Estimating Probability Distributions through Gradient Optimization over the Wasserstein Space

This work is dedicated to the task of estimating an absolutely continuous probability measure in context of online optimization in the Wasserstein space. Moreover, in the general statement of the considered problem it is assumed that a measure might insignificantly change in time between iterations. To achieve an effective algorithm for solving this task, the problem was restated in the optimal transport terms. The connection between the Wasserstein distance and optimal transport was used for that. During the problem statement entropical regularization was added to increase the efficiency of the algorithm. It was followed by the usage of the theorem about a duality of the optimal transport problem. After that, it became possible to use c-transforms of Kantorovich potentials. That led to the simplification of the task into a minimax optimization in finite-dimensional space. It was proven that this problem belongs to the class of Nonconvex-Concave Minimax optimization. Based on the gradient descent ascent algorithm, which is widely used in similar tasks, the algorithm for the estimation of a probability density in online optimization framework was developed. Further, this algorithms was programmed. To evaluate its quality a series of tests was held using different generated samples and true distributions. Particularly, tests were done for normal, exponential and power law distributions. Furthermore, algorithm’s behavior was tested using datasets generated by a normal distributions with monotonically changing means between iterations; and by an exponential distribution with the addition of a noise random variable into parameter’s value (each datapoint was generated with a parameter with different value of a noise random variable). Algorithm shown multiple significant results on a multiple used datasets. Keywords: estimating distribution from a sample, semidiscrete optimal transport, online learning, nonidentical data distribution.