Efficient implementation of Newton-raphson methods for sequential data prediction

We investigate the problem of sequential linear data prediction for real life big data applications. The second order algorithms, i.e., Newton-Raphson Methods, asymptotically achieve the performance of the 'best' possible linear data predictor much faster compared to the first order algorithms, e.g., Online Gradient Descent. However, implementation of these second order methods results in a computational complexity in the order of $O(M2)$ for an $M$ dimensional feature vector, where the first order methods offer complexity in the order of $O(M)$. Because of this extremely high computational need, their usage in real life big data applications is prohibited. To this end, in order to enjoy the outstanding performance of the second order methods, we introduce a highly efficient implementation where the computational complexity of these methods is reduced from $O(M2)$ to $O(M)$. The presented algorithm provides the well-known merits of the second order methods while offering a computational complexity similar to the first order methods. We do not rely on any statistical assumptions, hence, both regular and fast implementations achieve the same performance in terms of mean square error. We demonstrate the efficiency of our algorithm on several sequential big datasets. We also illustrate the numerical stability of the presented algorithm. © 1989-2012 IEEE

Efficient implementation of Newton-raphson methods for sequential data prediction

We investigate the problem of sequential linear data prediction for real life big data applications. The second order algorithms, i.e., Newton-Raphson Methods, asymptotically achieve the performance of the 'best' possible linear data predictor much faster compared to the first order algorithms, e.g., Online Gradient Descent. However, implementation of these second order methods results in a computational complexity in the order of $O(M2)$ for an $M$ dimensional feature vector, where the first order methods offer complexity in the order of $O(M)$. Because of this extremely high computational need, their usage in real life big data applications is prohibited. To this end, in order to enjoy the outstanding performance of the second order methods, we introduce a highly efficient implementation where the computational complexity of these methods is reduced from $O(M2)$ to $O(M)$. The presented algorithm provides the well-known merits of the second order methods while offering a computational complexity similar to the first order methods. We do not rely on any statistical assumptions, hence, both regular and fast implementations achieve the same performance in terms of mean square error. We demonstrate the efficiency of our algorithm on several sequential big datasets. We also illustrate the numerical stability of the presented algorithm. © 1989-2012 IEEE

Particle approximations of the score and observed information matrix for parameter estimation in state space models with linear computational cost

Poyiadjis et al. (2011) show how particle methods can be used to estimate both the score and the observed information matrix for state space models. These methods either suffer from a computational cost that is quadratic in the number of particles, or produce estimates whose variance increases quadratically with the amount of data. This paper introduces an alternative approach for estimating these terms at a computational cost that is linear in the number of particles. The method is derived using a combination of kernel density estimation, to avoid the particle degeneracy that causes the quadratically increasing variance, and Rao-Blackwellisation. Crucially, we show the method is robust to the choice of bandwidth within the kernel density estimation, as it has good asymptotic properties regardless of this choice. Our estimates of the score and observed information matrix can be used within both online and batch procedures for estimating parameters for state space models. Empirical results show improved parameter estimates compared to existing methods at a significantly reduced computational cost. Supplementary materials including code are available.Comment: Accepted to Journal of Computational and Graphical Statistic

Incremental Sparse Bayesian Ordinal Regression

Ordinal Regression (OR) aims to model the ordering information between different data categories, which is a crucial topic in multi-label learning. An important class of approaches to OR models the problem as a linear combination of basis functions that map features to a high dimensional non-linear space. However, most of the basis function-based algorithms are time consuming. We propose an incremental sparse Bayesian approach to OR tasks and introduce an algorithm to sequentially learn the relevant basis functions in the ordinal scenario. Our method, called Incremental Sparse Bayesian Ordinal Regression (ISBOR), automatically optimizes the hyper-parameters via the type-II maximum likelihood method. By exploiting fast marginal likelihood optimization, ISBOR can avoid big matrix inverses, which is the main bottleneck in applying basis function-based algorithms to OR tasks on large-scale datasets. We show that ISBOR can make accurate predictions with parsimonious basis functions while offering automatic estimates of the prediction uncertainty. Extensive experiments on synthetic and real word datasets demonstrate the efficiency and effectiveness of ISBOR compared to other basis function-based OR approaches