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Block coordinate descent (BCD) methods are widely used for large-scalenumerical optimization because of their cheap iteration costs, low memoryrequirements, amenability to parallelization, and ability to exploit problemstructure. Three main algorithmic choices influence the performance of BCDmethods: the block partitioning strategy, the block selection rule, and theblock update rule. In this paper we explore all three of these building blocksand propose variations for each that can significantly improve the progressmade by each BCD iteration. We (i) propose new greedy block-selectionstrategies that guarantee more progress per iteration than the Gauss-Southwellrule; (ii) explore practical issues like how to implement the new rules whenusing "variable" blocks; (iii) explore the use of message-passing to computematrix or Newton updates efficiently on huge blocks for problems with sparsedependencies between variables; and (iv) consider optimal active manifoldidentification, which leads to bounds on the "active-set complexity" of BCDmethods and leads to superlinear convergence for certain problems with sparsesolutions (and in some cases finite termination at an optimal solution). Wesupport all of our findings with numerical results for the classic machinelearning problems of least squares, logistic regression, multi-class logisticregression, label propagation, and L1-regularization.

Let's Make Block Coordinate Descent Converge Faster: Faster Greedy Rules, Message-Passing, Active-Set Complexity, and Superlinear Convergence