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Deep Mathematical Properties of Submodularity with Applications to Machine Learning
Jeffrey A Bilmes

Thu Dec 05 01:00 PM -- 03:00 PM (PST) @ Emerald Bay A
Event URL: http://melodi.ee.washington.edu/~bilmes/pgs/b2hd-bilmes2013-nips-tutorial.html »

Submodular functions have received significant attention in the mathematics community owing to their natural and wide ranging applicability. Submodularity has a very simple definition which belies a treasure trove of consequent mathematical richness. This tutorial will attempt to convey some of this richness.

We will start by defining submodularity and polymatroidality --- we will survey a surprisingly diverse set of functions that are submodular and operations that (sometimes remarkably) preserve submodularity. Next, we'll define the submodular polytope, and its relationship to the greedy algorithm and its exact and efficient solution to certain linear programs with an exponential number of constraints. We will see how submodularity shares certain properties with convexity (efficient minimization, discrete separation, subdifferentials, lattices and sub-lattices, and the convexity of the Lovasz extension), concavity (via its definition, submodularity via concave functions, superdifferentials), and neither (simultaneous sub- and super-differentials, efficient approximate maximization). The Lovasz extension will be given particular attention due to its growing use for structured convex norms and surrogates in relaxation methods. We will survey both constrained and unconstrained submodular optimization (including the minimum norm point algorithm), discussing what is currently known about hardness (both upper and lower bounds), and also when algorithms or instances are practical.

As to applications, it is interesting that a submodular function itself can often be seen as a parameter to instantiate a machine-learning instance --- this includes active/semi-supervised learning, structured sparsity inducing norms, combinatorial independence and generalized entropy, and rank-order based divergences. Other examples include feature selection, data subset (or core set) selection, inference in graphical models with high tree-width and global potentials in computer vision, and influence determination in social networks.

Author Information

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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