An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions

Abstract

The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in a number of related literature. However, Rosenblatt's analysis is only valid for infinitesimal bandwidth.In contrast, we propose in this paper an upper bound of the bias which holds for finite bandwidths. Moreover, contrarily to the classic analysis we allow for discontinuous first order derivative of the regression function, we extend our bounds for multidimensional domains and we include the knowledge of the bound of the regression function when it exists and if it is known, to obtain a tighter bound. We believe that this work has potential applications in those fields where some hard guarantees on the error are needed.

Batch Reinforcement Learning With a Nonparametric Off-Policy Policy Gradient

“…Term A is the bias of Nadaraya-Watson kernel regression, as it is possible to observe in [33], therefore Theorem 2 applies…”

“…However, this asymptotic behavior is valid only for infinitesimal bandwidth, infinite samples (h → 0, nh → ∞) and requires the knowledge of the regression function and of the sampling distribution. In a recent work, we propose an upper bound of the bias that is also valid for finite bandwidths [33]. We show under some Lipschitz conditions that the bound of the Nadaraya-Watson kernel regression bias does not depend on the samples' distribution, which is a desirable property in off-policy scenarios.…”