Collaborative Research: Langevin Markov Chain Monte Carlo Methods for Machine Learning

该项目的研究将重点关注机器学习和数据科学的一类特定算法。研究人员特别考虑了机器学习和数据科学中多种背景下出现的一大类马尔可夫链蒙特卡罗（MCMC）方法。该项目将在名为 Langevin MCMC 方法的子类中开发新算法。这些新算法将可扩展到高维度和大型数据集，并且比传统算法更快。可扩展性和快速收敛的特征对于贝叶斯统计推断以及机器学习的非凸随机优化方法非常重要。这些算法将允许根据大规模数据对预测机器学习模型进行高效训练和校准，并对从信息技术到计算机视觉的广泛数据驱动应用领域产生直接影响。 研究生将接受培训并参与研究。在这个项目中，PI 研究了 Langevin MCMC 方法类中的一类新算法。 这些算法可以应用于机器学习和数据科学的三种环境中。首先，它们可用于高维模型的贝叶斯（学习）推理问题，其目标是在给定参数空间上的先验分布和观测数据的可能性的情况下从后验分布中进行采样。其次，它们可用于解决随机非凸优化问题，包括深度学习中出现的挑战性问题。第三，它们出现在数据科学中的建模和近似主力算法中，例如随机梯度下降方法。通过利用随机梯度算法和 MCMC 算法之间的联系，所提出的方法产生了一类新的随机梯度算法，称为哈密顿加速随机梯度，它可以在深度学习实践中优于现有方法。该项目的第一个目标是进一步研究所提出算法的理论收敛特性，以填补当前理论与实践之间的差距，并开发可以扩展现有框架的新的可扩展算法。第二个目标是进一步研究现有的 Langevin 算法，以提供与机器学习和数据科学实践相关的非渐近严格性能保证。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

The research in this project will focus on a particular class of algorithms for machine learning and data science. In particular, the investigators consider the large class of Markov Chain Monte Carlo (MCMC) methods which arise in several contexts in machine learning and data science. The project will develop new algorithms within the subclass called Langevin MCMC methods. These new algorithms will be scalable to high dimensions and large datasets and will be faster than traditional ones. The features of scalability and fast convergence are important for use in Bayesian statistical inference as well as in non-convex stochastic optimization methods for machine learning. The algorithms will allow efficient training and calibration of predictive machine learning models from large-scale data and have a direct impact on a broad range of data-driven application areas from information technology to computer vision. Graduate students will be trained and involved in research. In this project, the PIs investigate a new class of algorithms within the class of Langevin MCMC methods. These algorithms can be applied in three contexts of machine learning and data science. First, they can be used for Bayesian (learning) inference problems with high-dimensional models, where the objective is to sample from a posterior distribution given a prior distribution on the parameter space and the likelihood of the observed data. Second, they can be used for solving stochastic non-convex optimization problems including the challenging problems arising in deep learning. Third, they arise in modeling and approximating workhorse algorithms in data science such as stochastic gradient descent methods. By leveraging out the connections between stochastic gradient algorithms and MCMC algorithms, the proposed approach results in a new class of stochastic gradient algorithms called Hamiltonian Accelerated Stochastic Gradient that can outperform existing methods in deep learning practice. A first goal of the project is to study theoretical convergence properties of the proposed algorithms further to fill out the current gap between theory and practice, as well as to develop new scalable algorithms that can extend the existing framework. A second goal is to investigate existing Langevin algorithms further to provide non-asymptotic rigorous performance guarantees relevant to machine learning and data science practice.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.