There are a variety of settings where vague prior information may be available on the importance of predictors in high-dimensional regression settings. Examples include ordering on the variables offered by their empirical variances (which is typically discarded through standardisation), the lag of predictors when fitting autoregressive models in time series settings, or the level of missingness of the variables. Whilst such orderings may not match the true importance of variables, we argue that there is little to be lost, and potentially much to be gained, by using them. We propose a simple scheme involving fitting a sequence of models indicated by the ordering. We show that the computational cost for fitting all models when ridge regression is used is no more than for a single fit of ridge regression, and describe a strategy for Lasso regression that makes use of previous fits to greatly speed up fitting the entire sequence of models. We propose to select a final estimator by cross-validation and provide a general result on the quality of the best performing estimator on a test set selected from among a number $M$ of competing estimators in a high-dimensional linear regression setting. Our result requires no sparsity assumptions and shows that only a $\log M$ price is incurred compared to the unknown best estimator. We demonstrate the effectiveness of our approach when applied to missing or corrupted data, and time series settings. An R package is available on github.
相關內容
We consider the problem of inferring the conditional independence graph (CIG) of a sparse, high-dimensional stationary multivariate Gaussian time series. A sparse-group lasso-based frequency-domain formulation of the problem based on frequency-domain sufficient statistic for the observed time series is presented. We investigate an alternating direction method of multipliers (ADMM) approach for optimization of the sparse-group lasso penalized log-likelihood. We provide sufficient conditions for convergence in the Frobenius norm of the inverse PSD estimators to the true value, jointly across all frequencies, where the number of frequencies are allowed to increase with sample size. This results also yields a rate of convergence. We also empirically investigate selection of the tuning parameters based on Bayesian information criterion, and illustrate our approach using numerical examples utilizing both synthetic and real data.
In many practices, scientists are particularly interested in detecting which of the predictors are truly associated with a multivariate response. It is more accurate to model multiple responses as one vector rather than separating each component one by one. This is particularly true for complex traits having multiple correlated components. A Bayesian multivariate variable selection (BMVS) approach is proposed to select important predictors influencing the multivariate response from a candidate pool with an ultrahigh dimension. By applying the sample-size-dependent spike and slab priors, the BMVS approach satisfies the strong selection consistency property under certain conditions, which represents the advantages of BMVS over other existing Bayesian multivariate regression-based approaches. The proposed approach considers the covariance structure of multiple responses without assuming independence and integrates the estimation of covariance-related parameters together with all regression parameters into one framework through a fast updating MCMC procedure. It is demonstrated through simulations that the BMVS approach outperforms some other relevant frequentist and Bayesian approaches. The proposed BMVS approach possesses the flexibility of wide applications, including genome-wide association studies with multiple correlated phenotypes and a large scale of genetic variants and/or environmental variables, as demonstrated in the real data analyses section. The computer code and test data of the proposed method are available as an R package.
We consider network autoregressive models for count data with a non-random time-varying neighborhood structure. The main methodological contribution is the development of conditions that guarantee stability and valid statistical inference. We consider both cases of fixed and increasing network dimension and we show that quasi-likelihood inference provides consistent and asymptotically normally distributed estimators. The work is complemented by simulation results and a data example.
A learning method is self-certified if it uses all available data to simultaneously learn a predictor and certify its quality with a statistical certificate that is valid on unseen data. Recent work has shown that neural network models trained by optimising PAC-Bayes bounds lead not only to accurate predictors, but also to tight risk certificates, bearing promise towards achieving self-certified learning. In this context, learning and certification strategies based on PAC-Bayes bounds are especially attractive due to their ability to leverage all data to learn a posterior and simultaneously certify its risk. In this paper, we assess the progress towards self-certification in probabilistic neural networks learnt by PAC-Bayes inspired objectives. We empirically compare (on 4 classification datasets) classical test set bounds for deterministic predictors and a PAC-Bayes bound for randomised self-certified predictors. We first show that both of these generalisation bounds are not too far from out-of-sample test set errors. We then show that in data starvation regimes, holding out data for the test set bounds adversely affects generalisation performance, while self-certified strategies based on PAC-Bayes bounds do not suffer from this drawback, proving that they might be a suitable choice for the small data regime. We also find that probabilistic neural networks learnt by PAC-Bayes inspired objectives lead to certificates that can be surprisingly competitive with commonly used test set bounds.
In many stochastic problems, the output of interest depends on an input random vector mainly through a single random variable (or index) via an appropriate univariate transformation of the input. We exploit this feature by proposing an importance sampling method that makes rare events more likely by changing the distribution of the chosen index. Further variance reduction is guaranteed by combining this single-index importance sampling approach with stratified sampling. The dimension-reduction effect of single-index importance sampling also enhances the effectiveness of quasi-Monte Carlo methods. The proposed method applies to a wide range of financial or risk management problems. We demonstrate its efficiency for estimating large loss probabilities of a credit portfolio under a normal and t-copula model and show that our method outperforms the current standard for these problems.
We propose confidence regions with asymptotically correct uniform coverage probability of parameters whose Fisher information matrix can be singular at important points of the parameter set. Our work is motivated by the need for reliable inference on scale parameters close or equal to zero in mixed models, which is obtained as a special case. The confidence regions are constructed by inverting a continuous extension of the score test statistic standardized by expected information, which we show exists at points of singular information under regularity conditions. Similar results have previously only been obtained for scalar parameters, under conditions stronger than ours, and applications to mixed models have not been considered. In simulations our confidence regions have near-nominal coverage with as few as $n = 20$ independent observations, regardless of how close to the boundary the true parameter is. It is a corollary of our main results that the proposed test statistic has an asymptotic chi-square distribution with degrees of freedom equal to the number of tested parameters, even if they are on the boundary of the parameter set.
We study sparse linear regression over a network of agents, modeled as an undirected graph (with no centralized node). The estimation problem is formulated as the minimization of the sum of the local LASSO loss functions plus a quadratic penalty of the consensus constraint -- the latter being instrumental to obtain distributed solution methods. While penalty-based consensus methods have been extensively studied in the optimization literature, their statistical and computational guarantees in the high dimensional setting remain unclear. This work provides an answer to this open problem. Our contribution is two-fold. First, we establish statistical consistency of the estimator: under a suitable choice of the penalty parameter, the optimal solution of the penalized problem achieves near optimal minimax rate $\mathcal{O}(s \log d/N)$ in $\ell_2$-loss, where $s$ is the sparsity value, $d$ is the ambient dimension, and $N$ is the total sample size in the network -- this matches centralized sample rates. Second, we show that the proximal-gradient algorithm applied to the penalized problem, which naturally leads to distributed implementations, converges linearly up to a tolerance of the order of the centralized statistical error -- the rate scales as $\mathcal{O}(d)$, revealing an unavoidable speed-accuracy dilemma.Numerical results demonstrate the tightness of the derived sample rate and convergence rate scalings.
The matrix normal model, the family of Gaussian matrix-variate distributions whose covariance matrix is the Kronecker product of two lower dimensional factors, is frequently used to model matrix-variate data. The tensor normal model generalizes this family to Kronecker products of three or more factors. We study the estimation of the Kronecker factors of the covariance matrix in the matrix and tensor models. We show nonasymptotic bounds for the error achieved by the maximum likelihood estimator (MLE) in several natural metrics. In contrast to existing bounds, our results do not rely on the factors being well-conditioned or sparse. For the matrix normal model, all our bounds are minimax optimal up to logarithmic factors, and for the tensor normal model our bound for the largest factor and overall covariance matrix are minimax optimal up to constant factors provided there are enough samples for any estimator to obtain constant Frobenius error. In the same regimes as our sample complexity bounds, we show that an iterative procedure to compute the MLE known as the flip-flop algorithm converges linearly with high probability. Our main tool is geodesic strong convexity in the geometry on positive-definite matrices induced by the Fisher information metric. This strong convexity is determined by the expansion of certain random quantum channels. We also provide numerical evidence that combining the flip-flop algorithm with a simple shrinkage estimator can improve performance in the undersampled regime.
Systems with stochastic time delay between the input and output present a number of unique challenges. Time domain noise leads to irregular alignments, obfuscates relationships and attenuates inferred coefficients. To handle these challenges, we introduce a maximum likelihood regression model that regards stochastic time delay as an "error" in the time domain. For a certain subset of problems, by modelling both prediction and time errors it is possible to outperform traditional models. Through a simulated experiment of a univariate problem, we demonstrate results that significantly improve upon Ordinary Least Squares (OLS) regression.
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.
小貼士
登錄享
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191