CANDECOMP/PARAFAC (CP) decomposition has been widely used to deal with multi-way data. For real-time or large-scale tensors, based on the ideas of randomized-sampling CP decomposition algorithm and online CP decomposition algorithm, a novel CP decomposition algorithm called randomized online CP decomposition (ROCP) is proposed in this paper. The proposed algorithm can avoid forming full Khatri-Rao product, which leads to boost the speed largely and reduce memory usage. The experimental results on synthetic data and real-world data show the ROCP algorithm is able to cope with CP decomposition for large-scale tensors with arbitrary number of dimensions. In addition, ROCP can reduce the computing time and memory usage dramatically, especially for large-scale tensors.
We propose accelerated randomized coordinate descent algorithms for stochastic optimization and online learning. Our algorithms have significantly less per-iteration complexity than the known accelerated gradient algorithms. The proposed algorithms for online learning have better regret performance than the known randomized online coordinate descent algorithms. Furthermore, the proposed algorithms for stochastic optimization exhibit as good convergence rates as the best known randomized coordinate descent algorithms. We also show simulation results to demonstrate performance of the proposed al…
Università degli Studi di Milano（米兰大学）；Microsoft Research（微软研究院）
Most traditional online learning algorithms are based on variants of mirror descent or follow-the-leader. In this paper, we present an online algorithm based on a completely different approach, tailored for transductive settings, which combines "random playout" and randomized rounding of loss subgradients. As an application of our approach, we present the first computationally efficient online algorithm for collaborative filtering with trace-norm constrained matrices. As a second application, we solve an open question linking batch learning and transductive online learning
We extend the randomized singular value decomposition (SVD) algorithm \citep{Halko2011finding} to estimate the SVD of a shifted data matrix without explicitly constructing the matrix in the memory. With no loss in the accuracy of the original algorithm, the extended algorithm provides for a more efficient way of matrix factorization. The algorithm facilitates the low-rank approximation and principal component analysis (PCA) of off-center data matrices. When applied to different types of data matrices, our experimental results confirm the advantages of the extensions made to the original algori…
Sharif University of Technology（谢里夫理工大学）；Ohio State University（俄亥俄州立大学）
With the increasing volume of data in the world, the best approach for learning from this data is to exploit an online learning algorithm. Online ensemble methods are online algorithms which take advantage of an ensemble of classifiers to predict labels of data. Prediction with expert advice is a well-studied problem in the online ensemble learning literature. The Weighted Majority algorithm and the randomized weighted majority (RWM) are the most well-known solutions to this problem, aiming to converge to the best expert. Since among some expert, the best one does not necessarily have the mini…
University of Oxford（牛津大学）；Cornell University（康奈尔大学）
The randomized singular value decomposition (SVD) is a popular and effective algorithm for computing a near-best rank $k$ approximation of a matrix $A$ using matrix-vector products with standard Gaussian vectors. Here, we generalize the randomized SVD to multivariate Gaussian vectors, allowing one to incorporate prior knowledge of $A$ into the algorithm. This enables us to explore the continuous analogue of the randomized SVD for Hilbert--Schmidt (HS) operators using operator-function products with functions drawn from a Gaussian process (GP). We then construct a new covariance kernel for GPs,…
Guangdong University of Technology（广东工业大学）；RIKEN（理化学研究所）
Efficient and accurate low-rank approximation (LRA) methods are of great significance for large-scale data analysis. Randomized tensor decompositions have emerged as powerful tools to meet this need, but most existing methods perform poorly in the presence of noise interference. Inspired by the remarkable performance of randomized block Krylov iteration (rBKI) in reducing the effect of tail singular values, this work designs an rBKI-based Tucker decomposition (rBKI-TK) for accurate approximation, together with a hierarchical tensor ring decomposition based on rBKI-TK for efficient compression …
We resolve the fundamental problem of online decoding with general $n^{th}$ order ergodic Markov chain models. Specifically, we provide deterministic and randomized algorithms whose performance is close to that of the optimal offline algorithm even when latency is small. Our algorithms admit efficient implementation via dynamic programs, and readily extend to (adversarial) non-stationary or time-varying settings. We also establish lower bounds for online methods under latency constraints in both deterministic and randomized settings, and show that no online algorithm can perform significantly …
Decision trees are widely-used classification and regression models because of their interpretability and good accuracy. Classical methods such as CART are based on greedy approaches but a growing attention has recently been devoted to optimal decision trees. We investigate the nonlinear continuous optimization formulation proposed in Blanquero et al. (EJOR, vol. 284, 2020; COR, vol. 132, 2021) for (sparse) optimal randomized classification trees. Sparsity is important not only for feature selection but also to improve interpretability. We first consider alternative methods to sparsify such tr…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
Low-rank Tucker and CP tensor decompositions are powerful tools in data analytics. The widely used alternating least squares (ALS) method, which solves a sequence of over-determined least squares subproblems, is costly for large and sparse tensors. We propose a fast and accurate sketched ALS algorithm for Tucker decomposition, which solves a sequence of sketched rank-constrained linear least squares subproblems. Theoretical sketch size upper bounds are provided to achieve $O(ε)$ relative error for each subproblem with two sketching techniques, TensorSketch and leverage score sampling. Experime…
Two types of low cost-per-iteration gradient descent methods have been extensively studied in parallel. One is online or stochastic gradient descent (OGD/SGD), and the other is randomzied coordinate descent (RBCD). In this paper, we combine the two types of methods together and propose online randomized block coordinate descent (ORBCD). At each iteration, ORBCD only computes the partial gradient of one block coordinate of one mini-batch samples. ORBCD is well suited for the composite minimization problem where one function is the average of the losses of a large number of samples and the other…
Learning big data by matrix decomposition always suffers from expensive computation, mixing of complicated structures and noise. In this paper, we study more adaptive models and efficient algorithms that decompose a data matrix as the sum of semantic components with incoherent structures. We firstly introduce "GO decomposition (GoDec)", an alternating projection method estimating the low-rank part $L$ and the sparse part $S$ from data matrix $X=L+S+G$ corrupted by noise $G$. Two acceleration strategies are proposed to obtain scalable unmixing algorithm on big data: 1) Bilateral random projecti…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；Shanghai Jiao Tong University（上海交通大学）；BIMSA（北京雁栖湖应用数学研究院）；AIR, Tsinghua University（清华大学智能产业研究院）；HeliXon Limited（华深智药）
Many practical applications of reinforcement learning require agents to learn from sparse and delayed rewards. It challenges the ability of agents to attribute their actions to future outcomes. In this paper, we consider the problem formulation of episodic reinforcement learning with trajectory feedback. It refers to an extreme delay of reward signals, in which the agent can only obtain one reward signal at the end of each trajectory. A popular paradigm for this problem setting is learning with a designed auxiliary dense reward function, namely proxy reward, instead of sparse environmental sig…
Randomized Tensor Ring Decomposition and Its Application to Large-scale Data Reconstruction
Graduate School of Engineering, Saitama Institute of Technology（埼玉工业大学工程研究生院）；RIKEN Center for Advanced Intelligence Project (AIP)（理化学研究所先进智能项目中心）；School of Automation, Guangdong University of Technology（广东工业大学自动化学院）；School of Computer Science and Technology, Hangzhou Dianzi University（杭州电子科技大学计算机学院）
We present a new fast online clustering algorithm that reliably recovers arbitrary-shaped data clusters in high throughout data streams. Unlike the existing state-of-the-art online clustering methods based on k-means or k-medoid, it does not make any restrictive generative assumptions. In addition, in contrast to existing nonparametric clustering techniques such as DBScan or DenStream, it gives provable theoretical guarantees. To achieve fast clustering, we propose to represent each cluster by a skeleton set which is updated continuously as new data is seen. A skeleton set consists of weighted…
Institute of Automation, Chinese Academy of Sciences（中国科学院自动化研究所）；University of Chinese Academy of Sciences（中国科学院大学）
We compared the regular Singular Value Decomposition (SVD), truncated SVD, Krylov method and Randomized PCA, in terms of time and space complexity. It is well-known that Krylov method and Randomized PCA only performs well when k << n, i.e. the number of eigenpair needed is far less than that of matrix size. We compared them for calculating all the eigenpairs. We also discussed the relationship between Principal Component Analysis and SVD.
University of California, Berkeley（加州大学伯克利分校）；Twitter（推特）；Max Planck Institute for Intelligent Systems（马克斯·普朗克智能系统研究所）
Online platforms regularly conduct randomized experiments to understand how changes to the platform causally affect various outcomes of interest. However, experimentation on online platforms has been criticized for having, among other issues, a lack of meaningful oversight and user consent. As platforms give users greater agency, it becomes possible to conduct observational studies in which users self-select into the treatment of interest as an alternative to experiments in which the platform controls whether the user receives treatment or not. In this paper, we conduct four large-scale within…
A data-driven analysis method known as dynamic mode decomposition (DMD) approximates the linear Koopman operator on projected space. In the spirit of Johnson-Lindenstrauss Lemma, we will use random projection to estimate the DMD modes in reduced dimensional space. In practical applications, snapshots are in high dimensional observable space and the DMD operator matrix is massive. Hence, computing DMD with the full spectrum is infeasible, so our main computational goal is estimating the eigenvalue and eigenvectors of the DMD operator in a projected domain. We will generalize the current algorit…
Decomposing a complex time series into trend, seasonality, and remainder components is an important primitive that facilitates time series anomaly detection, change point detection, and forecasting. Although numerous batch algorithms are known for time series decomposition, none operate well in an online scalable setting where high throughput and real-time response are paramount. In this paper, we propose OnlineSTL, a novel online algorithm for time series decomposition which is highly scalable and is deployed for real-time metrics monitoring on high-resolution, high-ingest rate data. Experime…
Online controlled experiments (A/B tests) have become the gold standard for learning the impact of new product features in technology companies. Randomization enables the inference of causality from an A/B test. The randomized assignment maps end users to experiment buckets and balances user characteristics between the groups. Therefore, experiments can attribute any outcome differences between the experiment groups to the product feature under experiment. Technology companies run A/B tests at scale -- hundreds if not thousands of A/B tests concurrently, each with millions of users. The large …
Efficient GPU implementation of randomized SVD and its applications
Jagiellonian University（雅盖隆大学）；Faculty of Mathematics and Computer Science（数学与计算机科学学院）；NVIDIA Corp.（英伟达公司）；AGH University of Science and Technology（AGH科技大学）；Faculty of Applied Mathematics（应用数学学院）
Matrix operations such as matrix inversion, eigenvalue decomposition, singular value decomposition are ubiquitous in real-world applications. Unfortunately, many of these matrix operations so time and memory expensive that they are prohibitive when the scale of data is large. In real-world applications, since the data themselves are noisy, machine-precision matrix operations are not necessary at all, and one can sacrifice a reasonable amount of accuracy for computational efficiency. In recent years, a bunch of randomized algorithms have been devised to make matrix computations more scalable. M…
Institute for Information Transmission Problems Russian Academy of Sciences（俄罗斯科学院信息传输问题研究所）
We present a universal algorithm for online trading in Stock Market which performs asymptotically at least as good as any stationary trading strategy that computes the investment at each step using a fixed function of the side information that belongs to a given RKHS (Reproducing Kernel Hilbert Space). Using a universal kernel, we extend this result for any continuous stationary strategy. In this learning process, a trader rationally chooses his gambles using predictions made by a randomized well-calibrated algorithm. Our strategy is based on Dawid's notion of calibration with more general che…
In a cognitive radio network, a secondary user learns the spectrum environment and dynamically accesses the channel where the primary user is inactive. At the same time, a primary user emulation (PUE) attacker can send falsified primary user signals and prevent the secondary user from utilizing the available channel. The best attacking strategies that an attacker can apply have not been well studied. In this paper, for the first time, we study optimal PUE attack strategies by formulating an online learning problem where the attacker needs to dynamically decide the attacking channel in each tim…
