We propose a new random pruning method (called "submodular sparsification (SS)") to reduce the cost of submodular maximization. The pruning is applied via a "submodularity graph" over the @math ground elements, where each directed edge is associated with a pairwise dependency defined by the submodular function. In each step, SS prunes a @math (for @math ) fraction of the nodes using weights on edges computed based on only a small number ( @math ) of randomly sampled nodes. The algorithm requires @math steps with a small and highly parallelizable per-step computation. An accuracy-speed tradeoff parameter @math , set as @math , leads to a fast shrink rate @math and small iteration complexity @math . Analysis shows that w.h.p., the greedy algorithm on the pruned set of size @math can achieve a guarantee similar to that of processing the original dataset. In news and video summarization tasks, SS is able to substantially reduce both computational costs and memory usage, while maintaining (or even slightly exceeding) the quality of the original (and much more costly) greedy algorithm.
Motivated by extremely large-scale machine learning problems, we introduce a new multi-stage algorithmic framework for submodular maximization (called MULTGREED), where at each stage we apply an approximate greedy procedure to maximize surrogate submodular functions. The surrogates serve as proxies for a target sub-modular function but require less memory and are easy to evaluate. We theoretically analyze the performance guarantee of the multi-stage framework and give examples on how to design instances of MULTGREED for a broad range of natural submodular functions. We show that MULTGREED performs very closely to the standard greedy algorithm given appropriate surrogate functions and argue how our framework can easily be integrated with distributive algorithms for further optimization. We complement our theory by empirically evaluating on several real-world problems, including data subset selection on millions of speech samples where MULTGREED yields at least a thousand times speedup and superior results over the state-of-the-art selection methods. Submodular function maximization has been studied extensively in recent years under various constraints and models. The problem plays a major role in various disciplines. We study a natural online variant of this problem in which elements arrive one-by-one and the algorithm has to maintain a solution obeying certain constraints at all times. Upon arrival of an element, the algorithm has to decide whether to accept the element into its solution and may preempt previously chosen elements. The goal is to maximize a submodular function over the set of elements in the solution. We study two special cases of this general problem and derive upper and lower bounds on the competitive ratio. Specifically, we design a @math -competitive algorithm for the unconstrained case in which the algorithm may hold any subset of the elements, and constant competitive ratio algorithms for the case where the algorithm may hold at most @math elements in its solution. Is it possible to maximize a monotone submodular function faster than the widely used lazy greedy algorithm (also known as accelerated greedy), both in theory and practice? In this paper, we develop the first linear-time algorithm for maximizing a general monotone submodular function subject to a cardinality constraint. We show that our randomized algorithm, STOCHASTIC-GREEDY, can achieve a @math approximation guarantee, in expectation, to the optimum solution in time linear in the size of the data and independent of the cardinality constraint. We empirically demonstrate the effectiveness of our algorithm on submodular functions arising in data summarization, including training large-scale kernel methods, exemplar-based clustering, and sensor placement. We observe that STOCHASTIC-GREEDY practically achieves the same utility value as lazy greedy but runs much faster. More surprisingly, we observe that in many practical scenarios STOCHASTIC-GREEDY does not evaluate the whole fraction of data points even once and still achieves indistinguishable results compared to lazy greedy. 