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Stochastic Submodular Maximization: The Case of Coverage Functions

Neural Information Processing Systems

Stochastic optimization of continuous objectives is at the heart of modern machine learning. However, many important problems are of discrete nature and often involve submodular objectives. We seek to unleash the power of stochastic continuous optimization, namely stochastic gradient descent and its variants, to such discrete problems. We first introduce the problem of stochastic submodular optimization, where one needs to optimize a submodular objective which is given as an expectation. Our model captures situations where the discrete objective arises as an empirical risk (e.g., in the case of exemplar-based clustering), or is given as an explicit stochastic model (e.g., in the case of influence maximization in social networks). By exploiting that common extensions act linearly on the class of submodular functions, we employ projected stochastic gradient ascent and its variants in the continuous domain, and perform rounding to obtain discrete solutions. We focus on the rich and widely used family of weighted coverage functions. We show that our approach yields solutions that are guaranteed to match the optimal approximation guarantees, while reducing the computational cost by several orders of magnitude, as we demonstrate empirically.


Non-monotone Submodular Maximization in Exponentially Fewer Iterations

Neural Information Processing Systems

In this paper we consider parallelization for applications whose objective can be expressed as maximizing a non-monotone submodular function under a cardinality constraint. Our main result is an algorithm whose approximation is arbitrarily close to 1/2e in O(log^2 n) adaptive rounds, where n is the size of the ground set. This is an exponential speedup in parallel running time over any previously studied algorithm for constrained non-monotone submodular maximization. Beyond its provable guarantees, the algorithm performs well in practice. Specifically, experiments on traffic monitoring and personalized data summarization applications show that the algorithm finds solutions whose values are competitive with state-of-the-art algorithms while running in exponentially fewer parallel iterations.


Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach

arXiv.org Machine Learning

We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm's memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance.


Streaming Robust Submodular Maximization: A Partitioned Thresholding Approach

Neural Information Processing Systems

We study the classical problem of maximizing a monotone submodular function subject to a cardinality constraint k, with two additional twists: (i) elements arrive in a streaming fashion, and (ii) m items from the algorithm’s memory are removed after the stream is finished. We develop a robust submodular algorithm STAR-T. It is based on a novel partitioning structure and an exponentially decreasing thresholding rule. STAR-T makes one pass over the data and retains a short but robust summary. We show that after the removal of any m elements from the obtained summary, a simple greedy algorithm STAR-T-GREEDY that runs on the remaining elements achieves a constant-factor approximation guarantee. In two different data summarization tasks, we demonstrate that it matches or outperforms existing greedy and streaming methods, even if they are allowed the benefit of knowing the removed subset in advance.


An Optimal Algorithm for Online Unconstrained Submodular Maximization

arXiv.org Machine Learning

We consider a basic problem at the interface of two fundamental fields: submodular optimization and online learning. In the online unconstrained submodular maximization (online USM) problem, there is a universe $[n]=\{1,2,...,n\}$ and a sequence of $T$ nonnegative (not necessarily monotone) submodular functions arrive over time. The goal is to design a computationally efficient online algorithm, which chooses a subset of $[n]$ at each time step as a function only of the past, such that the accumulated value of the chosen subsets is as close as possible to the maximum total value of a fixed subset in hindsight. Our main result is a polynomial-time no-$1/2$-regret algorithm for this problem, meaning that for every sequence of nonnegative submodular functions, the algorithm's expected total value is at least $1/2$ times that of the best subset in hindsight, up to an error term sublinear in $T$. The factor of $1/2$ cannot be improved upon by any polynomial-time online algorithm when the submodular functions are presented as value oracles. Previous work on the offline problem implies that picking a subset uniformly at random in each time step achieves zero $1/4$-regret. A byproduct of our techniques is an explicit subroutine for the two-experts problem that has an unusually strong regret guarantee: the total value of its choices is comparable to twice the total value of either expert on rounds it did not pick that expert. This subroutine may be of independent interest.