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This paper presents a methodology for selecting the mini-batch size that minimizes Stochastic Gradient Descent (SGD) learning time for single and multiple learner problems. By decoupling algorithmic analysis issues from hardware and software implementation details, we reveal a robust empirical inverse law between mini-batch size and the average number of SGD updates required to converge to a specified error threshold. Combining this empirical inverse law with measured system performance, we create an accurate, closed-form model of average training time and show how this model can be used to identify quantifiable implications for both algorithmic and hardware aspects of machine learning. We demonstrate the inverse law empirically, on both image recognition (MNIST, CIFAR10 and CIFAR100) and machine translation (Europarl) tasks, and provide a theoretic justification via proving a novel bound on mini-batch SGD training.

We present magneto-hydrodynamic simulation results for heterogeneous systems. Heterogeneous architectures combine high floating point performance many-core units hosted in conventional server nodes. Examples include Graphics Processing Units (GPU's) and Cell. They have potentially large gains in performance, at modest power and monetary cost. We implemented a magneto-hydrodynamic (MHD) simulation code on a variety of heterogeneous and multi-core architectures --- multi-core x86, Cell, Nvidia and ATI GPU --- in different languages, FORTRAN, C, Cell, CUDA and OpenCL. We present initial performance results for these systems. To our knowledge, this is the widest comparison of heterogeneous systems for MHD simulations. We review the different challenges faced in each architecture, and potential bottlenecks. We conclude that substantial gains in performance over traditional systems are possible, and in particular that is possible to extract a greater percentage of peak theoretical performance from some systems when compared to x86 architectures., Comment: 8 pages, 2 figures