AF: Small: Optimal algorithms and new models for statistical estimation

数据变得越来越有价值，但也越来越难以有效分析。在统计学和机器学习的许多领域，人们希望算法能够1)提供更准确的预测，2)需要更少的数据来操作，以及3)对真实数据中出现的各种错误具有鲁棒性。该项目着重于在两个原型设置中解决这一挑战。首先，该项目旨在开发新的算法和分析工具，用于最有用和最通用的统计原语之一：估计给定数据的概率分布的平均值。其次，它寻求开发新的模型来识别、统一并希望解决处理来自非理想采样过程的数据的挑战。这些问题是统计学、机器学习和计算机科学的交叉点。更广泛地说，这个项目将帮助建立未来研究人员的管道，通过为本科生创造新的研究和研究生院的途径，并通过与计算机科学专业的学生一起探索研究的角色、研究的思维过程、研究的机制和研究的影响。更详细地说，这个项目的第一个目标是“均值估计”的基本问题，通常被认为是统计学中最重要的经典估计问题：从一个或多个维度的概率分布中给定样本，尽可能准确和稳健地估计其均值。令人惊讶的是，在许多重要的环境中，这个基本问题仍然没有找到好的解决方案。该项目将主要关注高维数据的设置，并将致力于开发高效、准确和灵活的算法。第二个重点领域是从数据中做出准确统计推断的挑战，尽管数据来自不均匀的、有偏见的样本。该项目为如何利用数据收集过程的见解来提高统计数据的准确性开发了新的框架，这是计算机图形学、计量经济学、社会学和统计物理学等许多领域都在努力应对的挑战。为解决这些问题而开发的技术将揭示新的算法和微妙的概率现象，这将为有效使用有价值数据的挑战提供下一代解决方案。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Data is becoming both more valuable, yet harder to effectively analyze. Across many areas of statistics and machine learning, one desires algorithms that 1) give more accurate predictions, 2) require less data to operate, and 3) are robust to the wide array of errors that show up in real data. This project focuses on addressing this challenge in two archetypal settings. First, this project seeks to develop new algorithms and analysis tools for one of the most useful and versatile primitives of statistics: estimating the mean of a probability distribution, given data from it. Second, it seeks to develop new models that identify, unify, and hope to resolve the challenges of dealing with data that come from a non-ideal sampling process. These problems lie at the intersection of statistics, machine learning, and computer science. More broadly, this project will help build a pipeline of future researchers, by creating new pathways for undergraduates towards research and to graduate school, and by exploring with computer science students the role of research, the thought process of research, the mechanics of research, and the impact of research.In more detail, the first target of this project is the fundamental problem of "mean estimation", often considered the most important classical estimation problem in statistics: given samples from a probability distribution in one or more dimensions, estimate its mean as accurately and robustly as possible. Surprisingly, there are many important settings in which good solutions to this basic question remain to be found. This project will focus primarily on the setting of high-dimensional data, and will aim to develop algorithms that are efficient, accurate, and flexible. The second area of focus is the challenge of making accurate statistical inferences from data, despite the data originating from a non-uniform, biased sample. This project develops new frameworks for how to leverage insights about the data-collection process to improve the accuracy of statistics, a challenge that has been grappled with by many fields including computer graphics, econometrics, sociology, and statistical physics. The techniques developed to tackle these problems will reveal new algorithms and subtle probabilistic phenomena that will inform the next generation of solutions to the challenges of effectively using valuable data.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.