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# Efficient greedy coordinate descent via variable partitioning

*Proceedings of the Thirty-Seventh Conference on Uncertainty in Artificial Intelligence*, PMLR 161:547-557, 2021.

#### Abstract

Greedy coordinate descent (GCD) is an efficient optimization algorithm for a wide range of machine learning and data mining applications. GCD could be significantly faster than randomized coordinate descent (RCD) if they have similar per iteration cost. Nevertheless, in some cases, the greedy rule used in GCD cannot be efficiently implemented, leading to huge per iteration cost and making GCD slower than RCD. To alleviate the cost per iteration, the existing solutions rely on maximum inner product search (MIPS) as an approximate greedy rule. But it has been empirically shown that GCD with approximate greedy rule could suffer from slow convergence even with the state-of-the-art MIPS algorithms. We propose a hybrid coordinate descent algorithm with a simple variable partition strategy to tackle the cases when greedy rule cannot be implemented efficiently. The convergence rate and theoretical properties of the new algorithm are presented. The proposed method is shown to be especially useful when the data matrix has a group structure. Numerical experiments with both synthetic and real-world data demonstrate that our new algorithm is competitive against RCD, GCD, approximate GCD with MIPS and their accelerated variants.