New method speeds up low-temperature sampling in complex systems
Low-Temperature Sampling via Equilibrium Disagreements
Discrete MathematicsData Structures and Algorithms
Summary
Sampling complex systems at low temperatures can be very slow because the system can get stuck between multiple stable states or 'phases.' The authors develop a new way to understand and speed up this sampling by looking at how two independent samples from the same phase disagree. They apply this approach to several important models on random graphs and show that starting from certain natural configurations, the sampling process mixes quickly within each phase. They also provide a faster algorithm to combine these phase samples into samples representing the full system.
What this means in practice
- •For statistical physics modelers: Simulate low-temperature phases of spin and particle models on random graphs faster starting from natural initial configurations.
- •For probabilistic graphical model engineers: Improve the efficiency of sampling algorithms in systems with multiple stable states by exploiting structural disagreements between samples.
Authors
Andreas Galanis, Daniel Stefankovic
Abstract
We develop a new framework for sampling at low temperatures using Glauber dynamics. In this regime, the coexistence of competing modes (commonly referred to as phases) creates bottlenecks that can make worst-case mixing exponentially slow. The main new ingredient in our framework is to relate convergence within a phase to the geometry of disagreements between two independent configurations drawn from that phase. As applications of our approach, we consider three canonical models on random regular graphs that have been central to low-temperature sampling, and establish fast convergence from natural initialisations of Glauber dynamics for every fixed degree. We show: --- For the hard-core model on random bipartite $d$-regular graphs, Glauber dynamics initialised from either one-sided independent set mixes within the corresponding phase in $n^{3/2+o(1)}$ steps throughout the non-uniqueness regime $λ>λ_c(d)$. --- For the random-cluster model on random $d$-regular graphs, for every real $q>2$, Glauber dynamics initialised from the all-in edge set mixes within the ordered phase in $n^{3/2+o(1)}$ steps for all $β\geq β_c$. Similarly, Glauber dynamics initialised from the all-out edge set mixes within the disordered phase in $n^{3/2+o(1)}$ steps for $β\leq β_c$. --- For the ferromagnetic Ising model on random $d$-regular graphs, Glauber dynamics initialised from all-plus or all-minus mixes within the corresponding phase in $n^{3/2+o(1)}$ steps throughout the low-temperature regime $β>β_c(d)$. We also give an $n^{2+o(1)}$-time algorithm that combines phase samples into samples from the full distribution, avoiding in particular time-costly approximations of the corresponding partition functions to the target sampling accuracy.