  In this paper, 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 stucking to 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 to existing methods in terms of
estimation accuracy. Managerial insights for the decision maker to use the
algorithms are also provided.
