Point Process Estimation with Mirror Prox Algorithms

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.