Stochastic Perturbation Theory for Machine Learning

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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).