Weak Poincaré Inequalities, Simulated Annealing, and Sampling from Spherical Spin Glasses (2411.09075v2)

Abstract: There has been a recent surge of powerful tools to show rapid mixing of Markov chains, via functional inequalities such as Poincar\'e inequalities. In many situations, Markov chains fail to mix rapidly from a worst-case initialization, yet are expected to approximately sample from a random initialization. For example, this occurs if the target distribution has metastable states, small clusters accounting for a vanishing fraction of the mass that are essentially disconnected from the bulk of the measure. Under such conditions, a Poincar\'e inequality cannot hold, necessitating new tools to prove sampling guarantees. We develop a framework to analyze simulated annealing, based on establishing so-called weak Poincar\'e inequalities. These inequalities imply mixing from a suitably warm start, and simulated annealing provides a way to chain such warm starts together into a sampling algorithm. We further identify a local-to-global principle to prove weak Poincar\'e inequalities, mirroring the spectral independence and localization schemes frameworks for analyzing mixing times of Markov chains. As our main application, we prove that simulated annealing samples from the Gibbs measure of a spherical spin glass for inverse temperatures up to a natural threshold, matching recent algorithms based on algorithmic stochastic localization. This provides the first Markov chain sampling guarantee that holds beyond the uniqueness threshold for spherical spin glasses, where mixing from a worst-case initialization is provably slow due to the presence of metastable states. As an ingredient in our proof, we prove bounds on the operator norm of the covariance matrix of spherical spin glasses in the full replica-symmetric regime. Additionally, we resolve a question related to sampling using data-based initializations.

• The paper introduces a framework leveraging weak Poincaré inequalities and simulated annealing to enable efficient sampling from challenging probability distributions like spherical spin glasses beyond conventional limits.
• The methodology employs simulated annealing with a schedule of inverse temperatures and warm starts, proving efficient sampling for spherical spin glasses up to the eta_{SL} threshold.
• The findings extend to related problems like sampling from random initializations in the ferromagnetic Potts model and have broader implications for high-dimensional models and optimization.

Weak Poincaré Inequalities, Simulated Annealing, and Sampling from Spherical Spin Glasses

This paper investigates the challenge of efficiently sampling from complex probability distributions commonly encountered in statistical physics, computer science, and probability theory, focusing on Gibbs measures in particular. When sampling from such distributions, Markov Chain Monte Carlo (MCMC) methods are often employed. However, difficulties arise when metastable states exist, which inhibit rapid mixing from worst-case initializations due to their effectively segregated nature. These challenges necessitate novel approaches and tools, specifically weak Poincaré inequalities, to ensure effective sampling.

Weak Poincaré Inequalities and Simulated Annealing

The authors introduce a framework that leverages weak Poincaré inequalities as a means to guarantee effective sampling under certain initial distribution conditions. These inequalities allow for the establishment of a mixing time bound from a "warm start" rather than an arbitrary initial distribution. The core idea is to utilize simulated annealing—a process involving a schedule of inverse temperatures that incrementally adapts a distribution towards the target distribution, leveraging intermediate distributions as more informed starting points for its Markov chains.

The authors successfully apply this framework to demonstrate effective sampling from the Gibbs measure of spherical spin glasses across a range of inverse temperatures surpassing the classic uniqueness threshold, a domain previously inaccessible to MCMC methods due to slow mixing. A key introductory condition for the success of simulated annealing in this setting is outlined: ensuring state transitions that track closely enough under weak Poincaré characteristics.

Application to Spherical Spin Glasses

The paper achieves significant progress in proving that simulated annealing can efficiently sample spherical spin glass models within the regime defined by a well-posed threshold condition $\beta_{SL}$, which is asymptotically consistent with the conjectured shattering transition in statistical physics. This is further substantiated by rigorous quantitative bounds on the covariance matrices of the processes involved, which inform the spherical spin glass’s structural analysis and condition properties under the uniqueness threshold’s extension.

Mixing from Random Initializations and the Ferromagnetic Potts Model

Additionally, the insights garnered are extended to address related problems such as mixing from random initializations, including Glauber dynamics in a low-temperature ferromagnetic Potts model on expander graphs. Here, the synchronized use of a symmetric initialization tailored to inherent coloring symmetries resonates with the structural alignment emphasized throughout the paper.

Broader Implications and Future Directions

The broader implications of this research span theoretical advancements in the understanding of phase transitions in high-dimensional models and practical enhancements in efficient sampling methods critical for spin glass theory, optimization, and machine learning. The authors encourage further examination into extending these techniques to generalize simulated annealing guarantees beyond these paradigms, potentially impacting a range of complex distributed systems and optimization problems.

Overall, the paper provides a robust analytical framework bridging functional inequalities with effective sampling in challenging probabilistic models, spearheading advancements in both theoretical exploration and practical computational techniques.