Random environments, stochastic equations, and randomized algorithms

我的研究的首要目标是理解和利用随机性，因为它发生在不同的设置集合。这些技术广泛地借鉴了随机分析的数学学科，它基于随机过程、偏微分方程和动力系统的相互作用，以理解(I)随机均匀化、（II）随机偏微分方程和（III）机器学习中的问题。随机均匀化的基本观察是随机现象可以表现得好像它们是确定的。例如，金属的电导率受到微小杂质存在的显著影响，这些杂质可能来自制造过程中的缺陷或环境污染。这些杂质复杂的微观结构使其无法用标准的数值方法进行有效的模拟。在随机均质化中，我们确定了一个简单的确定性模型，它与原始材料非常接近。我们通过在一个有效的随机环境中建立一个复杂的非线性平均来做到这一点。本建议的目标是利用均质化理论来表征复杂材料和湍流的性质。虽然随机均匀化描述的是有效确定性的随机现象，但一些看似确定性的现象实际上是随机的。当股市对政治事件做出反应时，或者在森林大火蔓延期间，就会出现这种情况。这种随机性是由微观层面上本质上无法量化的波动驱动的，比如个人投资者的反应或森林地面的变化。我们使用随机噪声驱动的方程来模拟这些现象，这些方程被称为随机偏微分方程（SPDEs）。由于驱动噪声的存在，spde并没有得到经典的定义，因此理解它们的解决方案是一个难题。本提案的目标是为模拟罕见事件（如机械系统中热量或能量的极端集中）的spde类开发一个解决方案理论。这些结果将在spde和相互作用的粒子系统之间建立严格的长期非正式联系。机器学习的目标是识别大型数据集的基本特征，并创建具有预测能力的人工神经网络。例如，图像识别和人工智能技术的发展依赖于对大量信息的深度网络的训练。然而，现代数据的规模使得像梯度下降这样的经典训练技术在计算上不可行。我们通过故意在算法中引入随机性来克服这个问题。随机梯度下降（SGD）是一种随机过程，它在每个步骤上对一个小但随机的数据样本进行优化。SGD是训练神经网络最常用的方法，但它的收敛性没有严格的理由。本提案的目标是对SGD的收敛性进行定量理解，并描述深度学习中的损失情况。

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.