Monday, August 14, 2017

Randomization or Condensation?: Linear-Cost Matrix Sketching Via Cascaded Compression Sampling / A Bootstrap Method for Error Estimation in Randomized Matrix Multiplication / Effective sketching methods for value function approximation

We are starting the week with some sketching and randomized approaches ! 

Matrix sketching is aimed at finding compact representations of a matrix while simultaneously preserving most of its properties, which is a fundamental building block in modern scientific computing. Randomized algorithms represent state-of-the-art and have attracted huge interest from the fields of machine learning, data mining, and theoretic computer science. However, it still requires the use of the entire input matrix in producing desired factorizations, which can be a major computational and memory bottleneck in truly large problems. In this paper, we uncover an interesting theoretic connection between matrix low-rank decomposition and lossy signal compression, based on which a cascaded compression sampling framework is devised to approximate an m-by-n matrix in only O(m+n) time and space. Indeed, the proposed method accesses only a small number of matrix rows and columns, which significantly improves the memory footprint. Meanwhile, by sequentially teaming two rounds of approximation procedures and upgrading the sampling strategy from a uniform probability to more sophisticated, encoding-orientated sampling, significant algorithmic boosting is achieved to uncover more granular structures in the data. Empirical results on a wide spectrum of real-world, large-scale matrices show that by taking only linear time and space, the accuracy of our method rivals those state-of-the-art randomized algorithms consuming a quadratic, O(mn), amount of resources. 

In recent years, randomized methods for numerical linear algebra have received growing interest as a general approach to large-scale problems. Typically, the essential ingredient of these methods is some form of randomized dimension reduction, which accelerates computations, but also creates random approximation error. In this way, the dimension reduction step encodes a tradeoff between cost and accuracy. However, the exact numerical relationship between cost and accuracy is typically unknown, and consequently, it may be difficult for the user to precisely know (1) how accurate a given solution is, or (2) how much computation is needed to achieve a given level of accuracy. In the current paper, we study randomized matrix multiplication (sketching) as a prototype setting for addressing these general problems. As a solution, we develop a bootstrap method for {directly estimating} the accuracy as a function of the reduced dimension (as opposed to deriving worst-case bounds on the accuracy in terms of the reduced dimension). From a computational standpoint, the proposed method does not substantially increase the cost of standard sketching methods, and this is made possible by an "extrapolation" technique. In addition, we provide both theoretical and empirical results to demonstrate the effectiveness of the proposed method.

High-dimensional representations, such as radial basis function networks or tile coding, are common choices for policy evaluation in reinforcement learning. Learning with such high-dimensional representations, however, can be expensive, particularly for matrix methods, such as least-squares temporal difference learning or quasi-Newton methods that approximate matrix step-sizes. In this work, we explore the utility of sketching for these two classes of algorithms. We highlight issues with sketching the high-dimensional features directly, which can incur significant bias. As a remedy, we demonstrate how to use sketching more sparingly, with only a left-sided sketch, that can still enable significant computational gains and the use of these matrix-based learning algorithms that are less sensitive to parameters. We empirically investigate these algorithms, in four domains with a variety of representations. Our aim is to provide insights into effective use of sketching in practice.

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