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ABSTRACT The unadjusted Langevin algorithm is commonly used to sample probability distributions in extremely high‐dimensional settings. However, existing analyses of the algorithm for strongly log‐concave distributions suggest that, as the dimension of the problem increases, the number of iterations required to ensure convergence within a desired error in the metric scales in proportion to or . In this paper, we argue that, despite this poor scaling of the error for the full set of variables, the behavior for asmall numberof variables can be significantly better: A number of iterations proportional to , up to logarithmic terms in , often suffices for the algorithm to converge to within a desired error for all ‐marginals. We refer to this effect asdelocalization of bias. We show that the delocalization effect does not hold universally and prove its validity for Gaussian distributions and strongly log‐concave distributions with certain sparse interactions. Our analysis relies on a novel metric to measure convergence. A key technical challenge we address is the lack of a one‐step contraction property in this metric. Finally, we use asymptotic arguments to explore potential generalizations of the delocalization effect beyond the Gaussian and sparse interactions setting. 

We show how to obtain improved active learning methods in the agnostic (adversarial noise) setting by combining marginal leverage score sampling with non- independent sampling strategies that promote spatial coverage. In particular, we propose an easily implemented method based on the pivotal sampling algorithm, which we test on problems motivated by learning-based methods for parametric PDEs and uncertainty quantification. In comparison to independent sampling, our method reduces the number of samples needed to reach a given target accuracy by up to 50%. We support our findings with two theoretical results. First, we show that any non-independent leverage score sampling method that obeys a weak one-sided l∞ independence condition (which includes pivotal sampling) can actively learn d dimensional linear functions with O(d log d) samples, matching independent sampling. This result extends recent work on matrix Chernoff bounds under l∞ independence, and may be of interest for analyzing other sampling strategies beyond pivotal sampling. Second, we show that, for the important case of polynomial regression, our pivotal method obtains an improved bound on O(d) samples.