CIF: Medium: Collaborative Research: Theory of Optimization Geometry and Algorithms for Neural Networks

近年来，由于深度学习在计算机视觉、人工智能和自然语言处理方面的广泛应用，以及最近在自动驾驶方面的进展，深度学习引起了人们的极大兴趣。然而，这种成功背后的理论基础在很大程度上仍然难以捉摸，阻碍了它在其他应用中的进一步采用。该项目旨在从优化景观和算法功效方面推进训练神经网络的理论基础，进而通过为网络设计、算法选择、超参数调优和对抗训练提供指导原则，对深度学习的实践产生可衡量的影响。该项目采用跨学科的方法，融合了机器学习，优化，统计信号处理，高维统计，非参数统计和信息论的思想。该项目还将开发有关大规模机器学习理论基础的课程和教程，并为各级学生提供广泛的培训机会。本项目旨在发展一个全面的理论，以表征主要神经网络训练问题的损失函数和算法正则化的优化景观和几何形状，并探索网络架构（包括深度、宽度和激活函数）如何影响这些属性，从而为设计算法提供指导，从而在理论性能保证的情况下更有效地训练这些网络。该项目将探索几何性质及其对训练多层神经网络、自编码器、生成对抗网络和涉及非凸和鞍点问题的对抗训练的优化性能的影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Deep learning has attracted a significant amount of interest in recent years due to its widespread applicability in computer vision, artificial intelligence and natural language processing, alongside recent strides in autonomous driving. The theoretical underpinnings behind such success, however, remain elusive to a large extent, hindering its further adoption in other applications. This project aims to advance the theoretical foundations of training neural networks in terms of optimization landscape and algorithmic efficacy, which in turn should have a measurable impact on the practice of deep learning by providing guiding principles for network design, algorithm selection, hyperparameter tuning, and adversarial training. This project adopts an interdisciplinary approach fusing ideas from machine learning, optimization, statistical signal processing, high-dimensional statistics, nonparametric statistics, and information theory. This project will likewise develop courses and tutorials on theoretical foundations of large-scale machine learning and provide extensive training opportunities for students at all levels.This project aims to develop a comprehensive theory to characterize the optimization landscape and geometry of loss functions and algorithmic regularizations of major neural network training problems, and explore how the network architecture---including depth, width, and activation functions---affect these properties, thus providing guidelines for the design of algorithms to train these networks more efficiently with theoretical performance guarantees. The project will explore the geometric properties and their impact on the optimization performance in training multi-layer neural networks, auto-encoders, generative adversarial networks, and adversarial training involving non-convex and saddle-point problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.