In many applications of statistical estimation via sampling, one may wish to sample from a high-dimensional target distribution that is adaptively evolving to the samples already seen. We study an example of such dynamics, given by a Langevin diffusion for posterior sampling in a Bayesian linear regression model with i.i.d. regression design, whose prior continuously adapts to the Langevin trajectory via a maximum marginal-likelihood scheme. Results of dynamical mean-field theory (DMFT) developed in our companion paper establish a precise high-dimensional asymptotic limit for the joint evolution of the prior parameter and law of the Langevin sample. In this work, we carry out an analysis of the equations that describe this DMFT limit, under conditions of approximate time-translation-invariance which include, in particular, settings where the posterior law satisfies a log-Sobolev inequality. In such settings, we show that this adaptive Langevin trajectory converges on a dimension-independent time horizon to an equilibrium state that is characterized by a system of scalar fixed-point equations, and the associated prior parameter converges to a critical point of a replica-symmetric limit for the model free energy. As a by-product of our analyses, we obtain a new dynamical proof that this replica-symmetric limit for the free energy is exact, in models having a possibly misspecified prior and where a log-Sobolev inequality holds for the posterior law.

翻译：在通过抽样进行统计估计的许多应用中，人们可能希望从一个自适应演化至已观测样本的高维目标分布中抽样。我们研究此类动力学的一个示例，该示例由贝叶斯线性回归模型中的后验抽样朗之万扩散给出，该模型具有独立同分布的回归设计，其先验通过最大边际似然方案连续适应朗之万轨迹。我们在配套论文中发展的动力学平均场理论（DMFT）结果，为先验参数与朗之万样本律的联合演化建立了精确的高维渐近极限。在本工作中，我们在近似时间平移不变性条件下，对描述该DMFT极限的方程进行分析，该条件特别包括后验律满足对数索博列夫不等式的情形。在此类设定下，我们证明该自适应朗之万轨迹在维度无关的时间尺度上收敛至一个由标量固定点方程组表征的平衡态，且相关先验参数收敛至模型自由能的复本对称极限的临界点。作为我们分析的副产品，我们获得了一个新的动力学证明，表明该自由能的复本对称极限是精确的，适用于先验可能设定错误且后验律满足对数索博列夫不等式的模型。