödinger bridges for more generalized settings. Notably, the control perspective could inform better computational methodologies for both training and deploying generative models in practice.

By reframing generative modeling as an optimal control problem, this paper sets the stage for cross-pollination of ideas between distinct but related fields, potentially leading to more refined models and efficient inference methods. This work not only enriches the theory but also enhances practical algorithms for real-world applications where sampling and generative tasks play a crucial role.