论文标题
关于Sinkhorn算法的收敛速率
On the Convergence Rate of Sinkhorn's Algorithm
论文作者
论文摘要
我们研究了Sinkhorn的算法，用于解决熵正规化的最佳运输问题。它的迭代$π_{t} $显示满足$ h（π_{t} |π_ {*}）+h（π_{*} |π_{t}）= o（t^{ - 1}）$ h $表示相对入口和$π_{*} $ cou（这适用于大量的成本功能和边际，包括次要边缘的二次成本。我们还获得了双次优率的费率$ o（t^{ - 1}）$，边缘熵获得$ o（t^{ - 2}）$。更确切地说，我们得出了非反应性界限，并且与以前的线性收敛结果相反，该结果仅限于有限成本，我们的估计值不会因正则化参数而呈指数级别。我们还获得了$π_ {*} $作为边际函数的稳定性结果，该函数在相对熵中量化。
We study Sinkhorn's algorithm for solving the entropically regularized optimal transport problem. Its iterate $π_{t}$ is shown to satisfy $H(π_{t}|π_{*})+H(π_{*}|π_{t})=O(t^{-1})$ where $H$ denotes relative entropy and $π_{*}$ the optimal coupling. This holds for a large class of cost functions and marginals, including quadratic cost with subgaussian marginals. We also obtain the rate $O(t^{-1})$ for the dual suboptimality and $O(t^{-2})$ for the marginal entropies. More precisely, we derive non-asymptotic bounds, and in contrast to previous results on linear convergence that are limited to bounded costs, our estimates do not deteriorate exponentially with the regularization parameter. We also obtain a stability result for $π_{*}$ as a function of the marginals, quantified in relative entropy.
