# Intelligent Information Gathering and Submodular Function Optimization

## Description

A key problem in AI is to develop intelligent systems and services that actively gather most relevant information. In recent years, a fundamental problem structure has emerged as extremely useful for addressing this problem: Submodularity is an intuitive diminishing returns property, stating that adding an element to a smaller set helps more than adding it to a larger set. Applications where this property is useful include active learning and experimental design, informative path planning, multi-agent coordination, structure learning, clustering, influence maximization, weblog ranking, trading off utility and privacy as well as game theoretic applications. In contrast to most existing approaches, submodularity allows to efficiently find provably (near-)optimal solutions. In this tutorial, we will give an introduction to the concept of submodularity, discuss algorithms for optimizing submodular functions and illustrate their usefulness in solving difficult AI problems, with a special focus on active information gathering tasks. Since submodularity arises in so many different areas of AI, and since information gathering is central to many AI applications, we believe that this property is both of theoretical and practical interest to a large part of the AI community. This tutorial will not require prior knowledge beyond the basic concepts covered in an introductory AI class.## Materials

- Tutorial slides [ppt, 20 MB]
- MATLAB Toolbox for submodular function optimization [link].
- Previous tutorial given at ICML 2008.

## Outline

- Introduction
- Motivating Applications
- Why should the AI community care about submodularity?
- Submodular set functions
- Definitions
- Intuition: Why are submodular functions useful
- Examples of submodular functions (entropy [8] and mutual information [10, 14], influence in graphs [9], etc.) and examples of functions which are not submodular
- Operations preserving submodularity and relationship between submodularity and convexity [16]
- Minimizing submodular functions
- Overview about known results for minimization [22, 7]
- Queyranne’s algorithm for minimizing symmetric submodular functions [22], applications to factorizing distributions and clustering [20]
- Learning structure of graphical models [19, 1]
- Maximizing submodular functions
- The Greedy algorithm [21,32]
- Lazy evaluations and scaling up to large data sets [28]
- Applications of maximizing submodular functions
- Informative path planning and multi-agent coordination [24]
- Sensor placement and scheduling [39]
- Information gathering in the presence of adversaries [13]
- Online and stochastic optimization of submodular functions [5, 11]
- Conclusions
- Current research directions / Open questions
- Other pointers (submodularity in games / allocation
problems [4,35])

## Who should attend

The main objective of this tutorial is to introduce the concept of submodular function optimization and its emerging importance in solving complex AI problems. As a special focus, we illustrate the concept on a key AI task: Intelligent gathering of most relevant information, in a variety of complex real-world problems.Since submodularity arises in so many different areas of AI, and since information gathering is central to many AI applications, we believe that this property is both of theoretical and practical interest to a large part of the AI community..

## Presenters

Andreas Krause is an assistant professor of Computer Science at the California Institute of Technology. He received his Ph.D. from Carnegie Mellon University in 2008. He is a recipient of a Microsoft Research Graduate Fellowship, and his research on sensor placement and information acquisition received awards at several major conferences (KDD '07, IPSN '06, ICML '05 and UAI '05) and the ASCE journal of Water Resource Planning and Management.Carlos Guestrin is the Finmeccanica Assistant Professor in the Machine Learning and Computer Science Departments at Carnegie Mellon University. Previously, he was a senior researcher at Intel, and received his PhD from Stanford University. Carlos' work received awards at a number of major conferences and a journal. He is also a recipient of the ONR Young Investigator Award, the NSF Career Award, the Alfred P. Sloan Fellowship, the IBM Faculty Fellowship, and was named one of the 2008 Brilliant 10 by Popular Science Magazine. Carlos is currently a member of the Information Sciences and Technology (ISAT) advisory group for DARPA.

## References

The following references are used in the tutorial. A short high level summary is given, without any claim of completeness.- [1] A. Chechetka and C. Guestrin. Efficient principled learning of thin junction trees. In In Advances in Neural Information Processing Systems (NIPS), Vancouver, Canada, December 2007 [pdf]
- Uses submodular function optimization to speed up structure learning.
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- Slightly superpolynomial approximation algorithm for submodular path planning (orienteering).
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- Variance reduction in Gaussian models is submodular under certain conditions on the covariance.
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- Generalization of the submodular welfare problem to additive set functions.
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- Online algorithm for maximizing submodular functions, with application to scheduling.
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- Strongly polynomial algorithm for minimizing arbitrary submodular functions.
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- Proves that entropy is a submodular function.
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- Maximizing influence (viral marketing) in social networks is submodular.
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- Information gain in Naive Bayes models is submodular. Also gives (1-1/e) hardness of approximation result for information gain in such models.
- [11] A. Krause and C. Guestrin. Nonmyopic active learning of gaussian processes: An exploration—exploitation approach. In ICML, 2007 [pdf]
- Using submodularity to analyze sequential experimental design in Gaussian Processes with uncertain kernel parameters.
- [12] A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg. Near-optimal sensor placements: Maximizing information while minimizing communication cost. In Proceedings of the Fifth International Symposium on Information Processing in Sensor Networks (IPSN), 2006 [pdf]
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- Maximizing the minimum over a set of monotonic submodular functions, with applications to robust experimental design.
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- Proves approximate monotonicity of mutual information for experimental design in Gaussian Processes.
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- Proves monotonic submodularity for outbreak detection in networks, with applications to sensor placement and selecting informative blogs.
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- Discusses connections between submodular and convex functions. Introduces Lovasz extension.
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- Spatio-temporal submodular path planning.
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- Uses symmetric submodular function minimization to find separators, which allows efficient structure learning.
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- Shows how Queyranne's algorithm can be used for (near-)optimal clustering.
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- (1-1/e) guarantee for the greedy algorithm for maximizing submodular functions.
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- Strongly polynomial algorithm for minimizing symmetric submodular functions.
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- Uses lazy evaluations to construct D-optimal designs.
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- An efficient approximation algorithm for submodular path planning (orienteering), with extensions for multiple robots.
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- Shows that (1-1/e) guarantee can be achieved for budgeted maximization of submodular functions.
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- Proves guarantees for the greedy algorithm for submodular coverage.
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- Introduces the lazy greedy algorithm and online bounds for maximizing submodular functions.
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- Shows how the ellipsoid algorithm can minimize submodular functions in polynomial time.
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- Proves the existence of Gomory-Hu trees for arbitrary symmetric submodular functions.
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- Analyzes greedy splitting for submodular clustering.
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- Empirical study on the performance of the minimum norm algorithm.
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- Proves (1-1/e) guarantee for a continuous greedy algorithm for maximizing submodular functions subject to arbitrary matroid constraints.
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- Studies submodular functions, proves duality results, ...
- [37] A. Krause, C. Guestrin. A Note on the Budgeted Maximization on Submodular Functions. Technical Report CMU-CALD-05-103, 2005 [pdf]
- Guarantees for submodular function maximization if function is perturbed by noise / only approximately submodular.
- [38] M. Seeger. Greedy Forward Selection in the Informative Vector Machine. Manuscript 2004 [pdf]
- Proves that the greedy algorithm used in the IVM optimizes a submodular function.
- [39] A. Krause, R. Rajagopal, A. Gupta, C. Guestrin. Simultaneous Placement and Scheduling of Sensors, IPSN 09.