Regularization and approximation: statistical inference, model selection, and large data

从大量数据中提取信息需要计算处理能力和统计效率。这通常是通过近似或正则化来实现的，这两种方法中的任何一种都会启发式地平衡对数据的保真度与简约、平滑、稀疏或可解释性等科学目标。我的长期目标是通过发展和描述计算近似和统计正则化之间的联系，从而促进改进的推理，从而使基础科学进步成为可能。特别是，我的研究项目调查了统计上最优的决策如何取决于正则化或近似化的数量及其结构，这两者都必须与相关的科学问题捆绑在一起，并根据数据进行校准。计算机科学的研究集中于改进算法，使计算能够以最小的近似值进行。与此同时，统计学家已经开发出正则化技术，以便利用简单的结构，如果这些结构代表了真相，就会改进推理和预测。我的作品试图弥合这些观点之间的差距。我的研究计划旨在通过以下方式深化近似算法与估计和预测推理之间的理论联系：(1)开发和证明相依数据的近似技术；(2)通过合理的调整参数选择来实现应用；(3)精确表征非参数统计中的近似效果；以及(4)通过协同开发和与领域专家的合作确保科学适用性。

Extracting information from large data requires computational tractability and statistical efficiency. These are typically achieved through approximation or regularization, either of which heuristically balances fidelity to the data with scientific goals like parsimony, smoothness, sparsity, or interpretability. My long-term objective is to enable fundamental scientific progress by developing and characterizing the connections between computational approximation and statistical regularization, thereby facilitating improved inference. In particular, my research program investigates how statistically optimal decisions depend on the amount of regularization or approximation and their structures, both of which must be tied to the scientific questions at stake and calibrated according to the data. Research in computer science has focused on improving algorithms to enable computation with a minimum of approximation. Meanwhile, statisticians have developed regularization techniques in order to take advantage of simple structures that, if representative of the truth, will improve inference and prediction. My work seeks to bridge the gap between these perspectives. My research program aims to deepen the theoretical links between approximation algorithms and inference for estimation and prediction by: (1) developing and justifying approximation techniques for dependent data; (2) enabling application through reasoned tuning parameter selection; (3) precisely characterizing the effect of approximations in nonparametric statistics; and (4) ensuring scientific applicability through synergistic development and collaboration with domain experts.