Functional Geometric Monitoring for Distributed Streams

We introduce Functional Geometric Monitoring (FGM), a substan- tial theoretical and practical improvement on the core ideas of Geometric Monitoring. Instead of a binary constraint, each site is provided with a non-linear function, which, applied to its local summary vector, projects it to a real number. The sites collec- tively monitor the sum of these one-dimensional projections and as long as the global sum is subzero, the monitoring bounds are guaranteed. We demonstrate that FGM is as generally applicable as Geometric Monitoring, and provides substantial benefits in terms of performance, scalability, and robustness. In addition, in FGM it is possible to prove worst-case results, under standard monotonicity assumptions on the monitoring problem. In terms of performance, the salient quality of FGM is that it can adapt naturally to adverse changes in the monitored problem, such as lack of monotonicity or very tight monitoring bounds, where no method can deliver asymptotically good performance. We provide formal proofs for many of the properties of FGM, and present an extensive empirical performance evaluation under adverse conditions, on real data.