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Register a new userLeave-one-out cross-validation is risk consistent for lasso
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Daniel McDonald
Publication date
2012
fields
Mathematical Statistics
and research's language is
English
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The lasso procedure is ubiquitous in the statistical and signal processing literature, and as such, is the target of substantial theoretical and applied research. While much of this research focuses on the desirable properties that lasso possesses---predictive risk consistency, sign consistency, correct model selection---all of it has assumes that the tuning parameter is chosen in an oracle fashion. Yet, this is impossible in practice. Instead, data analysts must use the data twice, once to choose the tuning parameter and again to estimate the model. But only heuristics have ever justified such a procedure. To this end, we give the first definitive answer about the risk consistency of lasso when the smoothing parameter is chosen via cross-validation. We show that under some restrictions on the design matrix, the lasso estimator is still risk consistent with an empirically chosen tuning parameter.
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The lasso and related sparsity inducing algorithms have been the target of substantial theoretical and applied research. Correspondingly, many results are known about their behavior for a fixed or optimally chosen tuning parameter specified up to unknown constants. In practice, however, this oracle tuning parameter is inaccessible so one must use the data to select one. Common statistical practice is to use a variant of cross-validation for this task. However, little is known about the theoretical properties of the resulting predictions with such data-dependent methods. We consider the high-dimensional setting with random design wherein the number of predictors $p$ grows with the number of observations $n$. Under typical assumptions on the data generating process, similar to those in the literature, we recover oracle rates up to a log factor when choosing the tuning parameter with cross-validation. Under weaker conditions, when the true model is not necessarily linear, we show that the lasso remains risk consistent relative to its linear oracle. We also generalize these results to the group lasso and square-root lasso and investigate the predictive and model selection performance of cross-validation via simulation.
We present a simple algorithm for identifying and correcting real-valued noisy labels from a mixture of clean and corrupted sample points using Gaussian process regression. A heteroscedastic noise model is employed, in which additive Gaussian noise terms with independent variances are associated with each and all of the observed labels. Optimizing the noise model using maximum likelihood estimation leads to the containment of the GPR models predictive error by the posterior standard deviation in leave-one-out cross-validation. A multiplicative update scheme is proposed for solving the maximum likelihood estimation problem under non-negative constraints. While we provide proof of convergence for certain special cases, the multiplicative scheme has empirically demonstrated monotonic convergence behavior in virtually all our numerical experiments. We show that the presented method can pinpoint corrupted sample points and lead to better regression models when trained on synthetic and real-world scientific data sets.
The present work aims at deriving theoretical guaranties on the behavior of some cross-validation procedures applied to the $k$-nearest neighbors ($k$NN) rule in the context of binary classification. Here we focus on the leave-$p$-out cross-validation (L$p$O) used to assess the performance of the $k$NN classifier. Remarkably this L$p$O estimator can be efficiently computed in this context using closed-form formulas derived by cite{CelisseMaryHuard11}. We describe a general strategy to derive moment and exponential concentration inequalities for the L$p$O estimator applied to the $k$NN classifier. Such results are obtained first by exploiting the connection between the L$p$O estimator and U-statistics, and second by making an intensive use of the generalized Efron-Stein inequality applied to the L$1$O estimator. One other important contribution is made by deriving new quantifications of the discrepancy between the L$p$O estimator and the classification error/risk of the $k$NN classifier. The optimality of these bounds is discussed by means of several lower bounds as well as simulation experiments.
In the multiple testing context, a challenging problem is the estimation of the proportion $pi_0$ of true-null hypotheses. A large number of estimators of this quantity rely on identifiability assumptions that either appear to be violated on real data, or may be at least relaxed. Under independence, we propose an estimator $hat{pi}_0$ based on density estimation using both histograms and cross-validation. Due to the strong connection between the false discovery rate (FDR) and $pi_0$, many multiple testing procedures (MTP) designed to control the FDR may be improved by introducing an estimator of $pi_0$. We provide an example of such an improvement (plug-in MTP) based on the procedure of Benjamini and Hochberg. Asymptotic optimality results may be derived for both $hat{pi}_0$ and the resulting plug-in procedure. The latter ensures the desired asymptotic control of the FDR, while it is more powerful than the BH-procedure. Finally, we compare our estimator of $pi_0$ with other widespread estimators in a wide range of simulations. We obtain better results than other tested methods in terms of mean square error (MSE) of the proposed estimator. Finally, both asymptotic optimality results and the interest in tightly estimating $pi_0$ are confirmed (empirically) by results obtained with the plug-in MTP.
As the main workhorse for model selection, Cross Validation (CV) has achieved an empirical success due to its simplicity and intuitiveness. However, despite its ubiquitous role, CV often falls into the following notorious dilemmas. On the one hand, for small data cases, CV suffers a conservatively biased estimation, since some part of the limited data has to hold out for validation. On the other hand, for large data cases, CV tends to be extremely cumbersome, e.g., intolerant time-consuming, due to the repeated training procedures. Naturally, a straightforward ambition for CV is to validate the models with far less computational cost, while making full use of the entire given data-set for training. Thus, instead of holding out the given data, a cheap and theoretically guaranteed auxiliary/augmented validation is derived strategically in this paper. Such an embarrassingly simple strategy only needs to train models on the entire given data-set once, making the model-selection considerably efficient. In addition, the proposed validation approach is suitable for a wide range of learning settings due to the independence of both augmentation and out-of-sample estimation on learning process. In the end, we demonstrate the accuracy and computational benefits of our proposed method by extensive evaluation on multiple data-sets, models and tasks.
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