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Submodular Streaming in All its Glory: Tight Approximation, Minimum Memory and Low Adaptive Complexity

arXiv.org Machine Learning

Streaming algorithms are generally judged by the quality of their solution, memory footprint, and computational complexity. In this paper, we study the problem of maximizing a monotone submodular function in the streaming setting with a cardinality constraint $k$. We first propose Sieve-Streaming++, which requires just one pass over the data, keeps only $O(k)$ elements and achieves the tight $(1/2)$-approximation guarantee. The best previously known streaming algorithms either achieve a suboptimal $(1/4)$-approximation with $\Theta(k)$ memory or the optimal $(1/2)$-approximation with $O(k\log k)$ memory. Next, we show that by buffering a small fraction of the stream and applying a careful filtering procedure, one can heavily reduce the number of adaptive computational rounds, thus substantially lowering the computational complexity of Sieve-Streaming++. We then generalize our results to the more challenging multi-source streaming setting. We show how one can achieve the tight $(1/2)$-approximation guarantee with $O(k)$ shared memory while minimizing not only the required rounds of computations but also the total number of communicated bits. Finally, we demonstrate the efficiency of our algorithms on real-world data summarization tasks for multi-source streams of tweets and of YouTube videos.


Non-Monotone Adaptive Submodular Maximization

AAAI Conferences

A wide range of AI problems, such as sensor placement, active learning, and network influence maximization, require sequentially selecting elements from a large set with the goal of optimizing the utility of the selected subset. Moreover, each element that is picked may provide stochastic feedback, which can be used to make smarter decisions about future selections. Finding efficient policies for this general class of adaptive optimization problems can be extremely hard. However, when the objective function is adaptive monotone and adaptive submodular, a simple greedy policy attains a 1-1/e approximation ratio in terms of expected utility. Unfortunately, many practical objective functions are naturally non-monotone; to our knowledge, no existing policy has provable performance guarantees when the assumption of adaptive monotonicity is lifted. We propose the adaptive random greedy policy for maximizing adaptive submodular functions, and prove that it retains the aforementioned 1-1/e approximation ratio for functions that are also adaptive monotone, while it additionally provides a 1/e approximation ratio for non-monotone adaptive submodular functions. We showcase the benefits of adaptivity on three real-world network data sets using two non-monotone functions, representative of two classes of commonly encountered non-monotone objectives.


Do Less, Get More: Streaming Submodular Maximization with Subsampling

Neural Information Processing Systems

In this paper, we develop the first one-pass streaming algorithm for submodular maximization that does not evaluate the entire stream even once. By carefully subsampling each element of the data stream, our algorithm enjoys the tightest approximation guarantees in various settings while having the smallest memory footprint and requiring the lowest number of function evaluations. More specifically, for a monotone submodular function and a $p$-matchoid constraint, our randomized algorithm achieves a $4p$ approximation ratio (in expectation) with $O(k)$ memory and $O(km/p)$ queries per element ($k$ is the size of the largest feasible solution and $m$ is the number of matroids used to define the constraint). For the non-monotone case, our approximation ratio increases only slightly to $4p+2-o(1)$. To the best or our knowledge, our algorithm is the first that combines the benefits of streaming and subsampling in a novel way in order to truly scale submodular maximization to massive machine learning problems. To showcase its practicality, we empirically evaluated the performance of our algorithm on a video summarization application and observed that it outperforms the state-of-the-art algorithm by up to fifty-fold while maintaining practically the same utility. We also evaluated the scalability of our algorithm on a large dataset of Uber pick up locations.


Do Less, Get More: Streaming Submodular Maximization with Subsampling

Neural Information Processing Systems

In this paper, we develop the first one-pass streaming algorithm for submodular maximization that does not evaluate the entire stream even once. By carefully subsampling each element of the data stream, our algorithm enjoys the tightest approximation guarantees in various settings while having the smallest memory footprint and requiring the lowest number of function evaluations. More specifically, for a monotone submodular function and a $p$-matchoid constraint, our randomized algorithm achieves a $4p$ approximation ratio (in expectation) with $O(k)$ memory and $O(km/p)$ queries per element ($k$ is the size of the largest feasible solution and $m$ is the number of matroids used to define the constraint). For the non-monotone case, our approximation ratio increases only slightly to $4p+2-o(1)$. To the best or our knowledge, our algorithm is the first that combines the benefits of streaming and subsampling in a novel way in order to truly scale submodular maximization to massive machine learning problems. To showcase its practicality, we empirically evaluated the performance of our algorithm on a video summarization application and observed that it outperforms the state-of-the-art algorithm by up to fifty-fold while maintaining practically the same utility. We also evaluated the scalability of our algorithm on a large dataset of Uber pick up locations.


Streaming Non-Monotone Submodular Maximization: Personalized Video Summarization on the Fly

AAAI Conferences

The need for real time analysis of rapidly producing data streams (e.g., video and image streams) motivated the design of streaming algorithms that can efficiently extract and summarize useful information from massive data "on the fly." Such problems can often be reduced to maximizing a submodular set function subject to various constraints. While efficient streaming methods have been recently developed for monotone submodular maximization, in a wide range of applications, such as video summarization, the underlying utility function is non-monotone, and there are often various constraints imposed on the optimization problem to consider privacy or personalization. We develop the first efficient single pass streaming algorithm, Streaming Local Search, that for any streaming monotone submodular maximization algorithm with approximation guarantee α under a collection of independence systems I, provides a constant 1/(1+2/√α+1/α+2d(1+√α)) approximation guarantee for maximizing a non-monotone submodular function under the intersection of I and d knapsack constraints. Our experiments show that for video summarization, our method runs more than 1700 times faster than previous work, while maintaining practically the same performance.