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.
Reinforcement learning (RL) algorithms still suffer from high sample complexity despite outstanding recent successes. The need for intensive interactions with the environment is especially observed in many widely popular policy gradient algorithms that perform updates using on-policy samples. The price of such inefficiency becomes evident in real world scenarios such as interaction-driven robot learning, where the success of RL has been rather limited. We address this issue by building on the general sample efficiency of off-policy algorithms. With nonparametric regression and density estimation methods we construct a nonparametric Bellman equation in a principled manner, which allows us to obtain closed-form estimates of the value function, and to analytically express the full policy gradient. We provide a theoretical analysis of our estimate to show that it is consistent under mild smoothness assumptions and empirically show that our approach has better sample efficiency than state-of-the-art policy gradient methods.
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 several related literature. However, given its asymptotic nature, it gives no access to a hard bound. The increasing popularity of predictive tools for automated decision-making surges the need for hard (non-probabilistic) guarantees. To alleviate this issue, we propose an upper bound of the bias which holds for finite bandwidths using Lipschitz assumptions and mitigating some of the prerequisites of Rosenblatt’s analysis. Our bound has potential applications in fields like surgical robots or self-driving cars, where some hard guarantees on the prediction-error are needed.
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