Learning Time-Varying Coverage Functions

Coverage functions are an important class of discrete functions that capture the law of diminishing returns arising naturally from applications in social network analysis, machine learning, and algorithmic game theory. In this paper, we propose a new problem of learning time-varying coverage functions, and develop a novel parametrization of these functions using random features. Based on the connection between time-varying coverage functions and counting processes, we also propose an efficient parameter learning algorithm based on likelihood maximization, and provide a sample complexity analysis. We applied our algorithm to the influence function estimation problem in information diffusion in social networks, and show that with few assumptions about the diffusion processes, our algorithm is able to estimate influence significantly more accurately than existing approaches on both synthetic and real world data.

[1]  Le Song,et al.  Scalable Influence Estimation in Continuous-Time Diffusion Networks , 2013, NIPS.

[2]  Maria-Florina Balcan,et al.  Learning submodular functions , 2010, STOC '11.

[3]  M. L. Fisher,et al.  An analysis of approximations for maximizing submodular set functions—I , 1978, Math. Program..

[4]  Andreas Krause,et al.  Near-optimal sensor placements in Gaussian processes , 2005, ICML.

[5]  Christos Faloutsos,et al.  Kronecker Graphs: An Approach to Modeling Networks , 2008, J. Mach. Learn. Res..

[6]  Le Song,et al.  Estimating time-varying networks , 2008, ISMB 2008.

[7]  Bernhard Schölkopf,et al.  Uncovering the Temporal Dynamics of Diffusion Networks , 2011, ICML.

[8]  AI Koan,et al.  Weighted Sums of Random Kitchen Sinks: Replacing minimization with randomization in learning , 2008, NIPS.

[9]  Larry Wasserman,et al.  All of Nonparametric Statistics (Springer Texts in Statistics) , 2006 .

[10]  Jure Leskovec,et al.  Meme-tracking and the dynamics of the news cycle , 2009, KDD.

[11]  K. Alexander,et al.  Rates of growth and sample moduli for weighted empirical processes indexed by sets , 1987 .

[12]  W. Wong,et al.  Probability inequalities for likelihood ratios and convergence rates of sieve MLEs , 1995 .

[13]  S. Geer.  Exponential Inequalities for Martingales, with Application to Maximum Likelihood Estimation for Counting Processes , 1995 .

[14]  Daniel Lehmann,et al.  Combinatorial auctions with decreasing marginal utilities , 2001, EC '01.

[15]  Sujay Sanghavi,et al.  Learning the graph of epidemic cascades , 2012, SIGMETRICS '12.

[16]  S. Geer.  Hellinger-Consistency of Certain Nonparametric Maximum Likelihood Estimators , 1993 .

[17]  Jan Vondrák,et al.  Optimal Bounds on Approximation of Submodular and XOS Functions by Juntas , 2013, 2013 IEEE 54th Annual Symposium on Foundations of Computer Science.

[18]  Vitaly Feldman,et al.  Optimal bounds on approximation of submodular and XOS functions by juntas , 2014, ITA.

[19]  Pravesh Kothari,et al.  Learning Coverage Functions , 2013, ArXiv.

[20]  Éva Tardos,et al.  Maximizing the Spread of Influence through a Social Network , 2015, Theory Comput..

[21]  P. Massart,et al.  Minimum contrast estimators on sieves: exponential bounds and rates of convergence , 1998 .

[22]  Le Song,et al.  Time-Varying Dynamic Bayesian Networks , 2009, NIPS.

[23]  Stergios B. Fotopoulos,et al.  All of Nonparametric Statistics , 2007, Technometrics.

[24]  Le Song,et al.  Influence Function Learning in Information Diffusion Networks , 2014, ICML.

[25]  J. Wellner,et al.  Empirical Processes with Applications to Statistics , 2009 .

[26]  O. Aalen,et al.  Survival and Event History Analysis: A Process Point of View , 2008 .

[27]  Mark W. Schmidt,et al.  Optimizing Costly Functions with Simple Constraints: A Limited-Memory Projected Quasi-Newton Algorithm , 2009, AISTATS.

[28]  Tim Roughgarden,et al.  Sketching valuation functions , 2012, SODA.