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Regularized Submodular Maximization at Scale

Ehsan Kazemi · shervin minaee · Moran Feldman · Amin Karbasi

Tue Jul 20 06:20 PM -- 06:25 PM (PDT) @

in Optimization and Algorithms 2 »

In this paper, we propose scalable methods for maximizing a regularized submodular function $f \triangleq g-\ell$ expressed as the difference between a monotone submodular function $g$ and a modular function $\ell$. Submodularity is inherently related to the notions of diversity, coverage, and representativeness. In particular, finding the mode (i.e., the most likely configuration) of many popular probabilistic models of diversity, such as determinantal point processes and strongly log-concave distributions, involves maximization of (regularized) submodular functions. Since a regularized function $f$ can potentially take on negative values, the classic theory of submodular maximization, which heavily relies on the non-negativity assumption of submodular functions, is not applicable. To circumvent this challenge, we develop the first one-pass streaming algorithm for maximizing a regularized submodular function subject to a $k$-cardinality constraint. Furthermore, we develop the first distributed algorithm that returns a solution $S$ in $O(1/ \epsilon)$ rounds of MapReduce computation. We highlight that our result, even for the unregularized case where the modular term $\ell$ is zero, improves the memory and communication complexity of the state-of-the-art by a factor of $O(1/ \epsilon)$ while arguably provides a simpler distributed algorithm and a unifying analysis. We empirically study the performance of our scalable methods on a set of real-life applications, including finding the mode of negatively correlated distributions, vertex cover of social networks, and several data summarization tasks.

Keywords: Combinatorial Optimization

Author Information

Ehsan Kazemi (Google)

shervin minaee (Snap)

Moran Feldman (University of Haifa)

Amin Karbasi (Yale)

Amin Karbasi is currently an assistant professor of Electrical Engineering, Computer Science, and Statistics at Yale University. He has been the recipient of the National Science Foundation (NSF) Career Award 2019, Office of Naval Research (ONR) Young Investigator Award 2019, Air Force Office of Scientific Research (AFOSR) Young Investigator Award 2018, DARPA Young Faculty Award 2016, National Academy of Engineering Grainger Award 2017, Amazon Research Award 2018, Google Faculty Research Award 2016, Microsoft Azure Research Award 2016, Simons Research Fellowship 2017, and ETH Research Fellowship 2013. His work has also been recognized with a number of paper awards, including Medical Image Computing and Computer Assisted Interventions Conference (MICCAI) 2017, International Conference on Artificial Intelligence and Statistics (AISTAT) 2015, IEEE ComSoc Data Storage 2013, International Conference on Acoustics, Speech, and Signal Processing (ICASSP) 2011, ACM SIGMETRICS 2010, and IEEE International Symposium on Information Theory (ISIT) 2010 (runner-up). His Ph.D. thesis received the Patrick Denantes Memorial Prize 2013 from the School of Computer and Communication Sciences at EPFL, Switzerland.

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