We consider the population Wasserstein barycenter problem for random probability measures supported on a finite set of points and generated by an online stream of data. This leads to a complicated stochastic optimization problem where the objective is given as an expectation of a function given as a solution to a random optimization problem. We employ the structure of the problem and obtain a convex-concave stochastic saddle-point reformulation of this problem. In the setting when the distribution of random probability measures is discrete, we propose a stochastic optimization algorithm and estimate its complexity. The second result, based on kernel methods, extends the previous one to the arbitrary distribution of random probability measures. Moreover, this new algorithm has a total complexity better than the Stochastic Approximation approach combined with the Sinkhorn algorithm in many cases. We also illustrate our developments by a series of numerical experiments.

翻译：我们认为人口瓦森斯坦温温热点问题,即随机概率测量的瓦塞斯汀温温热中心问题,它由一组有限的点支持,由在线数据流生成。这导致一个复杂的随机优化问题,即目标被设定为对一个函数的预期,作为随机优化问题的解决办法。我们采用问题的结构,并获得一个对该问题的随机概率测量分布离散时,我们建议一种随机优化算法,并估计其复杂性。第二个结果以内核方法为基础,将前一个结果扩大到随机概率测量的任意分布。此外,这一新的算法比许多情况下与辛克霍恩算法相结合的斯托切斯特吸附方法复杂得多。我们还通过一系列数字实验来说明我们的发展情况。