Model selection is a crucial issue in machine-learning and a wide variety of penalisation methods (with possibly data dependent complexity penalties) have recently been introduced for this purpose. However their empirical performance is generally not well documented in the literature. It is the goal of this paper to investigate to which extent such recent techniques can be successfully used for the tuning of both the regularisation and kernel parameters in support vector regression (SVR) and the complexity measure in regression trees (CART). This task is traditionally solved via V-fold cross-validation (VFCV), which gives efficient results for a reasonable computational cost. A disadvantage however of VFCV is that the procedure is known to provide an asymptotically suboptimal risk estimate as the number of examples tends to infinity. Recently, a penalisation procedure called V-fold penalisation has been proposed to improve on VFCV, supported by theoretical arguments. Here we report on an extensive set of experiments comparing V-fold penalisation and VFCV for SVR/CART calibration on several benchmark datasets. We highlight cases in which VFCV and V-fold penalisation provide poor estimates of the risk respectively and introduce a modified penalisation technique to reduce the estimation error.

Most machine learning algorithms, such as classification or regression, treat the individual data point as the object of interest. Here we consider extending machine learning algorithms to operate on groups of data points. We suggest treating a group of data points as an i.i.d. sample set from an underlying feature distribution for that group. Our approach employs kernel machines with a kernel on i.i.d. sample sets of vectors. We define certain kernel functions on pairs of distributions, and then use a nonparametric estimator to consistently estimate those functions based on sample sets. The projection of the estimated Gram matrix to the cone of symmetric positive semi-definite matrices enables us to use kernel machines for classification, regression, anomaly detection, and low-dimensional embedding in the space of distributions. We present several numerical experiments both on real and simulated datasets to demonstrate the advantages of our new approach.

The accuracy of machine learning systems is a widely studied research topic. Established techniques such as cross-validation predict the accuracy on unseen data of the classifier produced by applying a given learning method to a given training data set. However, they do not predict whether incurring the cost of obtaining more data and undergoing further training will lead to higher accuracy. In this paper we investigate techniques for making such early predictions. We note that when a machine learning algorithm is presented with a training set the classifier produced, and hence its error, will depend on the characteristics of the algorithm, on training set's size, and also on its specific composition. In particular we hypothesise that if a number of classifiers are produced, and their observed error is decomposed into bias and variance terms, then although these components may behave differently, their behaviour may be predictable. We test our hypothesis by building models that, given a measurement taken from the classifier created from a limited number of samples, predict the values that would be measured from the classifier produced when the full data set is presented. We create separate models for bias, variance and total error. Our models are built from the results of applying ten different machine learning algorithms to a range of data sets, and tested with "unseen" algorithms and datasets. We analyse the results for various numbers of initial training samples, and total dataset sizes. Results show that our predictions are very highly correlated with the values observed after undertaking the extra training. Finally we consider the more complex case where an ensemble of heterogeneous classifiers is trained, and show how we can accurately estimate an upper bound on the accuracy achievable after further training.

In this paper we discuss the variable selection method from \ell0-norm constrained regression, which is equivalent to the problem of finding the best subset of a fixed size. Our study focuses on two aspects, consistency and computation. We prove that the sparse estimator from such a method can retain all of the important variables asymptotically for even exponentially growing dimensionality under regularity conditions. This indicates that the best subset regression method can efficiently shrink the full model down to a submodel of a size less than the sample size, which can be analyzed by well-developed regression techniques for such cases in a follow-up study. We provide an iterative algorithm, called orthogonalizing subset selection (OSS), to address computational issues in best subset regression. OSS is an EM algorithm, and thus possesses the monotonicity property. For any sparse estimator, OSS can improve its fit of the model by putting it as an initial point. After this improvement, the sparsity of the estimator is kept. Another appealing feature of OSS is that, similarly to an effective algorithm for a continuous optimization problem, OSS can converge to the global solution to the \ell0-norm constrained regression problem if the initial point lies in a neighborhood of the global solution. An accelerating algorithm of OSS and its combination with forward stepwise selection are also investigated. Simulations and a real example are presented to evaluate the performances of the proposed methods.

Ultrahigh-dimensional variable selection plays an increasingly important role in contemporary scientific discoveries and statistical research. Among others, Fan and Lv [J. R. Stat. Soc. Ser. B Stat. Methodol. 70 (2008) 849-911] propose an independent screening framework by ranking the marginal correlations. They showed that the correlation ranking procedure possesses a sure independence screening property within the context of the linear model with Gaussian covariates and responses. In this paper, we propose a more general version of the independent learning with ranking the maximum marginal likelihood estimates or the maximum marginal likelihood itself in generalized linear models. We show that the proposed methods, with Fan and Lv [J. R. Stat. Soc. Ser. B Stat. Methodol. 70 (2008) 849-911] as a very special case, also possess the sure screening property with vanishing false selection rate. The conditions under which the independence learning possesses a sure screening is surprisingly simple. This justifies the applicability of such a simple method in a wide spectrum. We quantify explicitly the extent to which the dimensionality can be reduced by independence screening, which depends on the interactions of the covariance matrix of covariates and true parameters. Simulation studies are used to illustrate the utility of the proposed approaches. In addition, we establish an exponential inequality for the quasi-maximum likelihood estimator which is useful for high-dimensional statistical learning.

Ambient activity monitoring systems produce large amounts of data, which can be used for health monitoring.The problem is that patterns in this data reflecting health status are not identified yet. In this paper the possibility is explored of predicting the functional health status (the motor score of AMPS = Assessment of Motor and Process Skills) of a person from data of binary ambient sensors. Data is collected of five independently living elderly people. Based on expert knowledge, features are extracted from the sensor data and several subsets are selected. We use standard linear regression and Gaussian processes for mapping the features to the functional status and predict the status of a test person using a leave-one-person-out cross validation. The results show that Gaussian processes perform better than the linear regression model, and that both models perform better with the basic feature set than with location or transition based features.Some suggestions are provided for better feature extraction and selection for the purpose of health monitoring.These results indicate that automated functional health assessment is possible, but some challenges lie ahead. The most important challenge is eliciting expert knowledge and translating that into quantifiable features.

Regression, unlike classification, has lacked a comprehensive and effective approach to deal with cost-sensitive problems by the reuse (and not a re-training) of general regression models. In this paper, a wide variety of cost-sensitive problems in regression (such as bids, asymmetric losses and rejection rules) can be solved effectively by a lightweight but powerful approach, consisting of: (1) the conversion of any traditional one-parameter crisp regression model into a two-parameter soft regression model, seen as a normal conditional density estimator, by the use of newly-introduced enrichment methods; and (2) the reframing of an enriched soft regression model to new contexts by an instance-dependent optimisation of the expected loss derived from the conditional normal distribution.

We propose the Bayesian bridge estimator for regularized regression and classification. Two key mixture representations for the Bayesian bridge model are developed: (1) a scale mixture of normals with respect to an alpha-stable random variable; and (2) a mixture of Bartlett--Fejer kernels (or triangle densities) with respect to a two-component mixture of gamma random variables. Both lead to MCMC methods for posterior simulation, and these methods turn out to have complementary domains of maximum efficiency. The first representation is a well known result due to West (1987), and is the better choice for collinear design matrices. The second representation is new, and is more efficient for orthogonal problems, largely because it avoids the need to deal with exponentially tilted stable random variables. It also provides insight into the multimodality of the joint posterior distribution, a feature of the bridge model that is notably absent under ridge or lasso-type priors. We prove a theorem that extends this representation to a wider class of densities representable as scale mixtures of betas, and provide an explicit inversion formula for the mixing distribution. The connections with slice sampling and scale mixtures of normals are explored. On the practical side, we find that the Bayesian bridge model outperforms its classical cousin in estimation and prediction across a variety of data sets, both simulated and real. We also show that the MCMC for fitting the bridge model exhibits excellent mixing properties, particularly for the global scale parameter. This makes for a favorable contrast with analogous MCMC algorithms for other sparse Bayesian models. All methods described in this paper are implemented in the R package BayesBridge. An extensive set of simulation results are provided in two supplemental files.

We address the problem of general supervised learning when data can only be accessed through an (indefinite) similarity function between data points. Existing work on learning with indefinite kernels has concentrated solely on binary/multi-class classification problems. We propose a model that is generic enough to handle any supervised learning task and also subsumes the model previously proposed for classification. We give a "goodness" criterion for similarity functions w.r.t. a given supervised learning task and then adapt a well-known landmarking technique to provide efficient algorithms for supervised learning using "good" similarity functions. We demonstrate the effectiveness of our model on three important super-vised learning problems: a) real-valued regression, b) ordinal regression and c) ranking where we show that our method guarantees bounded generalization error. Furthermore, for the case of real-valued regression, we give a natural goodness definition that, when used in conjunction with a recent result in sparse vector recovery, guarantees a sparse predictor with bounded generalization error. Finally, we report results of our learning algorithms on regression and ordinal regression tasks using non-PSD similarity functions and demonstrate the effectiveness of our algorithms, especially that of the sparse landmark selection algorithm that achieves significantly higher accuracies than the baseline methods while offering reduced computational costs.

Graphical models with bi-directed edges (<->) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood estimation in the case of continuous variables with a Gaussian joint distribution, sometimes termed a covariance graph model. We present a new fitting algorithm which exploits standard regression techniques and establish its convergence properties. Moreover, we contrast our procedure to existing estimation methods.