Fast Conditional Mixing of MCMC Algorithms for Non-log-concave Distributions
This addresses a theoretical bottleneck in MCMC sampling for non-log-concave distributions, which is incremental as it builds on existing Poincaré inequality concepts.
The paper tackles the slow mixing rate of MCMC algorithms for non-log-concave distributions by proving that conditional distributions on subsets of the state space can mix fast when a Poincaré-style inequality holds, even when global mixing is slow, with quantified rates.
MCMC algorithms offer empirically efficient tools for sampling from a target distribution $π(x) \propto \exp(-V(x))$. However, on the theory side, MCMC algorithms suffer from slow mixing rate when $π(x)$ is non-log-concave. Our work examines this gap and shows that when Poincaré-style inequality holds on a subset $\mathcal{X}$ of the state space, the conditional distribution of MCMC iterates over $\mathcal{X}$ mixes fast to the true conditional distribution. This fast mixing guarantee can hold in cases when global mixing is provably slow. We formalize the statement and quantify the conditional mixing rate. We further show that conditional mixing can have interesting implications for sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture models and Gibbs-sampling with well-connected local minima.