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

从大数据中提取信息需要计算可追溯性和统计效率。这些通常是通过近似或正则化来实现的，其中任何一种都可以启发式地平衡数据的保真度和科学目标，如简约性、平滑性、稀疏性或可解释性。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，从而实现基本的科学进步。特别是，我的研究计划调查了统计上最优的决策是如何依赖于正则化或近似的数量及其结构的，这两者都必须与利害攸关的科学问题联系起来，并根据数据进行校准。计算机科学研究的重点是改进算法，使计算具有最小近似值。与此同时，统计学家已经开发了正则化技术，以便利用简单的结构，如果代表事实，将改进推理和预测。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加深近似算法与估计和预测推理之间的理论联系：(1)开发和证明依赖数据的近似技术；(2)通过合理的调优参数选择，实现应用；(3)精确表征非参数统计中的近似效应；(4)通过协同开发和与领域专家的协作，确保科学的适用性。

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.