In this paper we propose a new algorithm for streaming principal component analysis. With limited memory, small devices cannot store all the samples in the high-dimensional regime. Streaming principal component analysis aims to find the @math -dimensional subspace which can explain the most variation of the @math -dimensional data points that come into memory sequentially. In order to deal with large @math and large @math (number of samples), most streaming PCA algorithms update the current model using only the incoming sample and then dump the information right away to save memory. However the information contained in previously streamed data could be useful. Motivated by this idea, we develop a new streaming PCA algorithm called History PCA that achieves this goal. By using @math memory with @math being the block size, our algorithm converges much faster than existing streaming PCA algorithms. By changing the number of inner iterations, the memory usage can be further reduced to @math while maintaining a comparable convergence speed. We provide theoretical guarantees for the convergence of our algorithm along with the rate of convergence. We also demonstrate on synthetic and real world data sets that our algorithm compares favorably with other state-of-the-art streaming PCA methods in terms of the convergence speed and performance.
Recently, Shamir @cite_6 @cite_2 proposed the VR-PCA method that outperforms power method for PCA computation. The main idea is to reformulate PCA as a stochastic optimization problem and apply the variance reduction techniques @cite_7 . Although VR-PCA looks similar to streaming PCA, it cannot obtain the first update until the second pass. So it is not suitable in the streaming setting.
@cite_7 @cite_6 @cite_2
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We describe and analyze a simple algorithm for principal component analysis and singular value decomposition, VR-PCA, which uses computationally cheap stochastic iterations, yet converges exponentially fast to the optimal solution. In contrast, existing algorithms suffer either from slow convergence, or computationally intensive iterations whose runtime scales with the data size. The algorithm builds on a recent variance-reduced stochastic gradient technique, which was previously analyzed for strongly convex optimization, whereas here we apply it to an inherently non-convex problem, using a very different analysis. Variance Reducing (VR) stochastic methods are fast-converging alternatives to the classical Stochastic Gradient Descent (SGD) for solving large-scale regularized finite sum problems, especially when a highly accurate solution is required. One critical step in VR is the function sampling. State-of-the-art VR algorithms such as SVRG and SAGA, employ either Uniform Probability (UP) or Importance Probability (IP), which is deficient in reducing the variance and hence leads to suboptimal convergence rate. In this paper, we propose a novel sampling scheme that explicitly computes some Adaptive Probability (AP) at each iteration. Analysis shows that, equipped with AP, both SVRG and SAGA yield provably better convergence rate than the ones with UP or IP, which is confirmed in experiments. Additionally, to cut down the per iteration computation load, an efficient variant is proposed by utilizing AP periodically, whose performance is empirically validated. We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications: Streaming PCA. A recent work of (NIPS 2013) gives a space-efficient algorithm for PCA in a streaming model where samples are drawn from a gaussian spiked covariance model. We give a simpler and more general analysis that applies to arbitrary distributions confirming experimental evidence of Moreover, even in the spiked covariance model our result gives quantitative improvements in a natural parameter regime. It is also notably simpler and follows easily from our general convergence analysis of the noisy power method together with a matrix Chernoff bound. Private PCA. We provide the first nearly-linear time algorithm for the problem of differentially private principal component analysis that achieves nearly tight worst-case error bounds. Complementing our worst-case bounds, we show that the error dependence of our algorithm on the matrix dimension can be replaced by an essentially tight dependence on the coherence of the matrix. This result resolves the main problem left open by Hardt and Roth (STOC 2013). The coherence is always bounded by the matrix dimension but often substantially smaller thus leading to strong average-case improvements over the optimal worst-case bound. We provide a new robust convergence analysis of the well-known power method for computing the dominant singular vectors of a matrix that we call the noisy power method. Our result characterizes the convergence behavior of the algorithm when a significant amount noise is introduced after each matrix-vector multiplication. The noisy power method can be seen as a meta-algorithm that has recently found a number of important applications in a broad range of machine learning problems including alternating minimization for matrix completion, streaming principal component analysis (PCA), and privacy-preserving spectral analysis. Our general analysis subsumes several existing ad-hoc convergence bounds and resolves a number of open problems in multiple applications including streaming PCA and privacy-preserving singular vector computation.