Fast and Robust Gaussian Process Inference for Bayesian Nonparametric Learning

现代技术的进步使研究人员能够收集大量数据进行推理和预测。随着可用观测值的丰富，传统的统计方法基于参数假设，即模型可以由预先指定的参数数量来表征，变得不够充分和不那么有吸引力。在这种情况下，贝叶斯非参数模型很有吸引力，它允许以数据驱动的方式确定分析的分辨率水平，并提供不确定性的自动表征。本项目的目标是开发基于高斯过程先验的贝叶斯非参数推理的新理论、新方法和新的计算工具。考虑到海量数据的可用性，非参数推理为灵活建模底层结构和提取有用信息提供了一个有吸引力的框架。例如，这样的挑战出现在化学物理、计算生物学、计算机视觉、工程和气象学中。本项目旨在为基于高斯过程的非参数推理奠定坚实的方法论、算法和理论基础。特别是，基于高斯过程的方法往往容易受到数据污染，并且具有较高的计算成本。为了缓解高斯过程推理过程的高计算代价，研究者提出了两种新的计算框架，它们的近似目标不同，要么是先验的，要么是后验的。为了增强高斯过程推理对数据污染的稳健性，研究人员提出了一类新的贝叶斯分层模型来引入这种额外的测量误差结构，从而得到一类鲁棒的高斯过程推理过程。这一新的理论发展为实验设计从业者提供了关于测量误差对预测和估计的影响的宝贵见解，并为计算复杂性和统计可学习性之间的深层次联系提供了证据。这些计算和理论框架也有利于其他学科，如应用数学、计算机科学和金融学，这些学科经常使用随机过程，如高斯过程。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

Advances in modern technology have empowered researchers to collect massive data to conduct inference and making predictions. With the abundance of available observations, traditional statistical methods under the parametric assumption that a model can be characterized by a pre-specified number of parameters become inadequate and less attractive. Bayesian nonparametric models are attractive in this context which allow the resolution level of the analysis to be determined in a data-driven manner, and provide automatic characterization of uncertainty. The goal of this project is to develop new theory, methodology and computational tools for Bayesian nonparametric inference via Gaussian process priors. Given the availability of massive data, nonparametric inference offers an attractive framework for flexibly modeling the underlying structure and extracting useful information. For instance, such challenges occur in chemical physics, computational biology, computer vision, engineering, and meteorology. This project aims to lay down a solid methodological, algorithmic, and theoretical foundation for nonparametric inference based on Gaussian processes. In particular, Gaussian process-based approaches tend to be vulnerable to data contamination and have heavy computational costs. To alleviate the high-computational cost of Gaussian process inference procedures, the investigator puts forward two novel computational frameworks which differ at their respective approximating targets as being either the prior or the posterior. To enhance the robustness of Gaussian process inference against data contamination, the investigator proposes a novel class of Bayesian hierarchical models for incorporating this extra measurement error structure, leading to a class of robust Gaussian process inference procedures. The new theoretical development offers valuable insight to experiment-design practitioners into the impact of measurement errors upon prediction and estimation, and provides evidence on the deep connection between computational complexity and statistical learnability. These computational and theoretical frameworks also benefit other disciplines such as applied mathematics, computer science and finance where stochastic processes such as Gaussian processes are routinely used.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.