This paper describes how to incorporate sampled curvature information in a Newton-CG method and in a limited memory quasi-Newton method for statistical learning. The motivation for this work stems from supervised machine learning applications involving a very large number of training points. We follow a batch approach, also known in the stochastic optimization literature as a sample average approximation (SAA) approach. Curvature information is incorporated in two sub-sampled Hessian algorithms, one based on a matrix-free inexact Newton iteration and one on a preconditioned limited memory BFGS iteration. A crucial feature of our technique is that Hessian-vector multiplications are carried out with a significantly smaller sample size than is used for the function and gradient. The efficiency of the proposed methods is illustrated using a machine learning application involving speech recognition.
The paper presents an empirical exploration of google.com query stream language modeling. We describe the normalization of the typed query stream resulting in out-of-vocabulary (OoV) rates below 1% for a one million word vocabulary. We present a comprehensive set of experiments that guided the design decisions for a voice search service. In the process we re-discovered a less known interaction between Kneser-Ney smoothing and entropy pruning, and found empirical evidence that hints at non-stationarity of the query stream, as well as strong dependence on various English locales-USA, Britain and Australia.
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