论文标题
信息牛顿的流程：概率空间中的二阶优化方法
Information Newton's flow: second-order optimization method in probability space
论文作者
论文摘要
我们介绍了牛顿在概率空间中使用信息指标的框架，名为牛顿流的命名。这里考虑了两个信息指标，包括Fisher-Rao指标和Wasserstein-2度量。一个已知的事实是，过度阻尼的Langevin动力学对应于Kullback-Leibler（KL）Divergence的Wasserstein梯度流动。将这个事实扩展到韦顿牛顿的流动，我们得出了牛顿的Langevin Dynamics。我们提供了牛顿在一维空间和高斯家庭中的Langevin Dynamics的例子。对于数值实现，我们在仿期模型中设计采样有效的变分方法，并将内核希尔伯特空间（RKHS）重现为近似Wasserstein Newton的方向。我们还建立了纽顿提出的信息牛顿方法的收敛结果。贝叶斯采样问题中的几个数值示例已证明了该方法的有效性。
We introduce a framework for Newton's flows in probability space with information metrics, named information Newton's flows. Here two information metrics are considered, including both the Fisher-Rao metric and the Wasserstein-2 metric. A known fact is that overdamped Langevin dynamics correspond to Wasserstein gradient flows of Kullback-Leibler (KL) divergence. Extending this fact to Wasserstein Newton's flows, we derive Newton's Langevin dynamics. We provide examples of Newton's Langevin dynamics in both one-dimensional space and Gaussian families. For the numerical implementation, we design sampling efficient variational methods in affine models and reproducing kernel Hilbert space (RKHS) to approximate Wasserstein Newton's directions. We also establish convergence results of the proposed information Newton's method with approximated directions. Several numerical examples from Bayesian sampling problems are shown to demonstrate the effectiveness of the proposed method.
