Many real-world prediction tasks have outcome (a.k.a. target or response) variables that have characteristic heavy-tail distributions. Examples include copies of books sold, auction prices of art pieces, etc. By learning heavy-tailed distributions, big and rare'' instances (e.g., the best-sellers) will have accurate predictions. Most existing approaches are not dedicated to learning heavy-tailed distribution; thus, they heavily under-predict such instances. To tackle this problem, we introduce ( L2P ), which exploits the pairwise relationships between instances to learn from a proportionally higher number of rare instances. L2P consists of two stages. In Stage 1, L2P learns a pairwise preference classifier: . In Stage 2, L2P learns to place a new instance into an ordinal ranking of known instances. Based on its placement, the new instance is then assigned a value for its outcome variable. Experiments on real data show that L2P outperforms competing approaches in terms of accuracy and capability to reproduce heavy-tailed outcome distribution. In addition, L2P can provide an interpretable model with explainable outcomes by placing each predicted instance in context with its comparable neighbors.
Regression problems are known to suffer from under-predicting rare instances @cite_17 . Approaches proposed to correct fitting models consider prior correction that introduces terms capturing a fraction of rare events in the observations and weighting the data to compensate for differences @cite_0 @cite_4 . Hsu and Sabato @cite_27 proposed a methodology for linear regression with possibly heavy-tailed responses. They split data into multiple pieces, repeat the estimation process several times, and select the estimators based on their performance. They analytically prove that their method can perform reasonably well on heavy-tailed datasets. Quantile regression related approaches are proposed as well. Wang @cite_24 proposed estimating the intermediate conditional quantiles using conventional quantile regression and extrapolating these estimates to capture the behavior at the tail of the distribution. Robust Regression for Asymmetric Tails (RRAT) @cite_33 was proposed to address the problem of asymmetric noise distribution by using conditional quantile estimators. Zhang and Zhou @cite_20 considered linear regression with heavy-tail distributions and showed that using @math loss with truncated minimization can have advantages over @math loss. Like all truncated based approaches, their method requires prior knowledge of distributional properties. None of these regression techniques can capture non-linear decision boundaries.
@cite_4 @cite_33 @cite_0 @cite_24 @cite_27 @cite_20 @cite_17
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Estimation of conditional quantiles at very high or low tails is of interest in numerous applications. Quantile regression provides a convenient and natural way of quantifying the impact of covariates at different quantiles of a response distribution. However, high tails are often associated with data sparsity, so quantile regression estimation can suffer from high variability at tails especially for heavy-tailed distributions. In this article, we develop new estimation methods for high conditional quantiles by first estimating the intermediate conditional quantiles in a conventional quantile regression framework and then extrapolating these estimates to the high tails based on reasonable assumptions on tail behaviors. We establish the asymptotic properties of the proposed estimators and demonstrate through simulation studies that the proposed methods enjoy higher accuracy than the conventional quantile regression estimates. In a real application involving statistical downscaling of daily precipitation in... In the presence of a heavy-tail noise distribution, regression becomes much more difficult. Traditional robust regression methods assume that the noise distribution is symmetric, and they downweight the influence of so-called outliers. When the noise distribution is asymmetric, these methods yield biased regression estimators. Motivated by data-mining problems for the insurance industry, we propose a new approach to robust regression tailored to deal with asymmetric noise distribution. The main idea is to learn most of the parameters of the model using conditional quantile estimators (which are biased but robust estimators of the regression) and to learn a few remaining parameters to combine and correct these estimators, to minimize the average squared error in an unbiased way. Theoretical analysis and experiments show the clear advantages of the approach. Results are on artificial data as well as insurance data, using both linear and neural network predictors. In this paper, we consider the problem of linear regression with heavy-tailed distributions. Different from previous studies that use the squared loss to measure the performance, we choose the absolute loss, which is more robust in the presence of large prediction errors. To address the challenge that both the input and output could be heavy-tailed, we propose a truncated minimization problem, and demonstrate that it enjoys an O( √ d n ) excess risk, where d is the dimensionality and n is the number of samples. Compared with traditional work on l1-regression, the main advantage of our result is that we achieve a high-probability risk bound without exponential moment conditions on the input and output. Furthermore, if the input is bounded, we show that the classical empirical risk minimization is competent for l1-regression even when the output is heavy-tailed. Abstract We study the behavior at infinity of the tail of the stationary solution of a multidimensional linear auto-regressive process with random coefficients. We exhibit an extended class of multiplicative coefficients satisfying a condition of irreducibility and proximality that yield to a heavy tail behavior. To cite this article: B. de , C. R. Acad. Sci. Paris, Ser. I 339 (2004). We consider the problem of robustly predicting as well as the best linear combination of d given functions in least squares regression, and variants of this problem including constraints on the parameters of the linear combination. For the ridge estimator and the ordinary least squares estimator, and their variants, we provide new risk bounds of order d n without logarithmic factor unlike some standard results, where n is the size of the training data. We also provide a new estimator with better deviations in presence of heavy-tailed noise. It is based on truncating differences of losses in a min-max framework and satisfies a d n risk bound both in expectation and in deviations. The key common surprising factor of these results is the absence of exponential moment condition on the output distribution while achieving exponential deviations. All risk bounds are obtained through a PAC-Bayesian analysis on truncated differences of losses. Experimental results strongly back up our truncated min-max estimator. Quantile regression has gained increasing popularity as it provides richer information than the regular mean regression, and variable selection plays an important role in the quantile regression model building process, as it improves the prediction accuracy by choosing an appropriate subset of regression predictors. Unlike the traditional quantile regression, we consider the quantile as an unknown parameter and estimate it jointly with other regression coefficients. In particular, we adopt the Bayesian adaptive Lasso for the maximum entropy quantile regression. A flat prior is chosen for the quantile parameter due to the lack of information on it. The proposed method not only addresses the problem about which quantile would be the most probable one among all the candidates, but also reflects the inner relationship of the data through the estimated quantile. We develop an efficient Gibbs sampler algorithm and show that the performance of our proposed method is superior than the Bayesian adaptive Lasso and Bayesian Lasso through simulation studies and a real data analysis. This work studies applications and generalizations of a simple estimation technique that provides exponential concentration under heavy-tailed distributions, assuming only bounded low-order moments. We show that the technique can be used for approximate minimization of smooth and strongly convex losses, and specifically for least squares linear regression. For instance, our d-dimensional estimator requires just O(d log(1 δ)) random samples to obtain a constant factor approximation to the optimal least squares loss with probability 1-δ, without requiring the covariates or noise to be bounded or subgaussian. We provide further applications to sparse linear regression and low-rank covariance matrix estimation with similar allowances on the noise and covariate distributions. The core technique is a generalization of the median-of-means estimator to arbitrary metric spaces. We provide frequentist estimates of aleatoric and epistemic uncertainty for deep neural networks. To estimate aleatoric uncertainty we propose simultaneous quantile regression, a loss function to learn all the conditional quantiles of a given target variable. These quantiles lead to well-calibrated prediction intervals. To estimate epistemic uncertainty we propose training certificates, a collection of diverse non-trivial functions that map all training samples to zero. These certificates map out-of-distribution examples to non-zero values, signaling high epistemic uncertainty. We compare our proposals to prior art in various experiments. Random forests were introduced as a machine learning tool in Breiman (2001) and have since proven to be very popular and powerful for high-dimensional regression and classification. For regression, random forests give an accurate approximation of the conditional mean of a response variable. It is shown here that random forests provide information about the full conditional distribution of the response variable, not only about the conditional mean. Conditional quantiles can be inferred with quantile regression forests, a generalisation of random forests. Quantile regression forests give a non-parametric and accurate way of estimating conditional quantiles for high-dimensional predictor variables. The algorithm is shown to be consistent. Numerical examples suggest that the algorithm is competitive in terms of predictive power. Network monitoring in cellular networks requires the tracking of quantiles for data distributions of many evolving network measurements (e.g. number of high signaling subscribers per minute). Most quantile estimation algorithms are based on a summary of the empirical data distribution, using either a representative sample or a global approximation of the entire distribution. In contrast, by viewing data as a quantity from a random distribution, the stochastic approximation (SA) for quantile estimation does not keep a global approximation, but rather local approximations at the quantiles of interest, and therefore uses negligible memory even for estimating tail quantiles. However, the current stochastic approximation algorithm for quantile estimation tracks each quantile separately, and this may lead to a violation of the monotone property of quantiles. In this paper, we propose a stochastic approximation technique that enables the simultaneous tracking of multiple quantiles. Our technique maintains the monotone property of different quantiles, and is adaptive to changes in the data distribution. We evaluate its performance using real cellular provider datasets. Our results show that the technique is very efficient. In underwater target tracking applications, measurement uncertainty and inaccuracies are usually modeled as additive Gaussian noise. The Gaussian model of noise may not be appropriate in many practical systems. The non-Gaussian noise and the model non-linearity arising in a tracking system will seriously affect the tracking performance. This paper discusses one way to create a robust version of the extended Kalman filter for enhanced underwater target tracking. State estimation in the filter is done through the robust regression approach and Welsch's proposal is used in the regression process. Monte Carlo simulation results with heavy-tailed contaminated observation noise demonstrate the robustness of the proposed estimation procedure. > This paper considers estimation and testing using location measures for regression m odels that are based on an asymmetric least-squares criterion functio n. These estimators have properties that are analogous to regression quantiles, but are easier to calculate, as are the corresponding test statistics. Asymmetric least-squares tests of homoskedasticity and s ymmetry compare quite favorably with other tests of these hypotheses in terms of asymptotic relative efficiency. Consequently, asymmetric least-squares estimation provides a convenient and relatively efficie nt method of characterizing the conditional distributi on of a dependent variable given some regressors. Copyright 1987 by The Econometric Society. Many complex networks in natural and social phenomena have often been characterized by heavy-tailed degree distributions. However, due to rapidly growing size of network data and concerns on privacy issues about using these data, it becomes more difficult to analyze complete data sets. Thus, it is crucial to devise effective and efficient estimation methods for heavy tails of degree distributions in large-scale networks only using local information of a small fraction of sampled nodes. Here we propose a tail-scope method based on local observational bias of the friendship paradox. We show that the tail-scope method outperforms the uniform node sampling for estimating heavy tails of degree distributions, while the opposite tendency is observed in the range of small degrees. In order to take advantages of both sampling methods, we devise the hybrid method that successfully recovers the whole range of degree distributions. Our tail-scope method shows how structural heterogeneities of large-scale complex networks can be used to effectively reveal the network structure only with limited local information. Server side measurements from several Wi-Fi hot-spots deployed in a nationwide network over different types of venues from small coffee shops to large enterprises are used to highlight differences in traffic volumes and patterns. We develop a common modeling framework for the number of simultaneously present customers. Our approach has many novel elements: (a) We combine statistical clustering with Poisson regression from Generalized Linear Models to fit a non-stationary Poisson process to the arrival counts and demonstrate its remarkable accuracy; (b) We model the heavy tailed distribution of connection durations through fitting a Phase Type distribution to its logarithm so that not only the tail but also the overall distribution is well matched; (c) We obtain the distribution of the number of simultaneously present customers from an M t G ∞ queuing model using a novel regenerative argument that is transparent and avoids the customarily made assumption of the queue starting empty at an infinite past; (d) Most importantly, we validate our models by comparison of their predictions and confidence intervals against test data that is not used in fitting the models. Research on universal and minimax wavelet shrinkage and thresholding methods has demonstrated near-ideal estimation performance in various asymptotic frameworks. However, image processing practice has shown that universal thresholding methods are outperformed by simple Bayesian estimators assuming independent wavelet coefficients and heavy-tailed priors such as generalized Gaussian distributions (GGDs). In this paper, we investigate various connections between shrinkage methods and maximum a posteriori (MAP) estimation using such priors. In particular, we state a simple condition under which MAP estimates are sparse. We also introduce a new family of complexity priors based upon Rissanen's universal prior on integers. One particular estimator in this class outperforms conventional estimators based on earlier applications of the minimum description length (MDL) principle. We develop analytical expressions for the shrinkage rules implied by GGD and complexity priors. This allows us to show the equivalence between universal hard thresholding, MAP estimation using a very heavy-tailed GGD, and MDL estimation using one of the new complexity priors. Theoretical analysis supported by numerous practical experiments shows the robustness of some of these estimates against mis-specifications of the prior-a basic concern in image processing applications. Sponsored search is one of the major sources of revenue for search engines on the World Wide Web. It has been observed that while showing ads for every query maximizes short-term revenue, irrelevant ads lead to poor user experience and less revenue in the long-term. Hence, it is in search engines' interest to place ads only for queries that are likely to attract ad-clicks. Many algorithms for estimating query advertisability exist in literature, but most of these methods have been proposed for and tested on the frequent or "head" queries. Since query frequencies on search engine are known to be distributed as a power-law, this leaves a huge fraction of the queries uncovered. In this paper we focus on the more challenging problem of estimating query advertisability for infrequent or "tail" queries. These require fundamentally different methods than head queries: for e.g., tail queries are almost all unique and require the estimation method to be online and inexpensive. We show that previously proposed methods do not apply to tail queries, and when modified for our scenario they do not work well. Further, we give a simple, yet effective, approach, which estimates query advertisability using only the words present in the queries. We evaluate our approach on a real-world dataset consisting of search engine queries and user clicks. Our results show that our simple approach outperforms a more complex one based on regularized regression. Least squares support vector machines (LS-SVM) is an SVM version which involves equality instead of inequality constraints and works with a least squares cost function. In this way, the solution follows from a linear Karush–Kuhn–Tucker system instead of a quadratic programming problem. However, sparseness is lost in the LS-SVM case and the estimation of the support values is only optimal in the case of a Gaussian distribution of the error variables. In this paper, we discuss a method which can overcome these two drawbacks. We show how to obtain robust estimates for regression by applying a weighted version of LS-SVM. We also discuss a sparse approximation procedure for weighted and unweighted LS-SVM. It is basically a pruning method which is able to do pruning based upon the physical meaning of the sorted support values, while pruning procedures for classical multilayer perceptrons require the computation of a Hessian matrix or its inverse. The methods of this paper are illustrated for RBF kernels and demonstrate how to obtain robust estimates with selection of an appropriate number of hidden units, in the case of outliers or non-Gaussian error distributions with heavy tails. c 2002 Elsevier Science B.V. All rights reserved. We consider the estimation of regression coefficients in a high-dimensional linear model. For regression coefficients in lr balls, we provide lower bounds for the minimax lq risk and minimax quantiles of the lq loss for all design matrices. Under an l0 sparsity condition on a target coefficient vector, we sharpen and unify existing oracle inequalities for the Lasso and Dantzig selector. We derive oracle inequalities for target coefficient vectors with many small elements and smaller threshold levels than the universal threshold. These oracle inequalities provide sufficient conditions on the design matrix for the rate minimaxity of the Lasso and Dantzig selector for the lq risk and loss in lr balls, 0≤ r≤ 1≤ q≤ ∞. By allowing q=∞, our risk bounds imply the variable selection consistency of threshold Lasso and Dantzig selectors. We obtain sharp bounds on the performance of Empirical Risk Minimization performed in a convex class and with respect to the squared loss, without assuming that class members and the target are bounded functions or have rapidly decaying tails. Rather than resorting to a concentration-based argument, the method used here relies on a small-ball' assumption and thus holds for classes consisting of heavy-tailed functions and for heavy-tailed targets. The resulting estimates scale correctly with the noise level' of the problem, and when applied to the classical, bounded scenario, always improve the known bounds. Suppose an n x d design matrix in a linear regression problem is given, but the response for each point is hidden unless explicitly requested. The goal is to sample only a small number k << n of the responses, and then produce a weight vector whose sum of squares loss over all points is at most 1+epsilon times the minimum. When k is very small (e.g., k=d), jointly sampling diverse subsets of points is crucial. One such method called "volume sampling" has a unique and desirable property that the weight vector it produces is an unbiased estimate of the optimum. It is therefore natural to ask if this method offers the optimal unbiased estimate in terms of the number of responses k needed to achieve a 1+epsilon loss approximation. Surprisingly we show that volume sampling can have poor behavior when we require a very accurate approximation -- indeed worse than some i.i.d. sampling techniques whose estimates are biased, such as leverage score sampling. We then develop a new rescaled variant of volume sampling that produces an unbiased estimate which avoids this bad behavior and has at least as good a tail bound as leverage score sampling: sample size k=O(d log d + d epsilon) suffices to guarantee total loss at most 1+epsilon times the minimum with high probability. Thus, we improve on the best previously known sample size for an unbiased estimator, k=O(d^2 epsilon). Our rescaling procedure leads to a new efficient algorithm for volume sampling which is based on a "determinantal rejection sampling" technique with potentially broader applications to determinantal point processes. Other contributions include introducing the combinatorics needed for rescaled volume sampling and developing tail bounds for sums of dependent random matrices which arise in the process.