Convex relaxation of problems in data science and efficient solution methods

数据科学中问题的凸松弛及高效解决方法

• 批准号: RGPIN-2020-04096
• 负责人: Vavasis, Stephen
• 金额: $ 2.99万
• 依托单位: University of Waterloo
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=739500
• 关键词: Convex; relaxation; problems; data; science

Optimization is a fundamental ingredient of machine learning, and furthermore, mathematical tools used to analyze optimization can give some assurance that machine learning methodology gives accurate answers. The proposed research program advances the state of art in optimization in two ways, namely, better modeling of machine learning problems via optimization and better solution algorithms for the resulting optimization problems. A thread running throughout the proposed research is mathematical guarantees for the resulting methods. Among application problems to be tackled are: overlapping community detection, which has applications to social networks, studies of scientific collaboration, and brain science; clustering and nonnegative matrix factorization, two methodologies that discover hidden factors that underlie large data sets; and multiscale computational mechanics, which is the key technology to allow prediction of longevity of large scale structures (buildings, vehicles, and so on) from fundamental principles of physics. Algorithmic development includes better understanding of generalization, that is, the ability of a learning model optimized to classify "training data" to correctly classify future unseen data. Generalization is the key reason why machine learning works at all. The proposed research advances the understanding of generalization by considering "implicit regularization", that is, features of the optimization algorithm such as when to stop iterating that were not specifically designed to improve generalization but nonetheless can be proved to have this effect. The impact of this research is measured in several ways. Students and postdoctoral fellows at Waterloo will receive advanced training in novel and rigorous uses of optimization in machine learning. Within the larger framework of the optimization and machine learning communities, the research will bring newer ideas from optimization to machine learning while at the same time making optimization researchers aware of some challenges in machine learning. The application of machine learning and optimization techniques to multiscale computational mechanics will improve the capability to predict material failures in large-scale structures, particularly for novel materials such as newer fiber composites for which there is not much historical experience. Finally, with regard to society at large, machine learning is proliferating rapidly, and many observers believe that greater understanding and perhaps government regulation is necessary to address concomitant ethical and legal issues. Using mathematical rigor to improve the quality of the computations carried out by machine learning will improve its reliability, which is important to all those affected by this new technology.

优化是机器学习的基本组成部分，此外，用于分析优化的数学工具可以确保机器学习方法能够给出准确的答案。拟议的研究计划在两个方面推进了最优化的最新技术水平，即通过优化更好地建模机器学习问题，以及为所产生的优化问题提供更好的求解算法。贯穿所提出的研究的线程是所得到的方法的数学保证。其中应用问题要解决的是：重叠社区检测，它有应用到社交网络，科学合作的研究，脑科学，聚类和非负矩阵分解，两种方法，发现隐藏在大数据集的因素;多尺度计算力学是预测大型结构寿命的关键技术（建筑物、车辆等）从物理学的基本原理。数学发展包括更好地理解泛化，即优化学习模型对“训练数据”进行分类的能力，以正确分类未来不可见的数据。泛化是机器学习成功的关键原因。所提出的研究通过考虑“隐式正则化”来推进对泛化的理解，即优化算法的特征，例如何时停止迭代，这些特征不是专门为提高泛化而设计的，但可以证明具有这种效果。这项研究的影响可以从几个方面来衡量。滑铁卢大学的学生和博士后研究员将接受机器学习优化的新颖和严格用途的高级培训。在优化和机器学习社区的更大框架内，这项研究将为优化和机器学习带来更新的想法，同时使优化研究人员意识到机器学习中的一些挑战。机器学习和优化技术在多尺度计算力学中的应用将提高预测大型结构中材料失效的能力，特别是对于新型材料，如没有太多历史经验的新型纤维复合材料。最后，就整个社会而言，机器学习正在迅速扩散，许多观察家认为，有必要加强理解，也许还有政府监管，以解决随之而来的道德和法律的问题。使用数学严谨性来提高机器学习的计算质量将提高其可靠性，这对所有受这项新技术影响的人来说都很重要。