[Paper Review] Mixed Robust/Average Submodular Partitioning: Fast Algorithms, Guarantees, and Applications

Kai Wei, Rishabh Iyer+3 more|arXiv (Cornell University)|Oct 29, 2015

Complexity and Algorithms in Graphs30 references20 citations

TL;DR

This paper introduces scalable algorithms for mixed robust/average-case submodular partitioning, combining the strengths of robust and average-case optimization. It proposes greedy, majorization-minimization, and relaxation-based methods that achieve state-of-the-art approximation guarantees while scaling to large real-world datasets, with applications in distributed ML, clustering, and image segmentation.

ABSTRACT

We investigate two novel mixed robust/average-case submodular data partitioning problems that we collectively call Submodular Partitioning. These problems generalize purely robust instances of the problem, namely max-min submodular fair allocation (SFA) [12] and min-max submodular load balancing (SLB) [25], and also average-case instances, that is the submodular welfare problem (SWP) [26] and submodular multiway partition (SMP) [5]. While the robust versions have been studied in the theory community [11, 12, 16, 25, 26], existing work has focused on tight approximation guarantees, and the resultant algorithms are not generally scalable to large real-world applications. This is in contrast to the average case, where most of the algorithms are scalable. In the present paper, we bridge this gap, by proposing several new algorithms (including greedy, majorization-minimization, minorization-maximization, and relaxation algorithms) that not only scale to large datasets but that also achieve theoretical approximation guarantees comparable to the state-of-the-art. We moreover provide new scalable algorithms that apply to additive combinations of the robust and average-case objectives. We show that these problems have many applications in machine learning (ML), including data partitioning and load balancing for distributed ML, data clustering, and image segmentation. We empirically demonstrate the efficacy of our algorithms on real-world problems involving data partitioning for distributed optimization (of convex and deep neural network objectives), and also purely unsupervised image segmentation.

Motivation & Objective

• Address the scalability gap in robust submodular partitioning problems, which traditionally offer tight approximation guarantees but lack efficiency for large-scale applications.
• Bridge the divide between theoretically optimal but impractical robust algorithms and scalable but less robust average-case methods in submodular optimization.
• Develop new algorithms that balance robustness and average-case performance through additive combinations of objectives, enabling practical deployment in real-world ML workloads.
• Demonstrate the applicability of the proposed framework to distributed machine learning, data clustering, and image segmentation tasks.
• Provide theoretical approximation guarantees for the new algorithms while ensuring computational efficiency on large datasets.

Proposed method

• Propose a novel formulation of submodular partitioning that blends robust (e.g., max-min fairness) and average-case (e.g., submodular welfare) objectives into a unified framework.
• Design a greedy algorithm that iteratively assigns elements to partitions to maximize a combined robust/average objective, with theoretical approximation bounds.
• Introduce majorization-minimization and minorization-maximization schemes to efficiently optimize the non-convex, non-smooth submodular objectives via sequential convex approximations.
• Develop a continuous relaxation approach that transforms the discrete partitioning problem into a continuous optimization problem, solvable via projected gradient methods.
• Leverage submodularity properties to ensure convergence and approximation guarantees under the proposed algorithmic frameworks.
• Integrate the algorithms into practical pipelines for distributed ML and unsupervised image segmentation, enabling end-to-end evaluation on real-world data.

Experimental results

Research questions

• RQ1 Can we design scalable algorithms for submodular partitioning that maintain strong theoretical approximation guarantees, even when combining robust and average-case objectives?
• RQ2 How do the proposed algorithms compare in performance and scalability to existing state-of-the-art methods in both robust and average-case submodular optimization?
• RQ3 To what extent can mixed robust/average-case objectives improve fairness and utility in real-world ML applications such as distributed training and image segmentation?
• RQ4 What are the theoretical approximation bounds of the proposed algorithms under different objective combinations?
• RQ5 Can the proposed methods be effectively applied to large-scale, real-world datasets without sacrificing solution quality?

Key findings

• The proposed algorithms achieve theoretical approximation guarantees comparable to the state-of-the-art in both robust and average-case submodular optimization.
• The greedy and relaxation-based algorithms scale effectively to large datasets, enabling practical deployment in real-world applications such as distributed deep learning and image segmentation.
• The majorization-minimization and minorization-maximization approaches provide stable convergence and high-quality solutions for non-convex submodular objectives.
• Empirical evaluation shows that the mixed robust/average-case formulation improves load balancing and fairness in distributed ML workloads without sacrificing convergence speed.
• The algorithms outperform purely robust or average-case baselines in terms of solution quality and scalability on real-world data partitioning tasks.
• The framework is effective for unsupervised image segmentation, achieving competitive results while maintaining computational efficiency.