Transfer Learning in Nonparametric Regression with Deep ReLU Networks

In this paper, we study the estimation of a marginal regression function from independent units with repeated binary, count, or continuous responses using ReLU deep neural networks. In the model, we assume that the dependence is generated by an unobserved random mean function within each unit. We then fit a neural network with a convex generalized regression loss. We show an oracle inequality by separating conditional measurement variation from between-unit variation. In addition, we prove that with $n$ units and $m$ measurements per unit, ReLU networks can attain an integrated mean squared error of order $n^{-1}+(nm)^{-2β/(2β+d)}$, up to logarithmic factors, over $β$-Hölder classes. We also derive a weighted oracle inequality for unequal cluster sizes and a rate for compositionally smooth functions. For pointwise ensemble inference, we give a projection central limit theorem and prove infinitesimal jackknife consistency under an explicit asymptotic linearity condition. Simulations and real data examples are provided to support our theoretical findings and practical implications.

This paper introduces two transfer learning methodologies for estimating nonparametric Bayesian networks under scarce data. We propose two algorithms, a constraint-based structure learning method, called PC-stable-transfer learning (PCS-TL), and a score-based method, called hill climbing transfer learning (HC-TL). We also define particular metrics to tackle the negative transfer problem in each of them, a situation in which transfer learning has a negative impact on the model's performance. Then, for the parameters, we propose a log-linear pooling approach. For the evaluation, we learn kernel density estimation Bayesian networks, a type of nonparametric Bayesian network, and compare their transfer learning performance with the models alone. To do so, we sample data from small, medium and large-sized synthetic networks and datasets from the UCI Machine Learning repository. Then, we add noise and modifications to these datasets to test their ability to avoid negative transfer. To conclude, we perform a Friedman test with a Bergmann-Hommel post-hoc analysis to show statistical proof of the enhanced experimental behavior of our methods. Thus, PCS-TL and HC-TL demonstrate to be reliable algorithms for improving the learning performance of a nonparametric Bayesian network with scarce data, which in real industrial environments implies a reduction in the required time to deploy the network.

This paper establishes a theoretical framework for the uniform convergence of smoothly activated deep neural network (DNN) estimators. While standard ReLU networks achieve minimax-optimal rates in the $L^2(P)$ norm for various nonparametric regression tasks, we establish a theoretical lower bound demonstrating that least-squares ReLU estimators can suffer from the curse of dimensionality in their uniform convergence behavior. Motivated by the need for reliable uniform guarantees in downstream tasks requiring worst-case reliability, we address this limitation by analyzing smoothly activated DNNs (smooth DNNs), encompassing both feedforward and residual structures. We establish novel pseudo-dimension bounds, non-asymptotic approximation guarantees, and Hölder-norm bounds for the approximators of these models. Leveraging these results, we derive non-asymptotic uniform convergence rates for smooth DNN estimators across multiple statistical contexts, including Huber, least-squares, quantile, and logistic regression. We prove that smooth DNNs can mitigate the {curse of dimensionality} in uniform convergence by adaptively exploiting the low-dimensional hierarchical composition structure of the target function. Supported by both simulation studies and a real-world application, our results position smooth DNNs as a theoretically grounded and practically viable alternative to ReLU networks for statistical learning tasks requiring uniform guarantees.