Efficient Sampling from High-Entropy Initializations for Mean-Field Potts and Random-Cluster Models

Core Concepts

High-entropy initializations, such as product measures, can overcome the slow mixing of Markov chains in multimodal energy landscapes by allowing the dynamics to quickly escape from saddle points separating dominant modes.

Abstract

The paper studies the convergence of Markov chains for the mean-field Potts and random-cluster models from high-entropy initializations, such as product measures. These models exhibit complex energy landscapes with discontinuous phase transitions and asymmetric metastable modes, posing significant challenges for efficient sampling. The authors analyze two canonical Markov chains - the Chayes-Machta (CM) dynamics for the mean-field random-cluster model and the Glauber dynamics for the mean-field Potts model. They characterize the sharp families of product initializations that lead to fast mixing, even in parameter regimes where the worst-case mixing time is exponentially slow. The key technical contributions involve carefully approximating the high-dimensional Markov chains by tractable 1-dimensional random processes near the unstable saddle points separating the dominant modes. This allows the authors to understand the competition between the drift and fluctuations driving the dynamics away from the saddles, and precisely tune the initialization parameters for fast mixing. The results provide insights into the benefits and limitations of high-entropy initializations for overcoming metastability and phase coexistence in sampling from complex high-dimensional distributions, with connections to simulated annealing and tempering schemes.

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Mean-field Potts and random-cluster dynamics from high-entropy initializations

by Antonio Blan... at arxiv.org 04-22-2024

https://arxiv.org/pdf/2404.13014.pdf

Deeper Inquiries

What are some other high-dimensional spin system models where similar techniques could be applied to understand the benefits of high-entropy initializations

In addition to the Potts and random-cluster models discussed in the context above, similar techniques could be applied to understand the benefits of high-entropy initializations in other high-dimensional spin system models. One such model is the XY model, which describes spins on a lattice with continuous degrees of freedom. By applying the concept of high-entropy initializations, researchers could investigate how the dynamics of the XY model evolve from initial states with high entropy, potentially shedding light on the mixing behavior and convergence properties of the system. Another model where these techniques could be valuable is the Heisenberg model, which extends the Potts model to spins with continuous values and interactions in multiple dimensions. Analyzing the Heisenberg model from high-entropy initializations could provide insights into the dynamics of complex spin systems with continuous symmetries.

How do the insights from this work on product measure initializations translate to the analysis of more sophisticated initialization schemes, such as those used in simulated annealing or tempering algorithms

The insights gained from the analysis of product measure initializations in the context of spin system models can be directly applied to the study of more sophisticated initialization schemes, such as those used in simulated annealing or tempering algorithms. In simulated annealing, the temperature parameter plays a crucial role in controlling the exploration of the energy landscape, similar to how the inverse temperature parameter influences the dynamics in spin models. By understanding the convergence properties of high-entropy initializations in spin systems, researchers can optimize the initialization schemes in simulated annealing to efficiently explore multimodal energy landscapes. The concept of spreading mass across the space and diffusing away from saddle points separating dominant modes, as observed in the spin system models, can be leveraged to design effective initialization strategies in simulated annealing algorithms. Similarly, in simulated tempering, where multiple replicas of the system are run at different temperatures, insights from high-entropy initializations can guide the selection of initial states to enhance mixing and convergence rates across the temperature range.

Are there any fundamental limits on the ability of high-entropy initializations to overcome slow mixing in multimodal energy landscapes, or are there broader classes of models where such initializations can be shown to be universally effective

While high-entropy initializations offer a promising approach to overcoming slow mixing in multimodal energy landscapes, there may be fundamental limits to their effectiveness in certain scenarios. In highly complex energy landscapes with intricate topological features and multiple metastable states, the benefits of high-entropy initializations may be limited by the presence of deep energy minima and narrow transition regions. In such cases, the dynamics of the system may still get trapped in local optima or struggle to escape from metastable states, even with high-entropy initializations. However, for a broader class of models with well-separated energy basins and smoother energy landscapes, high-entropy initializations can be shown to be universally effective in promoting exploration and convergence. By carefully analyzing the structure of the energy landscape and the dynamics of the system, researchers can identify the regimes where high-entropy initializations provide significant advantages and where they may face limitations in facilitating efficient sampling and convergence.