Random environments, stochastic equations, and randomized algorithms

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The overarching aim of my research is to understand and exploit randomness as it occurs in a diverse collection of settings. The techniques draw broadly from the mathematical discipline of stochastic analysis, which is based on the interplay of random processes, partial differential equations, and dynamical systems, to understand problems in (I) stochastic homogenization, (II) stochastic partial differential equations, and (III) machine learning.I. The fundamental observation of stochastic homogenization is that random phenomena can behave as though they are deterministic. For example, the conductance of a metal is significantly affected by the presence of microscopic impurities, which may arise from flaws in a manufacturing process or from environmental contamination. The complicated microstructure of these impurities makes them impossible to simulate efficiently with standard numerical methods. In stochastic homogenization, we instead identify a simple deterministic model that closely approximates the original material. We do this by establishing a complicated nonlinear averaging in what is effectively a random environment. The objectives of this proposal will use homogenization theory to characterize the properties of complex materials and turbulent fluids.II. While stochastic homogenization describes random phenomena that are effectively deterministic, some apparently deterministic phenomena are effectively random. This is the case when the stock market reacts to political events, or during the growth of a forest fire. The randomness is driven by essentially unquantifiable fluctuations at the microscopic level, like the reactions of individual investors or variations in the forest floor. We model these phenomena using equations driven by random noise, which are called stochastic partial differential equations (SPDEs). Because of the driving noise, SPDEs are not classically defined and making sense of their solutions is a difficult problem. The objectives of this proposal will develop a solution theory for classes of SPDEs that model rare events, like the extreme concentration of heat or energy in a mechanical system. The results will make rigorous long-standing informal connections between SPDEs and interacting particle systems.III. The goal of machine learning is to identify the essential features of large data sets and to create artificial neural networks with predictive power. For example, the development of image recognition and artificial intelligence technologies relies on the training of deep networks over an enormous amount of information. However, the scale of modern data makes the implementation of classical training techniques like gradient descent computationally infeasible. We overcome this problem by deliberately introducing randomness into the algorithm. Stochastic gradient descent (SGD) is a randomized process that optimizes at each step over a small but random sample of the data. SGD is the most common way to train neural networks, yet there is no rigorous justification for its convergence. The objectives of this proposal will develop a quantitative understanding of convergence for SGD and will characterize the loss landscape in deep learning.