Torpid mixing of simulated tempering on the Potts model

Simulated tempering and swapping are two families of sampling algorithms in which a parameter representing temperature varies during the simulation. The hope is that this will overcome bottlenecks that cause sampling algorithms to be slow at low temperatures. Madras and Zheng demonstrate that the swapping and tempering algorithms allow efficient sampling from the low-temperature mean-field Ising model, a model of magnetism, and a class of symmetric bimodal distributions [10]. Local Markov chains fail on these distributions due to the existence of bad cuts in the state space.Bad cuts also arise in the q-state Potts model, another fundamental model for magnetism that generalizes the Ising model. Glauber (local) dynamics and the Swendsen-Wang algorithm have been shown to be prohibitively slow for sampling from the Potts model at some temperatures [1, 2, 6]. It is reasonable to ask whether tempering or swapping can overcome the bottlenecks that cause these algorithms to converge slowly on the Potts model.We answer this in the negative, and give the first example demonstrating that tempering can mix slowly. We show this for the 3-state ferromagnetic Potts model on the complete graph, known as the mean-field model. The slow convergence is caused by a first-order (discontinuous) phase transition in the underlying system. Using this insight, we define a variant of the swapping algorithm that samples efficiently from a class of bimodal distributions, including the mean-field Potts model.