Fast Mixing Markov Chains for Strongly Rayleigh Measures, DPPs, and Constrained Sampling
This addresses the challenge of efficient constrained sampling for researchers and practitioners in machine learning and statistics, offering incremental improvements with theoretical guarantees.
The paper tackles the problem of rapidly sampling from constrained probability measures, such as Strongly Rayleigh measures and Determinantal Point Processes, by developing fast Markov chain Monte Carlo samplers with provable polynomial mixing time bounds, including the first such sampler for DPPs in over four decades.
We study probability measures induced by set functions with constraints. Such measures arise in a variety of real-world settings, where prior knowledge, resource limitations, or other pragmatic considerations impose constraints. We consider the task of rapidly sampling from such constrained measures, and develop fast Markov chain samplers for them. Our first main result is for MCMC sampling from Strongly Rayleigh (SR) measures, for which we present sharp polynomial bounds on the mixing time. As a corollary, this result yields a fast mixing sampler for Determinantal Point Processes (DPPs), yielding (to our knowledge) the first provably fast MCMC sampler for DPPs since their inception over four decades ago. Beyond SR measures, we develop MCMC samplers for probabilistic models with hard constraints and identify sufficient conditions under which their chains mix rapidly. We illustrate our claims by empirically verifying the dependence of mixing times on the key factors governing our theoretical bounds.