Preconditioned Data Sparsification for Big Data with Applications to PCA and K-means

We analyze a compression scheme for large data sets that randomly keeps a small percentage of the components of each data sample. The benefit is that the output is a sparse matrix and therefore subsequent processing, such as PCA or K-means, is significantly faster, especially in a distributed-data setting. Furthermore, the sampling is single-pass and applicable to streaming data. The sampling mechanism is a variant of previous methods proposed in the literature combined with a randomized preconditioning to smooth the data. We provide guarantees for PCA in terms of the covariance matrix, and guarantees for K-means in terms of the error in the center estimators at a given step. We present numerical evidence to show both that our bounds are nearly tight and that our algorithms provide a real benefit when applied to standard test data sets, as well as providing certain benefits over related sampling approaches.Comment: 28 pages, 10 figure

Efficient Benchmarking of Algorithm Configuration Procedures via Model-Based Surrogates

The optimization of algorithm (hyper-)parameters is crucial for achieving peak performance across a wide range of domains, ranging from deep neural networks to solvers for hard combinatorial problems. The resulting algorithm configuration (AC) problem has attracted much attention from the machine learning community. However, the proper evaluation of new AC procedures is hindered by two key hurdles. First, AC benchmarks are hard to set up. Second and even more significantly, they are computationally expensive: a single run of an AC procedure involves many costly runs of the target algorithm whose performance is to be optimized in a given AC benchmark scenario. One common workaround is to optimize cheap-to-evaluate artificial benchmark functions (e.g., Branin) instead of actual algorithms; however, these have different properties than realistic AC problems. Here, we propose an alternative benchmarking approach that is similarly cheap to evaluate but much closer to the original AC problem: replacing expensive benchmarks by surrogate benchmarks constructed from AC benchmarks. These surrogate benchmarks approximate the response surface corresponding to true target algorithm performance using a regression model, and the original and surrogate benchmark share the same (hyper-)parameter space. In our experiments, we construct and evaluate surrogate benchmarks for hyperparameter optimization as well as for AC problems that involve performance optimization of solvers for hard combinatorial problems, drawing training data from the runs of existing AC procedures. We show that our surrogate benchmarks capture overall important characteristics of the AC scenarios, such as high- and low-performing regions, from which they were derived, while being much easier to use and orders of magnitude cheaper to evaluate

Spatial Random Sampling: A Structure-Preserving Data Sketching Tool

Random column sampling is not guaranteed to yield data sketches that preserve the underlying structures of the data and may not sample sufficiently from less-populated data clusters. Also, adaptive sampling can often provide accurate low rank approximations, yet may fall short of producing descriptive data sketches, especially when the cluster centers are linearly dependent. Motivated by that, this paper introduces a novel randomized column sampling tool dubbed Spatial Random Sampling (SRS), in which data points are sampled based on their proximity to randomly sampled points on the unit sphere. The most compelling feature of SRS is that the corresponding probability of sampling from a given data cluster is proportional to the surface area the cluster occupies on the unit sphere, independently from the size of the cluster population. Although it is fully randomized, SRS is shown to provide descriptive and balanced data representations. The proposed idea addresses a pressing need in data science and holds potential to inspire many novel approaches for analysis of big data

Randomized Dimensionality Reduction for k-means Clustering

We study the topic of dimensionality reduction for $k$-means clustering. Dimensionality reduction encompasses the union of two approaches: \emph{feature selection} and \emph{feature extraction}. A feature selection based algorithm for $k$-means clustering selects a small subset of the input features and then applies $k$-means clustering on the selected features. A feature extraction based algorithm for $k$-means clustering constructs a small set of new artificial features and then applies $k$-means clustering on the constructed features. Despite the significance of $k$-means clustering as well as the wealth of heuristic methods addressing it, provably accurate feature selection methods for $k$-means clustering are not known. On the other hand, two provably accurate feature extraction methods for $k$-means clustering are known in the literature; one is based on random projections and the other is based on the singular value decomposition (SVD). This paper makes further progress towards a better understanding of dimensionality reduction for $k$-means clustering. Namely, we present the first provably accurate feature selection method for $k$-means clustering and, in addition, we present two feature extraction methods. The first feature extraction method is based on random projections and it improves upon the existing results in terms of time complexity and number of features needed to be extracted. The second feature extraction method is based on fast approximate SVD factorizations and it also improves upon the existing results in terms of time complexity. The proposed algorithms are randomized and provide constant-factor approximation guarantees with respect to the optimal $k$-means objective value.Comment: IEEE Transactions on Information Theory, to appea

Space-Efficient Parallel Algorithms for Combinatorial Search Problems

We present space-efficient parallel strategies for two fundamental combinatorial search problems, namely, backtrack search and branch-and-bound, both involving the visit of an $n$-node tree of height $h$ under the assumption that a node can be accessed only through its father or its children. For both problems we propose efficient algorithms that run on a $p$-processor distributed-memory machine. For backtrack search, we give a deterministic algorithm running in $O(n/p+h\log p)$ time, and a Las Vegas algorithm requiring optimal $O(n/p+h)$ time, with high probability. Building on the backtrack search algorithm, we also derive a Las Vegas algorithm for branch-and-bound which runs in $O((n/p+h\log p \log n)h\log^2 n)$ time, with high probability. A remarkable feature of our algorithms is the use of only constant space per processor, which constitutes a significant improvement upon previous algorithms whose space requirements per processor depend on the (possibly huge) tree to be explored.Comment: Extended version of the paper in the Proc. of 38th International Symposium on Mathematical Foundations of Computer Science (MFCS