【直播】【青年科学半月谈】On the Statistical analysis of Scientific……

活动名称：

On the Statistical analysis of Scientific Machine learning

活动时间：

2023年4月13日（周四）10:00

报告嘉宾：

Yiping Lu

主办单位：

蔻享学术

直播通道

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报告人介绍

Yiping Lu

Yiping Lu is a 4th year Ph.D. student at Stanford University working with Jose Blanchet and Lexing Ying. Before joining Stanford, he received his B.Sc. degree in Math from Peking University. Yiping is broadly interested in non-parametric statistics, applied probability, and stochastic simulation. He is particularly interested in applying machine learning and experimental design techniques to learn and solve models from applied probability and industrial engineering models.

报告简介

Massive data collection and computational capabilities have enabled data-driven scientific discoveries and control of engineering systems. However, there are still several questions that should be answered to understand the fundamental limits of just how much can be discovered with data and what is the value of additional information. For example, 1) How can we learn a physics law or economic principle purely from data? 2) How hard is this task, both computationally and statistically? 3) What’s the impact on hardness when we add further information (e.g., adding data, model information)? I’ll answer these three questions in this talk in two learning tasks. A key insight in both two cases is that using direct plug-in estimators can result in statistically suboptimal inference.For the first learning task, we focus on variational formulations for differential equation models. We discuss a prototypical Poisson equation. We provide a minimax lower bound for this problem. Based on the lower bounds, we discover that the variance in the direct plug-in estimator makes sample complexity suboptimal. We also consider the optimization dynamic for different variational forms. Finally, based on our theory, we explain an implicit acceleration of using a Sobolev norm as the objective function for training.The second learning task I’ll discuss is (linear) operator learning, which has wide applications in causal inference, time series modeling, and conditional probability learning. We build the first min-max lower bound for this problem. The min-max rate has a particular structure where the more challenging parts of the input and output spaces determine the hardness of learning a linear operator. Our analysis also shows that an intuitive discretization of the infinite-dimensional operator could lead to a sub-optimal statistical learning rate. Then, I’ll discuss how, by suitably trading-off bias and variance, we can construct an estimator with an optimal learning rate for learning a linear operator between infinite dimension spaces. We also illustrate how this theory can inspire a multilevel machine-learning algorithm of potential practical use.

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编辑：吴良秀

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