Optimal Global Rates of Convergence for Nonparametric Regression

By C. Stone. 1982

Subfields: Statistics and Probability, Statistical Learning Theory

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Citations in the corpus

Benchmarking Neural Network Robustness to Common Corruptions and Perturbations (2019, background): “For many nonparametric regression functions, adversarial robustness can require an increase in the training set that is exponential in d (Stone, 1982).”
Deep Neural Networks for Estimation and Inference: Application to Causal Effects and Other Semiparametric Estimands (2018, background): “Turning to special cases, we ﬁrst show that the optimal rate of Stone (1982) can be attained, up to log factors.”
Metalearners for estimating heterogeneous treatment effects using machine learning (2017, background): “[18] Stone CJ (1982) Optimal global rates of convergence for nonparametric regression.”
Divide and conquer kernel ridge regression: a distributed algorithm with minimax optimal rates (2013, background): “Then the mean-squared error is bounded as The upper bound (19) is unimprovable up to constant factors, as shown by known minimax bounds on estimation error in Sobolev spaces (Stone, 1982; Tsybakov, 2009); see also Theorem 2(b) of Raskutti et al. (2012).”
Additive Regression and Other Nonparametric Models (1985, background): “Model dimensionality, as just defined, is relevant to optimal rates of convergence for nonparametric function estimation (see Stone, 1982).”
Massachusetts Institute of Technology Department of Economics Working Paper Series Conditional Quantile Processes Based on Series or Many Regressors Conditional Quantile Processes Based on Series or Many Regressors (unknown): “In particular, we rely on the work of Stone [35], Andrews [2], Newey [29], Chen and Shen [13], Chen [12] and others that rigorously motivated the series framework as an approximation scheme and gave pointwise normality results for least squares estimators, and on Chen [12] and van de Geer [36] that ”
Smoothing splines estimators for functional linear regression (2009, background): “∗(τ)−S n(αe∗)(τ))2 dτ ≥c n ·n−(2m+2q+1)/(2m+2q+2) →1. Convergence of the last probability to 1 follows from well-known results on optimal rates of convergence in nonparametric regression (cf. Stone [27]). 6.5. Proof of Proposition 2. We ﬁrst consider (3.11). The set {H ρ} ρ>0 constitutes an orde”
Optimal rates and adaptation in the single-index model using aggregation (2007, methodology): “It answers in particular to Question 2 from Stone (1982) .”
Lower bounds on minimax rates for nonparametric regression with additive sparsity and smoothness (2009, unknown): “Stone [13]).”
Bivariate Penalized Splines for Regression (2013, background): “A convergence rate for the penalized spline estimators is derived that achieves the optimal nonparametric convergence rate established by Stone (1982). The asymptotic normality of the proposed estimators is established and shown to hold uniformly over the points where the regression function is esti”
Fast learning rates for plug-in classifiers (2005, methodology): “This has been shown by Stone [18] under the additional strong density assumption and by Yang [26] with no assumption on PX .”
Divide and Conquer Kernel Ridge Regression (2013, unknown): “Since the firstorder Sobolev kernel has eigenvalues (Gu, 2002) that scale as µ j ≃ (1/j) 2 , the minimax convergence rate in terms of squared L 2 (P)-error is N −2/3 (e.g. Tsybakov, 2009; Stone, 1982; Caponnetto and De Vito, 2007).”
Kernel-Based Reinforcement Learning (2004, background): “This is because the lower bounds for the complexity of non-linear regression are exponential (Stone, 1982) and reinforcement learning is at least as hard as nonlinear regression (an arbitrary regression”
On Semiparametric estimation in Single-Index Regression (2006, methodology): “Finally, when selecting the smoothing parameter h in a semiparametric M−estimation procedure, one may look for a bandwidth with the optimal rate of decay n−1/(2k+1), where 5 k is the number of derivatives required for the regression (Stone (1982)).”
Robust and efficient variable selection for semiparametric partially linear varying coefficient model based on modal regression (2013, methodology): “Therefore, from Theorems 1 and 2, it is clear that, by choosing proper tuning parameters, our proposed variable selection method is consistent and the estimators of nonparametric components achieve the optimal convergence rate as if the subset of true zero coefficients is already known (Stone 1982).”