Least-mean squares (LMS) solvers such as Linear Ridge Lasso-Regression, SVD and Elastic-Net not only solve fundamental machine learning problems, but are also the building blocks in a variety of other methods, such as decision trees and matrix factorizations. We suggest an algorithm that gets a finite set of @math @math -dimensional real vectors and returns a weighted subset of @math vectors whose sum is the same. The proof in Caratheodory's Theorem (1907) computes such a subset in @math time and thus not used in practice. Our algorithm computes this subset in @math time, using @math calls to Caratheodory's construction on small but "smart" subsets. This is based on a novel paradigm of fusion between different data summarization techniques, known as sketches and coresets. As an example application, we show how it can be used to boost the performance of existing LMS solvers, such as those in scikit-learn library, up to x100. Generalization for streaming and distributed (big) data is trivial. Extensive experimental results and complete open source code are also provided.
A main advantage of a coreset over a sketch is that it preserves sparsity of the input rows @cite_35 , which usually reduces theoretical running time. Our experiments show, as expected, that coresets can also be used to significantly improve the numerical stability of existing algorithms, even if the running time is the same. Another advantage is that the same coreset can be used for parameter tuning over a large set of candidates. In addition to other reasons, this significantly reduced the running time of such algorithms in our experiments; see .
@cite_35
2963274201 2003895866 2953116466 2151683004
This paper resolves one of the longest standing basic problems in the streaming computational model. Namely, optimal construction of quantile sketches. An e approximate quantile sketch receives a stream of items x1,·,xn and allows one to approximate the rank of any query item up to additive error e n with probability at least 1-δ.The rank of a query x is the number of stream items such that xi ≤ x. The minimal sketch size required for this task is trivially at least 1 e.Felber and Ostrovsky obtain a O((1 e)log(1 e)) space sketch for a fixed δ.Without restrictions on the nature of the stream or the ratio between e and n, no better upper or lower bounds were known to date. This paper obtains an O((1 e)log log (1 δ)) space sketch and a matching lower bound. This resolves the open problem and proves a qualitative gap between randomized and deterministic quantile sketching for which an Ω((1 e)log(1 e)) lower bound is known. One of our contributions is a novel representation and modification of the widely used merge-and-reduce construction. This modification allows for an analysis which is both tight and extremely simple. The same technique was reported, in private communications, to be useful for improving other sketching objectives and geometric coreset constructions. We suggest a generic data reduction technique with provable guarantees for computing the low rank approximation of a matrix under some $ellz error, and constrained factorizations, such as the Non-negative Matrix Factorization (NMF). Our main algorithm reduces a given n x d matrix into a small, e-dependent, weighted subset C of its rows (known as a coreset), whose size is independent of both n and d. We then prove that applying existing algorithms on the resulting coreset can be turned into (1+e)-approximations for the original (large) input matrix. In particular, we provide the first linear time approximation scheme (LTAS) for the rank-one NMF. The coreset C can be computed in parallel and using only one pass over a possibly unbounded stream of row vectors. In this sense we improve the result in [4] (Best paper of STOC 2013). Moreover, since C is a subset of these rows, its construction time, as well as its sparsity (number of non-zeroes entries) and the sparsity of the resulting low rank approximation depend on the maximum sparsity of an input row, and not on the actual dimension d. In this sense, we improve the result of Libery [21](Best paper of KDD 2013) and answer affirmably, and in a more general setting, his open question of computing such a coreset. Source code is provided for reproducing the experiments and integration with existing and future algorithms. Coresets are compact representations of data sets such that models trained on a coreset are provably competitive with models trained on the full data set. As such, they have been successfully used to scale up clustering models to massive data sets. While existing approaches generally only allow for multiplicative approximation errors, we propose a novel notion of lightweight coresets that allows for both multiplicative and additive errors. We provide a single algorithm to construct lightweight coresets for k-means clustering as well as soft and hard Bregman clustering. The algorithm is substantially faster than existing constructions, embarrassingly parallel, and the resulting coresets are smaller. We further show that the proposed approach naturally generalizes to statistical k-means clustering and that, compared to existing results, it can be used to compute smaller summaries for empirical risk minimization. In extensive experiments, we demonstrate that the proposed algorithm outperforms existing data summarization strategies in practice. Sketching is a powerful dimensionality reduction tool for accelerating statistical learning algorithms. However, its applicability has been limited to a certain extent since the crucial ingredient, the so-called oblivious subspace embedding, can only be applied to data spaces with an explicit representation as the column span or row span of a matrix, while in many settings learning is done in a high-dimensional space implicitly defined by the data matrix via a kernel transformation. We propose the first fast oblivious subspace embeddings that are able to embed a space induced by a non-linear kernel without explicitly mapping the data to the high-dimensional space. In particular, we propose an embedding for mappings induced by the polynomial kernel. Using the subspace embeddings, we obtain the fastest known algorithms for computing an implicit low rank approximation of the higher-dimension mapping of the data matrix, and for computing an approximate kernel PCA of the data, as well as doing approximate kernel principal component regression.