We introduce a new non parametric method that allows for a direct, fast and efficient estimation of the matrix of kernel norms of a multivariate Hawkes process, also called branching ratio matrix. We demonstrate the capabilities of this method by applying it to high-frequency order book data from the EUREX exchange. We show that it is able to uncover (or recover) various relationships between all the first level order book events associated with some asset when mapped to a 12-dimensional process. We then scale up the model so as to account for events on two assets simultaneously and we discuss the joint high-frequency dynamics.

We introduce a doubly stochastic proximal gradient algorithm for optimizing a finite average of smooth convex functions, whose gradients depend on numerically expensive expectations. Indeed, the effectiveness of SGD-like algorithms relies on the assumption that the computation of a subfunction's gradient is cheap compared to the computation of the total function's gradient. This is true in the Empirical Risk Minimization (ERM) setting, but can be false when each subfunction depends on a sequence of examples. Our main motivation is the acceleration of the optimization of the regularized Cox partial-likelihood (the core model in survival analysis), but other settings can be considered as well.The proposed algorithm is doubly stochastic in the sense that gradient steps are done using stochastic gradient descent (SGD) with variance reduction, and the inner expectations are approximated by a Monte-Carlo Markov-Chain (MCMC) algorithm. We derive conditions on the MCMC number of iterations guaranteeing convergence, and obtain a linear rate of convergence under strong convexity and a sublinear rate without this assumption.We illustrate the fact that our algorithm improves the state-of-the-art solver for regularized Cox partial-likelihood on several datasets from survival analysis.

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