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Constrained Robust Submodular Partitioning
Shengjie Wang · Tianyi Zhou · Chandrashekhar Lavania · Jeff A Bilmes

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In the robust submodular partitioning problem, we aim to allocate a set of items into $m$ blocks, so that the evaluation of the minimum block according to a submodular function is maximized. Robust submodular partitioning promotes the diversity of every block in the partition. It has many applications in machine learning, e.g., partitioning data for distributed training so that the gradients computed on every block are consistent. We study an extension of the robust submodular partition problem with additional constraints (e.g., cardinality, multiple matroids, and/or knapsack) on every block. For example, when partitioning data for distributed training, we can add a constraint that the number of samples of each class is the same in each partition block, ensuring data balance. We present two classes of algorithms, i.e., Min-Block Greedy based algorithms (with an $\Omega(1/m)$ bound), and Round-Robin Greedy based algorithms (with a constant bound) and show that under various constraints, they still have good approximation guarantees. Interestingly, while normally the latter runs in only weakly polynomial time, we show that using the two together yields strongly polynomial running time while preserving the approximation guarantee. Lastly, we apply the algorithms on a real-world machine learning data partitioning problem showing good results.

#### Author Information

##### Tianyi Zhou (University of Washington, Seattle)

Tianyi Zhou is a Ph.D. student in Computer Science at University of Washington and a member of MELODI lab led by Prof. Jeff A. Bilmes. He will be joining University of Maryland, College Park as a tenure-track assistant professor at the Department of Computer Science and affiliated with UMIACS in 2022. His research interests are in machine learning, optimization, and natural language processing. He has published ~60 papers at NeurIPS, ICML, ICLR, AISTATS, EMNLP, NAACL, COLING, KDD, ICDM, AAAI, IJCAI, ISIT, Machine Learning (Springer), IEEE TIP/TNNLS/TKDE, etc. He is the recipient of the Best Student Paper Award at ICDM 2013 and the 2020 IEEE TCSC Most Influential Paper Award.