Proceedings of the 20th International Conference on Artificial Intelligence and Statistics, PMLR 54:794-802, 2017.
In this paper, we describe an algorithm for estimating the provenance of hacks on websites. That is, given properties of sites and the temporal occurrence of attacks, we are able to attribute individual attacks to joint causes and vulnerabilities, as well as estimating the evolution of these vulnerabilities over time. Specifically, we use hazard regression with a time-varying additive hazard function parameterized in a generalized linear form. The activation coefficients on each feature are continuous-time functions over time. We formulate the problem of learning these functions as a constrained variational maximum likelihood estimation problem with total variation penalty and show that the optimal solution is a 0th order spline (a piecewise constant function) with a finite number of adaptively chosen knots. This allows the inference problem to be solved efficiently and at scale by solving a finite dimensional optimization problem. Extensive experiments on real data sets show that our method significantly outperforms Cox’s proportional hazard model. We also conduct case studies and verify that the fitted functions are indeed recovering vulnerable features.