Second-Order Filtering Algorithm for Streaming Optimization Problems

We consider the minimization of the sum of convex objectives with overlapping decision variables. These types of objectives occur in streaming signal reconstruction applications, where the observations of the signal occur in frames, and the loss functions for each frame are connected through overlapping basis functions. We show that a Newton-type algorithm for solving this type of problem can be extremely efficient when the coupling between the terms is “weak,” which we characterize with a certain notion of block-diagonal dominance of the Hessian. We apply the algorithm to the streaming reconstruction of the intensity function of a nonhomogeneous Poisson process and show that almost nothing is lost if the updates are performed using a finite buffer.