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. 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 questions related to the mixing of Glauber dynamics in the ferromagnetic Potts model from a uniform monochromatic coloring, and sampling using data-based initializations.

翻译：近期涌现了一系列基于Poincaré不等式等泛函不等式证明马尔可夫链快速混合的强大工具。然而在许多场景中，马尔可夫链虽无法在最坏初始化条件下实现快速混合，却能在随机初始化条件下实现近似采样。例如，当目标分布存在亚稳态——即质量占比趋于零且与测度主体基本隔离的小规模聚类时，就会出现这种情况。在此条件下，Poincaré不等式不再成立，因此需要新的工具来证明采样保证。我们建立了一个基于弱Poincaré不等式的模拟退火分析框架。这类不等式保证了从适当"热启动"状态开始的混合过程，而模拟退火能够将此类热启动状态串联成完整的采样算法。我们进一步提出了证明弱Poincaré不等式的局部-全局对应原理，该原理与谱独立性和局部化方案框架中分析马尔可夫链混合时间的思路相呼应。作为主要应用，我们证明了模拟退火能够在逆温度达到自然阈值前对球面自旋玻璃的Gibbs测度进行采样，这与近期基于算法随机局部化的研究成果相符。这首次实现了超越球面自旋玻璃唯一性阈值的马尔可夫链采样保证，而在该阈值之上，最坏初始化条件下的混合过程已被证明是缓慢的。作为证明的关键要素，我们给出了全复本对称区域内球面自旋玻璃协方差矩阵算子范数的界。此外，我们还解决了铁磁Potts模型中Glauber动力学从均匀单色着色的混合问题，以及基于数据初始化的采样问题。