Online non-convex learning for river pollution source identification

In this article, novel gradient-based online learning algorithms are developed to investigate an important environmental application: real-time river pollution source identification, which aims at estimating the released mass, location, and time of a river pollution source based on downstream sensor data monitoring the pollution concentration. The pollution is assumed to be instantaneously released once. The problem can be formulated as a non-convex loss minimization problem in statistical learning, and our online algorithms have vectorized and adaptive step sizes to ensure high estimation accuracy in three dimensions which have different magnitudes. In order to keep the algorithm from sticking in the saddle points of non-convex loss, the "escaping from saddle points" module and multi-start setting are derived to further improve the estimation accuracy by searching for the global minimizer of the loss functions. This can be shown theoretically and experimentally as the O(N) local regret of the algorithms and the high probability cumulative regret bound O(N) under a particular error bound condition in loss functions. A real-life river pollution source identification example shows the superior performance of our algorithms compared with existing methods in terms of estimation accuracy. Managerial insights for the decision maker to use the algorithms are also provided.