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We consider the task of learning a parametric Continuous Time Markov Chain (CTMC) sequence model without examples of sequences, where the training data consists entirely of aggregate steady-state statistics. Making the problem harder, we assume that the states we wish to predict are unobserved in the training data. Specifically, given a parametric model over the transition rates of a CTMC and some known transition rates, we wish to extrapolate its steady state distribution to states that are unobserved. A technical roadblock to learn a CTMC from its steady state has been that the chain rule to compute gradients will not work over the arbitrarily long sequences necessary to reach steady state —from where the aggregate statistics are sampled. To overcome this optimization challenge, we propose ∞-SGD, a principled stochastic gradient descent method that uses randomly-stopped estimators to avoid infinite sums required by the steady state computation, while learning even when only a subset of the CTMC states can be observed. We apply ∞-SGD to a real-world testbed and synthetic experiments showcasing its accuracy, ability to extrapolate the steady state distribution to unobserved states under unobserved conditions (heavy loads, when training under light loads), and succeeding in difficult scenarios where even a tailor-made extension of existing methods fails. 

An inverted index is the basic data structure used in most current large-scale information retrieval systems. It can be modeled as a collection of sorted sequences of integers. Many compression techniques for inverted indexes have been studied in the past, with some of them reaching tremendous decompression speeds through the use of SIMD instructions available on modern CPUs. While there has been some work on query processing algorithms for Graphics Processing Units (GPUs), little of it has focused on how to efficiently access compressed index structures, and we see some potential for significant improvements in decompression speed. In this paper, we describe and implement two encoding schemes for index decompression on GPU architectures. Their format and decoding algorithm is adapted from existing CPU-based compression methods to exploit the execution model and memory hierarchy offered by GPUs. We show that our solutions, GPU-BP and GPU-VByte, achieve significant speedups over their already carefully optimized CPU counterparts.