Mastering Blue Prince: Embrace Clever Tactics

Diffusion processes have become a valuable tool for sampling from complex distributions, but they encounter difficulties with multimodal targets. Traditional methods relying on overdamped Langevin dynamics often struggle with slow convergence when moving between different distribution modes. Although underdamped Langevin dynamics introduce an additional momentum variable that offers empirical improvements, they still face fundamental challenges. The indirect coupling of Brownian motion to the space variable in these models results in smoother paths but complicates theoretical analysis.

Methods like Annealed Importance Sampling (AIS) connect prior and target distributions using transition kernels, while Unadjusted Langevin Annealing (ULA) applies uncorrected overdamped Langevin dynamics in this context. Monte Carlo Diffusion (MCD) focuses on minimizing marginal likelihood variance, whereas Controlled Monte Carlo Diffusion (CMCD) and Sequential Controlled Langevin Diffusion (SCLD) emphasize kernel optimization with resampling strategies. Other techniques prescribe backward transition kernels, including the Path Integral Sampler (PIS), the Time-Reversed Diffusion Sampler (DIS), and the Denoising Diffusion Sampler (DDS). Some approaches, like the Diffusion Bridge Sampler (DBS), independently learn both forward and backward kernels.

A collaborative effort by researchers from the Karlsruhe Institute of Technology, NVIDIA, Zuse Institute Berlin, dida Datenschmiede GmbH, and FZI Research Center for Information Technology has introduced a generalized framework for learning diffusion bridges that transport prior distributions to target distributions. This framework incorporates both existing diffusion models and underdamped versions with degenerate diffusion matrices, where noise affects only specific dimensions. It lays a solid theoretical foundation by demonstrating that score-matching in underdamped cases corresponds to maximizing a likelihood lower bound, thereby addressing the challenge of sampling from unnormalized densities when direct samples are not available.

This framework facilitates a comparative analysis of five major diffusion-based sampling methods: ULA, MCD, CMCD, DIS, and DBS. The underdamped variants of DIS and DBS provide novel contributions. The evaluation methodology involves a diverse suite of tests, including seven real-world benchmarks for Bayesian inference tasks (Credit, Cancer, Ionosphere, Sonar), parameter inference problems (Seeds, Brownian), and high-dimensional sampling using the Log Gaussian Cox process (LGCP) with 1600 dimensions. Additionally, synthetic benchmarks include the challenging Funnel distribution, which presents regions of varying concentration levels, offering a rigorous test for sampling methods across diverse dimensionality and complexity profiles.

The findings reveal that underdamped Langevin dynamics consistently outperform overdamped methods across both real-world and synthetic benchmarks. The underdamped DBS excels over competing methods even with as few as 8 discretization steps, leading to significant computational savings while maintaining high sampling quality. For numerical integration schemes, specialized integrators exhibit significant improvements over traditional Euler methods for underdamped dynamics. The OBAB and BAOAB schemes enhance performance without additional computational costs, and the OBABO scheme achieves the best results overall, despite requiring double evaluations of control parameters per discretization step.

In summary, this work establishes a comprehensive framework for diffusion bridges involving degenerate stochastic processes. The underdamped diffusion bridge sampler achieves state-of-the-art results across various sampling tasks with minimal hyperparameter tuning and limited discretization steps. Detailed ablation studies confirm that performance gains result from the combination of underdamped dynamics,