Online non-convex learning for river pollution source identification

Abstract

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.

Keywords:

• Online learning
• non-convex optimization
• gradient descent
• pollution source identification

Acknowledgments

The authors are grateful to Professor William B. Haskell and Professor Yao Cheng for their valuable comments and suggestions during the preparation of the manuscript. The authors are thankful to the anonymous referees for their careful reading and constructive comments of the manuscript.

Additional information

Funding

This research was supported by the National Research Foundation, Prime Minister’s Office, Singapore, under its Campus for Research Excellence and Technological Enterprise (CREATE) program, National Natural Science Foundation of China (Grants 72150002), and “HKU-100 Scholars” Research Start-up Funds.

Notes on contributors

Wenjie Huang

Dr. Wenjie Huang received PhD degree from the Department of Industrial Systems Engineering and Management, National University of Singapore in 2019. Currently, he is Research Assistant Professor in Department of Industrial and Manufacturing Systems Engineering, and Musketeers Foundation Institute of Data Science, The University of Hong Kong (HKU). Before joining HKU, he held joint postdoc positions at School of Data Science, The Chinese University of Hong Kong, Shenzhen and Group for Research in Decision Analysis (GERAD), Canada. His research interests are quantitative methodologies in decision-making under uncertainty, data-driven decision-making and sequential decision-making, with applications in smart society and operations management.

Jing Jiang

Dr. Jing Jiang received the PhD degree in Industry Engineering from Shanghai Jiao Tong University, Shanghai, PR China, in 2020. Currently, she is a logistics network design and optimization engineer at JD.com.

Xiao Liu

Dr. Xiao Liu received the MSc degree in system engineering from the Northeastern University, China, in 1999, and the Ph.D. degree in industry system from the Université de technologie de Troyes (UTT), Troyes, France, in 2004. Currently, she is Professor in Industrial Engineering at Shanghai Jiao Tong University. Her research focuses on systems modelling and analysis, environ-mental and energy infrastructures resilient design method and optimisation, risk assessment, prediction, and robust optimisation. She has authored or coauthored more than one hundred technical papers, and currently serves on more than ten international research Journals.