Stochastic Perturbation Theory for Machine Learning

随着大数据的出现和人工智能（AI）在科学技术中的日益普及，基于机器学习的推理和分类系统的可靠性已成为一个主要问题。数字错误的后果可能是灾难性的，就像自动驾驶汽车发生事故的例子一样。虽然机器学习在处理远远超出人类能力的复杂任务方面非常成功，但有时只需要一个小的、人类无法检测到的数据扰动就可以欺骗分类系统并导致它失败。这样的扰动可能产生戏剧性的后果；它们可能导致医疗误诊，语音识别或语音认证的误解，或者只是降低对预测和决策支持系统的信心。重要的是要了解对抗性扰动的性质和普遍性，这种扰动是否可能偶然发生，以及如何提高推理和分类系统的鲁棒性。本项目旨在通过数值条件反射理论和几何概率论的视角研究数据扰动对深度学习的影响。条件数是冯·诺伊曼和图灵在开创性工作中引入的，用来衡量计算问题的解决方案对数据扰动的敏感性。通过将深度学习中的鲁棒性问题表述为条件问题，我们揭示了一系列用于分析条件数的方法，以获得机器学习问题鲁棒性的更好界限。至于应用，这项工作将特别感兴趣的时间序列问题。其中一个问题是通过凌日系外行星测量卫星（TESS）记录的光度数据来探测系外平面。

With the advent of big data and the increased prevalence of Artificial Intelligence (AI) in science and technology, the reliability of machine learning-based inference and classification systems has become a major concern. The consequences of numerical mistakes can be catastrophic, as seen in the example of an accident involving a self-driving car. While machine learning has been very successful at tackling complex tasks that vastly exceed human capability, it sometimes only takes a small, humanly undetectable data perturbation to fool a classification system and cause it to fail. Such perturbations can have dramatic consequences; they can lead to medical misdiagnosis, misinterpretation in speech recognition or voice authentication, or simply to reduced confidence in prediction and decision support systems. It is important to understand the nature and prevalence of adversarial perturbations, whether such perturbations are likely to occur by accident, and how to improve the robustness of inference and classification systems.This project aims to study the effect of data perturbations in deep learning through the lens of numerical conditioning theory and geometric probability. The condition number, introduced in ground-breaking work by von Neuman and Turing, measures the sensitivity of a solution to a computational problem to perturbations in the data. By formulating robustness problems in deep learning as conditioning problems, we unlock a range of methods that have been employed in the analysis of condition numbers in order to obtain better bounds on the robustness of machine learning problems. As for applications, this work will be specifically interested in time series problems. One such problem is the detection of exoplanes from photometric data recorded by the Transiting Exoplanet Survey Satellite (TESS).