66,947 research outputs found

    A Convex Framework for Confounding Robust Inference

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    We study policy evaluation of offline contextual bandits subject to unobserved confounders. Sensitivity analysis methods are commonly used to estimate the policy value under the worst-case confounding over a given uncertainty set. However, existing work often resorts to some coarse relaxation of the uncertainty set for the sake of tractability, leading to overly conservative estimation of the policy value. In this paper, we propose a general estimator that provides a sharp lower bound of the policy value using convex programming. The generality of our estimator enables various extensions such as sensitivity analysis with f-divergence, model selection with cross validation and information criterion, and robust policy learning with the sharp lower bound. Furthermore, our estimation method can be reformulated as an empirical risk minimization problem thanks to the strong duality, which enables us to provide strong theoretical guarantees of the proposed estimator using techniques of the M-estimation.Comment: This is an extension of the following work https://proceedings.mlr.press/v206/ishikawa23a.html. arXiv admin note: text overlap with arXiv:2302.1334

    Multiplicative local linear hazard estimation and best one-sided cross-validation

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    This paper develops detailed mathematical statistical theory of a new class of cross-validation techniques of local linear kernel hazards and their multiplicative bias corrections. The new class of cross-validation combines principles of local information and recent advances in indirect cross-validation. A few applications of cross-validating multiplicative kernel hazard estimation do exist in the literature. However, detailed mathematical statistical theory and small sample performance are introduced via this paper and further upgraded to our new class of best one-sided cross-validation. Best one-sided cross-validation turns out to have excellent performance in its practical illustrations, in its small sample performance and in its mathematical statistical theoretical performance

    Optimal cross-validation in density estimation with the L2L^2-loss

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    We analyze the performance of cross-validation (CV) in the density estimation framework with two purposes: (i) risk estimation and (ii) model selection. The main focus is given to the so-called leave-pp-out CV procedure (Lpo), where pp denotes the cardinality of the test set. Closed-form expressions are settled for the Lpo estimator of the risk of projection estimators. These expressions provide a great improvement upon VV-fold cross-validation in terms of variability and computational complexity. From a theoretical point of view, closed-form expressions also enable to study the Lpo performance in terms of risk estimation. The optimality of leave-one-out (Loo), that is Lpo with p=1p=1, is proved among CV procedures used for risk estimation. Two model selection frameworks are also considered: estimation, as opposed to identification. For estimation with finite sample size nn, optimality is achieved for pp large enough [with p/n=o(1)p/n=o(1)] to balance the overfitting resulting from the structure of the model collection. For identification, model selection consistency is settled for Lpo as long as p/np/n is conveniently related to the rate of convergence of the best estimator in the collection: (i) p/n→1p/n\to1 as n→+∞n\to+\infty with a parametric rate, and (ii) p/n=o(1)p/n=o(1) with some nonparametric estimators. These theoretical results are validated by simulation experiments.Comment: Published in at http://dx.doi.org/10.1214/14-AOS1240 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

    Model selection consistency from the perspective of generalization ability and VC theory with an application to Lasso

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    Model selection is difficult to analyse yet theoretically and empirically important, especially for high-dimensional data analysis. Recently the least absolute shrinkage and selection operator (Lasso) has been applied in the statistical and econometric literature. Consis- tency of Lasso has been established under various conditions, some of which are difficult to verify in practice. In this paper, we study model selection from the perspective of generalization ability, under the framework of structural risk minimization (SRM) and Vapnik-Chervonenkis (VC) theory. The approach emphasizes the balance between the in-sample and out-of-sample fit, which can be achieved by using cross-validation to select a penalty on model complexity. We show that an exact relationship exists between the generalization ability of a model and model selection consistency. By implementing SRM and the VC inequality, we show that Lasso is L2-consistent for model selection under assumptions similar to those imposed on OLS. Furthermore, we derive a probabilistic bound for the distance between the penalized extremum estimator and the extremum estimator without penalty, which is dominated by overfitting. We also propose a new measurement of overfitting, GR2, based on generalization ability, that converges to zero if model selection is consistent. Using simulations, we demonstrate that the proposed CV-Lasso algorithm performs well in terms of model selection and overfitting control
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