[Paper Review] Do Less, Get More: Streaming Submodular Maximization with Subsampling

Moran Feldman, Amin Karbasi+1 more|arXiv (Cornell University)|Jan 1, 2018

Complexity and Algorithms in Graphs28 citations

TL;DR

This paper introduces a novel one-pass streaming algorithm for submodular maximization that uses subsampling to drastically reduce function evaluations and memory usage while achieving tight approximation guarantees. For monotone submodular functions under a $p$-matchoid constraint, it achieves a $4p$ approximation with $O(k)$ memory and $O(km/p)$ queries per element, outperforming state-of-the-art methods by up to fifty-fold in video summarization and scaling efficiently on large datasets.

ABSTRACT

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.

Motivation & Objective

• To address the scalability bottleneck of submodular maximization in massive machine learning applications.
• To reduce the number of function evaluations and memory usage in streaming submodular optimization without sacrificing approximation quality.
• To develop a practical, one-pass algorithm that processes data in a single pass with minimal evaluation of the stream.
• To achieve tight approximation ratios under both monotone and non-monotone submodular functions with $p$-matchoid constraints.

Proposed method

• The algorithm employs a novel subsampling strategy that selectively evaluates only a fraction of the stream elements, reducing computational overhead.
• It uses a randomized selection process to maintain a core set of candidate elements that are likely to contribute to the optimal solution.
• The method integrates a $p$-matchoid constraint model to ensure feasibility while maintaining approximation guarantees.
• It dynamically adjusts the sampling rate based on the current state of the solution to balance accuracy and efficiency.
• The algorithm operates in a single pass, storing only $O(k)$ elements in memory, where $k$ is the size of the largest feasible solution.
• It achieves a $4p$ approximation ratio in expectation for monotone functions and $4p+2-o(1)$ for non-monotone functions.

Experimental results

Research questions

• RQ1 Can a one-pass streaming algorithm for submodular maximization be designed that avoids evaluating the entire stream?
• RQ2 What is the best possible approximation ratio achievable with minimal memory and function evaluations in a streaming setting?
• RQ3 How can subsampling be leveraged to reduce computational cost while preserving solution quality in submodular optimization?
• RQ4 Can the algorithm scale to massive datasets like Uber pickup locations while maintaining high utility?

Key findings

• The algorithm achieves a $4p$ approximation ratio in expectation for monotone submodular functions under a $p$-matchoid constraint.
• For non-monotone functions, the approximation ratio is $4p+2-o(1)$, with only a slight degradation compared to the monotone case.
• The algorithm uses $O(k)$ memory and requires only $O(km/p)$ function queries per element, significantly reducing computational cost.
• In video summarization, the algorithm outperforms the state-of-the-art by up to fifty-fold in runtime while maintaining comparable utility.
• The algorithm scales effectively on a large dataset of Uber pickup locations, demonstrating practical applicability to real-world machine learning workloads.