Lower Bounds on the Convergence Rates of Adaptive Mcmc Methods

We consider the convergence properties of recently proposed adaptive Markov chain Monte Carlo (MCMC) algorithms for approximation of high-dimensional integrals arising in Bayesian analysis and statistical mechanics. Despite their name, in the general case these algorithms produce non-Markovian, time-inhomogeneous, irreversible stochastic processes. Nevertheless, we show that lower bounds on the mixing times of these processes can be obtained using familiar ideas of hitting times and conductance from the theory of reversible Markov chains. While loose in some cases, the bounds obtained are sufficient to demonstrate slow mixing of several recently proposed algorithms including the adaptive Metropolis algorithm of Haario et al. (2001), the equi-energy sampler (Kou et al., 2006), and the importance-resampling MCMC algorithm (Atchadé, 2009) on some multimodal target distributions including mixtures of normal distributions and the mean-field Potts model. These results appear to be the first non-trivial bounds on the mixing times of adaptive MCMC samplers, and suggest that the adaptive methods considered may not provide qualitative improvements in mixing over the simpler Markov chain algorithms on which they are based. Our bounds also indicate properties which adaptive MCMC algorithms must have to achieve exponential speed-ups, suggesting directions for further research in these methods.