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Submodular Maximization beyond Non-negativity: Guarantees, Fast Algorithms, and Applications
Christopher Harshaw · Moran Feldman · Justin Ward · Amin Karbasi

Thu Jun 13 06:30 PM -- 09:00 PM (PDT) @ Pacific Ballroom #139
It is generally believed that submodular functions--and the more general class of $\gamma$-weakly submodular functions--may only be optimized under the non-negativity assumption $f(S) \geq 0$. In this paper, we show that once the function is expressed as the difference $f = g - c$, where $g$ is monotone, non-negative, and $\gamma$-weakly submodular and $c$ is non-negative modular, then strong approximation guarantees may be obtained. We present an algorithm for maximizing $g - c$ under a $k$-cardinality constraint which produces a random feasible set $S$ such that $\mathbb{E}[g(S) -c(S)] \geq (1 - e^{-\gamma} - \epsilon) g(\opt) - c(\opt)$, whose running time is $O (\frac{n}{\epsilon} \log^2 \frac{1}{\epsilon})$, independent of $k$. We extend these results to the unconstrained setting by describing an algorithm with the same approximation guarantees and faster $O(n \frac{1}{\epsilon} \log\frac{1}{\epsilon})$ runtime. The main techniques underlying our algorithms are two-fold: the use of a surrogate objective which varies the relative importance between $g$ and $c$ throughout the algorithm, and a geometric sweep over possible $\gamma$ values. Our algorithmic guarantees are complemented by a hardness result showing that no polynomial-time algorithm which accesses $g$ through a value oracle can do better. We empirically demonstrate the success of our algorithms by applying them to experimental design on the Boston Housing dataset and directed vertex cover on the Email EU dataset.

Author Information

Chris Harshaw (Yale University)
Moran Feldman (The Open University of Israel)
Justin Ward (Queen Mary University of London)
Amin Karbasi (Yale)
Amin Karbasi

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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