Mixed Robust/Average Submodular Partitioning: Fast Algorithms, Guarantees, and Applications to Parallel Machine Learning and Multi-Label Image Segmentation
This work addresses scalability issues in submodular partitioning for machine learning practitioners, offering incremental improvements by bridging the gap between theoretical guarantees and practical efficiency.
The paper tackles the problem of scalable algorithms for mixed robust/average-case submodular partitioning, which generalizes existing robust and average-case problems, by proposing new algorithms that achieve close to state-of-the-art theoretical guarantees and scale to large real-world applications. The result includes empirical demonstrations on data partitioning for distributed machine learning and unsupervised multi-label image segmentation, showing efficacy in these domains.
We study two mixed robust/average-case submodular partitioning problems that we collectively call Submodular Partitioning. These problems generalize both purely robust instances of the problem (namely max-min submodular fair allocation (SFA) and min-max submodular load balancing (SLB) and also generalize average-case instances (that is the submodular welfare problem (SWP) and submodular multiway partition (SMP). While the robust versions have been studied in the theory community, existing work has focused on tight approximation guarantees, and the resultant algorithms are not, in general, scalable to very 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 those based on greedy, majorization-minimization, minorization-maximization, and relaxation algorithms) that not only scale to large sizes but that also achieve theoretical approximation guarantees close to the state-of-the-art, and in some cases achieve new tight bounds. We also provide new scalable algorithms that apply to additive combinations of the robust and average-case extreme objectives. We show that these problems have many applications in machine learning (ML). This includes: 1) data partitioning and load balancing for distributed machine algorithms on parallel machines; 2) data clustering; and 3) multi-label image segmentation with (only) Boolean submodular functions via pixel partitioning. We empirically demonstrate the efficacy of our algorithms on real-world problems involving data partitioning for distributed optimization of standard machine learning objectives (including both convex and deep neural network objectives), and also on purely unsupervised (i.e., no supervised or semi-supervised learning, and no interactive segmentation) image segmentation.