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

Block coordinate descent (BCD) methods are widely-used for large-scale numerical optimization because of their cheap iteration costs, low memory requirements, amenability to parallelization, and ability to exploit problem structure. Three main algorithmic choices influence the performance of BCD methods: the block partitioning strategy, the block selection rule, and the block update rule. In this paper we explore all three of these building blocks and propose variations for each that can lead to significantly faster BCD methods. We (i) propose new greedy block-selection strategies that guarantee more progress per iteration than the Gauss-Southwell rule; (ii) explore practical issues like how to implement the new rules when using "variable" blocks; (iii) explore the use of message-passing to compute matrix or Newton updates efficiently on huge blocks for problems with a sparse dependency between variables; and (iv) consider optimal active manifold identification, which leads to bounds on the "active set complexity" of BCD methods and leads to superlinear convergence for certain problems with sparse solutions (and in some cases finite termination at an optimal solution). We support all of our findings with numerical results for the classic machine learning problems of least squares, logistic regression, multi-class logistic regression, label propagation, and L1-regularization

Introduction: unmarried and unknown: urban men and women in the low countries since the early modern period

This essay introduces a special issue on The Lure of the City that examines the attraction of towns to unmarried men and women in the Low Countries during the early modern period and the nineteenth century. The issue has the relation between singles and cities as its main focus. Singles were present in disproportionately large numbers in urban areas, but the question is why? This essay sets out the historiographical framework for the contributions in the issue, discusses the sources and methodologies used, and provides a brief overview of the evolution of singleness in the Low Countries. The contributions all demonstrate the relevance of a comparative approach. It is revealed that towns and cities not only attracted but also created singles, that they offered different opportunities for different groups of unmarried people and that they affected men and women differently. Finally, it appears that not every town and city was attractive to men and women without a spouse