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Submodular Bregman Divergences with Applications
Rishabh K Iyer · Jeffrey A Bilmes

Mon Dec 03 07:00 PM -- 12:00 AM (PST) @ Harrah’s Special Events Center 2nd Floor

We introduce a class of discrete divergences on sets (equivalently binary vectors) that we call the submodular Bregman divergences. We consider two kinds, defined either from tight modular upper or tight modular lower bounds of a submodular function. We show that the properties of these divergences are analogous to the (standard continuous) Bregman divergence. Further, we demonstrate how they generalize many useful divergences, including the weighted Hamming distance, squared weighted Hamming, weighted precision, recall, conditional mutual information, and a generalized KL-divergence on sets. We also show that the lower bound submodular Bregman is actually a special case of the generalized Bregman divergence on the \lovasz{} extension of a submodular function which we call the \lovasz{} Bregman divergence. We then point out a number of applications of the submodular Bregman divergences, and in particular show that a proximal algorithm defined through the submodular Bregman divergences provides a framework for many mirror-descent style algorithms related to submodular function optimization. We also show that a generalization of the k-means algorithm using the \lovasz{} Bregman divergence is natural in clustering scenarios where the ordering is important. A unique property of this algorithm is that computing the mean ordering is extremely efficient unlike the other order based distance measures. \extendedv{Finally we provide a clustering framework for the submodular Bregman, and we derive fast algorithms for clustering sets of binary vectors (equivalently sets of sets).

Author Information

Rishabh K Iyer (University of Texas, Dallas)
Jeffrey A Bilmes (University of Washington, Seattle)

Jeffrey A. Bilmes is a professor at the Department of Electrical and Computer Engineering at the University of Washington, Seattle Washington. He is also an adjunct professor in Computer Science & Engineering and the department of Linguistics. Prof. Bilmes is the founder of the MELODI (MachinE Learning for Optimization and Data Interpretation) lab here in the department. Bilmes received his Ph.D. from the Computer Science Division of the department of Electrical Engineering and Computer Science, University of California in Berkeley and a masters degree from MIT. He was also a researcher at the International Computer Science Institute, and a member of the Realization group there. Prof. Bilmes is a 2001 NSF Career award winner, a 2002 CRA Digital Government Fellow, a 2008 NAE Gilbreth Lectureship award recipient, and a 2012/2013 ISCA Distinguished Lecturer. Prof. Bilmes was, along with Andrew Ng, one of the two UAI (Conference on Uncertainty in Artificial Intelligence) program chairs (2009) and then the general chair (2010). He was also a workshop chair (2011) and the tutorials chair (2014) at NIPS/NeurIPS (Neural Information Processing Systems), and is a regular senior technical chair at NeurIPS/NIPS since then. He was an action editor for JMLR (Journal of Machine Learning Research). Prof. Bilmes's primary interests lie in statistical modeling (particularly graphical model approaches) and signal processing for pattern classification, speech recognition, language processing, bioinformatics, machine learning, submodularity in combinatorial optimization and machine learning, active and semi-supervised learning, and audio/music processing. He is particularly interested in temporal graphical models (or dynamic graphical models, which includes HMMs, DBNs, and CRFs) and ways in which to design efficient algorithms for them and design their structure so that they may perform as better structured classifiers. He also has strong interests in speech-based human-computer interfaces, the statistical properties of natural objects and natural scenes, information theory and its relation to natural computation by humans and pattern recognition by machines, and computational music processing (such as human timing subtleties). He is also quite interested in high performance computing systems, computer architecture, and software techniques to reduce power consumption. Prof. Bilmes has also pioneered (starting in 2003) the development of submodularity within machine learning, and he received a best paper award at ICML 2013, a best paper award at NIPS 2013, and a best paper award at ACMBCB in 2016, all in this area. In 2014, Prof. Bilmes also received a most influential paper in 25 years award from the International Conference on Supercomputing, given to a paper on high-performance matrix optimization. Prof. Bilmes has authored the graphical models toolkit (GMTK), a dynamic graphical-model based software system widely used in speech, language, bioinformatics, and human-activity recognition.

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