Point process models have been extensively used in many areas of science and engineering, from quantitative sociology to medical imaging. Computing the maximum likelihood estimator of a point process model often leads to a convex optimization problem displaying a challenging feature, namely the lack of Lipschitz-continuity of the objective function. This feature can be a barrier to the application of common first order convex optimization methods. We present an approach where the estimation of a point process model is framed as a saddle point problem instead. This formulation allows us to develop Mirror Prox algorithms to efficiently solve the saddle point problem. We introduce a general Mirror Prox algorithm, as well as a variant appropriate for large-scale problems, and establish worst-case complexity guarantees for both algorithms. We illustrate the performance of the proposed algorithms for point process estimation on real datasets from medical imaging, social networks, and recommender systems. [ABSTRACT FROM AUTHOR]
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