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Explore the mathematical foundations of adaptive sampling methods through a conference talk examining Langevin diffusions that evolve with continuously updating priors. Delve into the theoretical framework of dynamical mean-field theory (DMFT) as applied to Bayesian linear regression models where the prior adapts via maximum marginal-likelihood schemes based on observed Langevin trajectory samples. Learn how DMFT techniques provide precise characterization of high-dimensional asymptotic limits governing the joint evolution of prior parameters and sample distributions. Examine the analysis of DMFT limit equations under approximate time-translation-invariance conditions, particularly in settings where posterior distributions satisfy log-Sobolev inequalities. Discover how adaptive Langevin trajectories converge to equilibrium states characterized by replica-symmetric fixed-point equations on dimension-independent time horizons, with prior parameters converging to critical points of replica-symmetric free energy limits. Investigate the landscape properties of free energy functions and their critical points through concrete examples, exploring scenarios where critical points may exhibit uniqueness or multiplicity. Gain insights into the intersection of statistical sampling theory, adaptive algorithms, and mean-field analysis techniques relevant to modern machine learning and statistical inference applications.
