Reproduce “Targeted Separation and Convergence with Kernel Discrepancies” on OpenResearch

About this paper

Alessandro Barp, Carl-Johann Simon-Gabriel, Mark Girolami, Lester Mackey

Researchers from University College London, Mirelo AI, University of Cambridge, and Microsoft Research significantly expanded the theoretical guarantees for Maximum Mean Discrepancies (MMDs) and Kernel Stein Discrepancies (KSDs), relaxing prior restrictive assumptions on separation and convergence. The work introduces methods for unbounded kernels and identifies the first bounded Stein kernels that metrize weak convergence, extending their applicability to a broader range of distributions and scenarios.

Key insights

• KSDs with standard translation-invariant kernels achieve general P-separation and control tight P-convergence under significantly relaxed conditions on the target score function, such as at most root-exponential growth.
• The first KSDs were constructed that exactly metrize weak convergence to a target distribution, utilizing bounded convergence-controlling Stein kernels.
• The research establishes that standard KSDs fail to control P-convergence for heavy-tailed target distributions (e.g., Student's t) due to the decay of their Stein kernels.

Summary by alphaXiv.