Generalized Schrödinger Bridge Matching (2310.02233v2)

Abstract: Modern distribution matching algorithms for training diffusion or flow models directly prescribe the time evolution of the marginal distributions between two boundary distributions. In this work, we consider a generalized distribution matching setup, where these marginals are only implicitly described as a solution to some task-specific objective function. The problem setup, known as the Generalized Schr\"odinger Bridge (GSB), appears prevalently in many scientific areas both within and without machine learning. We propose Generalized Schr\"odinger Bridge Matching (GSBM), a new matching algorithm inspired by recent advances, generalizing them beyond kinetic energy minimization and to account for task-specific state costs. We show that such a generalization can be cast as solving conditional stochastic optimal control, for which efficient variational approximations can be used, and further debiased with the aid of path integral theory. Compared to prior methods for solving GSB problems, our GSBM algorithm better preserves a feasible transport map between the boundary distributions throughout training, thereby enabling stable convergence and significantly improved scalability. We empirically validate our claims on an extensive suite of experimental setups, including crowd navigation, opinion depolarization, LiDAR manifolds, and image domain transfer. Our work brings new algorithmic opportunities for training diffusion models enhanced with task-specific optimality structures. Code available at https://github.com/facebookresearch/generalized-schrodinger-bridge-matching

• The paper introduces the GSBM algorithm, recasting the bridge problem as conditional stochastic optimal control to integrate task-specific state costs.
• It employs a dual-stage optimization approach that alternates between refining drift functions and conditional distributions for improved stability.
• Empirical results in high-dimensional tasks like crowd navigation and image domain translation validate GSBM’s scalability and robust performance.

Generalized Schrödinger Bridge Matching: Algorithmic Innovations and Experimental Validation

The paper presents a significant advancement in the development of distribution matching algorithms, specifically within the paradigm of the Generalized Schrödinger Bridge (GSB) problem. By introducing the Generalized Schrödinger Bridge Matching (GSBM) algorithm, the research extends current methodologies beyond standard kinetic energy minimization to incorporate task-specific state costs. The fundamental contribution of this work is in recasting the GSB problem as a conditional stochastic optimal control (CondSOC) problem, thereby allowing for efficient variational approximations using contemporary stochastic control techniques.

Core Contributions and Methodological Innovation

1. GSB Problem Revisited: The GSB problem is a generalization of distribution matching, wherein the marginal distributions between two boundaries are not prescribed explicitly but are instead derived as solutions to specific task-based objectives. The authors propose addressing this through a novel matching algorithm that maintains fidelity to the boundary distributions throughout training, assuring stable convergence and scalability improvements over existing methods.
2. The GSBM Algorithm: This new algorithm fundamentally processes the GSB problem by alternating optimizations—one targeting the drift $u_t^\theta$ given the fixed distribution paths, and the other optimizing the conditional distributions $p_{t|0,1}$ given the coupling $p^\theta_{0,1}$. This dual-stage optimization ensures both feasibility and optimality, with advancements in accuracy and convergence stability compared to previous approaches like DeepGSB.
3. Variational Approach and Path Integral Theory: For cases where direct solutions are impractical, especially when dealing with nontrivial $V_t$, the authors employ variational approximations. This approach utilizes Gaussian path approximations, bolstered by path integral theory, to achieve more faithful adherence to the desired transport map characteristics and results in more stable convergence.

Empirical Validation and Scalability

GSBM's efficacy is validated through a series of experiments spanning synthetic tasks to more complex, high-dimensional scenarios such as crowd navigation over LiDAR manifolds and unpaired image domain translation. The empirical evaluations clearly demonstrate GSBM's superior scalability and feasibility in maintaining the boundary conditions while optimizing for the predefined task-specific costs. Specifically:

• In crowd navigation tasks, GSBM successfully manages distribution paths that account for complex geometries and population dynamics with mean-field interactions. This is crucial in applications where agents interact with each other and the environment in intricate and non-linear ways.
• The image domain transfer experiments highlight GSBM's capability to handle high-dimensional, real-world data, showcasing its potential applicability in fields such as computer vision and robotic path planning where domain adaptation and style transfer are vital.

Implications and Future Directions

The implications of this work are profound for theoretical and practical applications in AI and machine learning. The ability to effectively generalize stochastic bridge problems to account for state costs opens new pathways for developing algorithms that interface more naturally with real-world dynamics under uncertainty. Further, the integration of variational methods and path integral theory underscores a critical advance in how noise and uncertainty are managed in optimizing dynamic systems, offering a more robust framework for future research in stochastic processes and generative modeling.

Conclusion

The introduction of the Generalized Schrödinger Bridge Matching algorithm signifies a substantial forward leap in distribution matching methodologies. Its robust performance across a wide range of tasks, alongside its efficient handling of high-dimensional spaces and nontrivial state costs, marks an essential contribution to the domain. This research lays a strong foundation for future investigations into more adaptive and intelligent control mechanisms in uncertain and dynamic environments.