Weak Poincaré Inequalities via Approximate Stochastic Localization: Application to Sampling the Sherrington-Kirkpatrick Model

Technical Summary: Weak Poincaré Inequalities via Approximate Stochastic Localization

Problem Statement
The paper addresses the challenge of proving functional inequalities, specifically Weak Poincaré Inequalities (WPI), for Gibbs measures of spin glass models, with a primary focus on the Sherrington–Kirkpatrick (SK) model. While Stochastic Localization (SL) has been a powerful framework for proving such inequalities and establishing rapid mixing of Markov chains (e.g., Glauber dynamics), it traditionally requires strong regularity conditions on the true magnetization process that are difficult to verify for complex models like the SK model.

Prior work established efficient sampling for the SK model at inverse temperatures $\beta < 1/2$ using Algorithmic Stochastic Localization (ASL) combined with Jarzynski's equality and rejection sampling. However, these results relied on weaker error metrics (Wasserstein distance) or complex algorithmic constructions. A significant open problem remains whether the standard Glauber dynamics mixes rapidly in the replica-symmetric regime ($\beta < 1/2$) and whether this can be proven via functional inequalities under conditions weaker than those required for exact SL.

Methodology
The authors develop a new method to prove a WPI for a target distribution $\mu$ by leveraging an "Approximate Stochastic Localization" (ASL) process. The core strategy involves three main components:

1. Approximate Stochastic Localization (ASL): Instead of relying on the ideal SL process (which requires the true magnetization $m(y)$ to be Lipschitz), the authors utilize an ASL process driven by an approximation $\hat{m}(y)$ that is uniformly bounded and Lipschitz. They assume the ASL process tracks the ideal SL process with bounded error (Assumption 1.1) and that the ASL drift is Lipschitz (Assumption 1.2).
2. Decomposition Theorem: The authors prove a general decomposition theorem for WPIs (Theorem 3.1). This theorem reduces the task of proving a WPI for the target distribution $\mu$ to proving WPIs for two simpler components:
   • The localized distribution $\mu_y$ (the conditional distribution given the localization state $y$).
   • The proximal sampler (a Gibbs chain alternating between sampling $X|Y$ and $Y|X$).
     This allows the authors to bypass proving a WPI for the complex joint distribution directly.
3. Scaffolding via Gaussian Smoothing: To prove a WPI for the ASL distribution $p_{T_0}$, the authors employ a "scaffolding" argument. They relate $p_{T_0}$ to a later-time distribution $p_{T_1}$ (where $T_1 > T_0$). While $p_{T_1}$ satisfies a Log-Sobolev Inequality (LSI) due to the Lipschitz drift of the ASL, $p_{T_0}$ is a Gaussian noising of $p_{T_1}$. Using coupling arguments and small-distance matching (Lemmas 4.1 and 4.2), they show that the Gaussian smoothing "lifts" the density of $p_{T_0}$ away from zero in regions where $p_{T_1}$ might be sparse, thereby establishing a WPI for $p_{T_0}$ despite the potential lack of strong regularity in the ideal process.
4. Warm Starts via Annealed Distributions: To convert the WPI into an algorithmic sampling result, the authors construct a "warm start" (an initial distribution with bounded density ratio relative to the target). They define explicit "annealed" distributions $\varrho_t$ based on the TAP free energy and Jarzynski's equality. They prove these distributions satisfy an LSI (Theorem 6.2) and are warm starts to the SL distribution (Theorem 7.2), enabling efficient initialization.

Key Contributions and Results

• General WPI Framework: The paper establishes that if an ASL process satisfies specific regularity conditions (bounded magnetization error and Lipschitz drift) and the localized distributions satisfy a WPI with high probability, then the target distribution satisfies a WPI (Theorem 1.4). This holds under significantly weaker assumptions than required for exact SL.
• Application to the SK Model: Applying this framework to the SK model with $\beta < 1/2$, the authors prove that the Gibbs measure satisfies a WPI with parameters $(c, \delta) = (n^{-1}e^{-C/\epsilon}, \epsilon)$ (Corollary 1.5). This result relies on the fact that the ASL process for the SK model satisfies the necessary regularity conditions, as shown in their prior work [DLSS26].
• Algorithmic Sampling: The paper demonstrates that the WPI implies that Glauber dynamics, when started from a "warm start," efficiently samples the Gibbs measure of the SK model for $\beta < 1/2$ (Theorem 1.6).
  • The algorithm consists of two phases: (1) obtaining a warm start using Langevin dynamics on an explicit annealed distribution (or ASL), and (2) running Glauber dynamics for $O(n^2 + n e^{O(1/\epsilon)})$ steps.
  • This simplifies the previous state-of-the-art algorithm [DLSS26], which required complex ASL-based sampling, Jarzynski reweighting, and polarized walks. The new method replaces the complex sampling phase with a standard Glauber dynamics run, provided a warm start is available.

Significance and Claims
The authors claim this work represents a significant structural step toward resolving the conjecture that Glauber dynamics mixes fast in the replica-symmetric regime of the SK model.

• Weaker Conditions: The method proves functional inequalities under conditions (second-moment bounds on covariance, Lipschitz ASL drift) that are known to hold for the SK model, whereas previous SL-based proofs required stronger conditions (e.g., uniform bounds on the operator norm of the covariance) that are currently out of reach for the SK model.
• Algorithmic Simplification: The result shows that a much simpler algorithm (Glauber dynamics with a warm start) suffices for efficient sampling, provided the warm start can be obtained. While the runtime dependence on $\epsilon$ is worse than in [DLSS26] (exponential in $1/\epsilon$ vs. polynomial), the conceptual simplification is substantial.
• Limitations: The authors are modest about the strength of the result. They note that the quantitative strength of the derived WPI is insufficient to support simulated annealing (which would require a stronger WPI to handle the temperature schedule). Furthermore, while the algorithm is simpler, the proof of its correctness still relies on the deep analytical results from [DLSS26] regarding the behavior of the ASL process. The paper does not claim to prove rapid mixing from a cold start, but rather from a warm start, which is a standard requirement for WPI-based mixing guarantees.

In summary, the paper bridges the gap between algorithmic stochastic localization and functional inequalities, demonstrating that approximate processes can be used to prove structural properties (WPI) that imply the efficiency of standard Markov chains, specifically for the SK model in the high-temperature regime.