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Register a new userStreaming Kernel PCA with $tilde{O}(sqrt{n})$ Random Features
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Added by
Enayat Ullah
Publication date
2018
fields
Informatics Engineering
and research's language is
English
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We study the statistical and computational aspects of kernel principal component analysis using random Fourier features and show that under mild assumptions, $O(sqrt{n} log n)$ features suffices to achieve $O(1/epsilon^2)$ sample complexity. Furthermore, we give a memory efficient streaming algorithm based on classical Ojas algorithm that achieves this rate.
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The Neural Tangent Kernel (NTK) has discovered connections between deep neural networks and kernel methods with insights of optimization and generalization. Motivated by this, recent works report that NTK can achieve better performances compared to training neural networks on small-scale datasets. However, results under large-scale settings are hardly studied due to the computational limitation of kernel methods. In this work, we propose an efficient feature map construction of the NTK of fully-connected ReLU network which enables us to apply it to large-scale datasets. We combine random features of the arc-cosine kernels with a sketching-based algorithm which can run in linear with respect to both the number of data points and input dimension. We show that dimension of the resulting features is much smaller than other baseline feature map constructions to achieve comparable error bounds both in theory and practice. We additionally utilize the leverage score based sampling for improved bounds of arc-cosine random features and prove a spectral approximation guarantee of the proposed feature map to the NTK matrix of two-layer neural network. We benchmark a variety of machine learning tasks to demonstrate the superiority of the proposed scheme. In particular, our algorithm can run tens of magnitude faster than the exact kernel methods for large-scale settings without performance loss.
Analysis of large-scale sequential data has been one of the most crucial tasks in areas such as bioinformatics, text, and audio mining. Existing string kernels, however, either (i) rely on local features of short substructures in the string, which hardly capture long discriminative patterns, (ii) sum over too many substructures, such as all possible subsequences, which leads to diagonal dominance of the kernel matrix, or (iii) rely on non-positive-definite similarity measures derived from the edit distance. Furthermore, while there have been works addressing the computational challenge with respect to the length of string, most of them still experience quadratic complexity in terms of the number of training samples when used in a kernel-based classifier. In this paper, we present a new class of global string kernels that aims to (i) discover global properties hidden in the strings through global alignments, (ii) maintain positive-definiteness of the kernel, without introducing a diagonal dominant kernel matrix, and (iii) have a training cost linear with respect to not only the length of the string but also the number of training string samples. To this end, the proposed kernels are explicitly defined through a series of different random feature maps, each corresponding to a distribution of random strings. We show that kernels defined this way are always positive-definite, and exhibit computational benefits as they always produce emph{Random String Embeddings (RSE)} that can be directly used in any linear classification models. Our extensive experiments on nine benchmark datasets corroborate that RSE achieves better or comparable accuracy in comparison to state-of-the-art baselines, especially with the strings of longer lengths. In addition, we empirically show that RSE scales linearly with the increase of the number and the length of string.
Kernel method has been developed as one of the standard approaches for nonlinear learning, which however, does not scale to large data set due to its quadratic complexity in the number of samples. A number of kernel approximation methods have thus been proposed in the recent years, among which the random features method gains much popularity due to its simplicity and direct reduction of nonlinear problem to a linear one. The Random Binning (RB) feature, proposed in the first random-feature paper cite{rahimi2007random}, has drawn much less attention than the Random Fourier (RF) feature. In this work, we observe that the RB features, with right choice of optimization solver, could be orders-of-magnitude more efficient than other random features and kernel approximation methods under the same requirement of accuracy. We thus propose the first analysis of RB from the perspective of optimization, which by interpreting RB as a Randomized Block Coordinate Descent in the infinite-dimensional space, gives a faster convergence rate compared to that of other random features. In particular, we show that by drawing $R$ random grids with at least $kappa$ number of non-empty bins per grid in expectation, RB method achieves a convergence rate of $O(1/(kappa R))$, which not only sharpens its $O(1/sqrt{R})$ rate from Monte Carlo analysis, but also shows a $kappa$ times speedup over other random features under the same analysis framework. In addition, we demonstrate another advantage of RB in the L1-regularized setting, where unlike other random features, a RB-based Coordinate Descent solver can be parallelized with guaranteed speedup proportional to $kappa$. Our extensive experiments demonstrate the superior performance of the RB features over other random features and kernel approximation methods. Our code and data is available at { url{https://github.com/teddylfwu/RB_GEN}}.
Principal Component Analysis (PCA) is a popular method for dimension reduction and has attracted an unfailing interest for decades. Recently, kernel PCA has emerged as an extension of PCA but, despite its use in practice, a sound theoretical understanding of kernel PCA is missing. In this paper, we contribute lower and upper bounds on the efficiency of kernel PCA, involving the empirical eigenvalues of the kernel Gram matrix. Two bounds are for fixed estimators, and two are for randomized estimators through the PAC-Bayes theory. We control how much information is captured by kernel PCA on average, and we dissect the bounds to highlight strengths and limitations of the kernel PCA algorithm. Therefore, we contribute to the better understanding of kernel PCA. Our bounds are briefly illustrated on a toy numerical example.
Random features provide a practical framework for large-scale kernel approximation and supervised learning. It has been shown that data-dependent sampling of random features using leverage scores can significantly reduce the number of features required to achieve optimal learning bounds. Leverage scores introduce an optimized distribution for features based on an infinite-dimensional integral operator (depending on input distribution), which is impractical to sample from. Focusing on empirical leverage scores in this paper, we establish an out-of-sample performance bound, revealing an interesting trade-off between the approximated kernel and the eigenvalue decay of another kernel in the domain of random features defined based on data distribution. Our experiments verify that the empirical algorithm consistently outperforms vanilla Monte Carlo sampling, and with a minor modification the method is even competitive to supervised data-dependent kernel learning, without using the output (label) information.
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