  In this paper, we design a nonparametric online algorithm for estimating the
triggering functions of multivariate Hawkes processes. Unlike parametric
estimation, where evolutionary dynamics can be exploited for fast computation
of the gradient, and unlike typical function learning, where representer
theorem is readily applicable upon proper regularization of the objective
function, nonparametric estimation faces the challenges of (i) inefficient
evaluation of the gradient, (ii) lack of representer theorem, and (iii)
computationally expensive projection necessary to guarantee positivity of the
triggering functions. In this paper, we offer solutions to the above
challenges, and design an online estimation algorithm named NPOLE-MHP that
outputs estimations with a $\mathcal{O}(1/T)$ regret, and a $\mathcal{O}(1/T)$
stability. Furthermore, we design an algorithm, NPOLE-MMHP, for estimation of
multivariate marked Hawkes processes. We test the performance of NPOLE-MHP on
various synthetic and real datasets, and demonstrate, under different
evaluation metrics, that NPOLE-MHP performs as good as the optimal maximum
likelihood estimation (MLE), while having a run time as little as parametric
online algorithms.
