AF: Small: Faster Algorithms for High-Dimensional Robust Statistics

随着机器学习在我们的社会中扮演着越来越重要的角色，我们需要可靠且鲁棒的学习算法。在现代机器学习中，人们经常需要处理高维和嘈杂的数据。最近的工作为几个基本的统计问题提供了第一个有效的鲁棒估计，从那时起，有一系列的研究为许多机器学习问题获得了有效的鲁棒算法。然而，文献中现有算法的一个主要缺点是，与非鲁棒算法相比，它们往往要慢得多，或者它们通常涉及需要仔细调优的参数。为了解决这些问题，本项目旨在(i)为广泛的高维统计和学习任务设计更快且可证明的鲁棒算法，以及（ii）探索鲁棒估计的非凸公式并分析其优化前景。这个项目将推动计算机科学和统计学领域的发展，也可能为其他领域带来有用的工具。追求更快、更简单的算法将有助于加速技术转化为实践，激发系统的鲁棒性方法，并从长远来看提供积极的社会影响。该项目的教育计划包括将该项目产生的材料纳入伊利诺伊大学芝加哥分校（UIC）的研究生课程，以及UIC的研究生和本科生培训，UIC是一所拥有多样化学生群体的城市大学。设计高维鲁棒算法是一项非常具有挑战性的任务。即使对于均值估计的基本问题，当一小部分输入被对抗性破坏时，直到最近才发现有效的算法。第一个具有维无关误差保证的多项式时间估计器于2016年被发现。然而，考虑到今天可用的数据量，多项式时间在实践中不再转化为可伸缩性。由于需要更快和更实用的算法，该项目主要关注两个重点，以扩展算法高维鲁棒统计领域。首先，研究者希望加快现有算法的速度，并针对更广泛的问题和更丰富的分布族开发新的鲁棒算法，最终目标是匹配最快的非鲁棒算法的运行时间。其次，研究者希望设计可以通过标准一阶优化方法计算的稳健估计量。主要的挑战是找到一个可以用基本矩阵运算求出梯度的目标函数，同时证明该目标没有坏的局部最优的结构结果。具体来说，研究者计划通过针对以下问题的各个方面来研究这两个重点：(1)鲁棒随机优化，(2)鲁棒稀疏均值估计和稀疏PCA，(3)鲁棒协方差估计，(4)列表可解码学习，(5)贝叶斯网络的鲁棒学习。这个项目是跨学科的，将依赖于直觉和统计学、概率、线性代数、离散和连续优化以及非凸优化的技术。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。

As machine learning plays a more prominent role in our society, there is a need for learning algorithms that are reliable and robust. In modern machine learning, one often needs to work with data that are high-dimensional and noisy. Recent work gave the first efficient robust estimators for several basic statistical problems, and since then, there has been a flurry of research that obtained efficient robust algorithms for many machine-learning problems. However, one major drawback of existing algorithms in the literature is that they tend to be much slower when compared to their non-robust counterparts, or they often involve parameters that require careful tuning. To address these issues, this project aims to (i) design faster and provably robust algorithms for a wide range of high-dimensional statistical and learning tasks, and (ii) explore non-convex formulations of robust estimation and analyze their optimization landscape. This project will advance the fields of computer science and statistics, and also potentially lead to useful tools for other areas. The pursuit of faster and simpler algorithms will help accelerate technology transfer into practice, stimulate systematic approaches to robustness, and provide a positive societal impact in the long run. The education plan of this project includes incorporating the materials generated from this project into graduate-level courses at the University of Illinois at Chicago (UIC), as well as training graduate and undergraduate students at UIC, which is an urban university with a diverse student population.Designing robust algorithms in high dimensions is a very challenging task. Even for the basic problem of mean estimation, when a small fraction of the input is adversarially corrupted, no efficient algorithms were known until recently. The first polynomial-time estimators with dimension-independent error guarantees were discovered in 2016. However, given the amount of data available today, polynomial-time no longer translates to scalability in practice. Motivated by the need for faster and more practical algorithms, this project focuses on two main thrusts to expand the area of algorithmic high-dimensional robust statistics. First, the investigator would like to speed up existing algorithms and develop new robust algorithms for a broader range of problems and richer families of distributions, with the ultimate goal of matching the runtime of the fastest non-robust algorithms. Second, the investigator wants to design robust estimators that can be computed via standard first-order optimization methods. The main challenge is to find an objective function whose gradient can be evaluated using basic matrix operations while proving the structural result that this objective has no bad local optima. Concretely, the investigator plans to work on these two thrusts by targeting various aspects of the following problems: (1) robust stochastic optimization, (2) robust sparse mean estimation and sparse PCA, (3) robust covariance estimation, (4) list-decodable learning, and (5) robust learning of Bayesian networks. This project is interdisciplinary and will rely on intuition and techniques from statistics, probability, linear algebra, discrete and continuous optimization, and non-convex optimization.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.