Regression neural networks (NNs) are most commonly trained by minimizing the mean squared prediction error, which is highly sensitive to outliers and data contamination. Existing robust training methods for regression NNs are often limited in scope and rely primarily on empirical validation, with only a few offering partial theoretical guarantees. In this paper, we propose a new robust learning framework for regression NNs based on the $β$-divergence (also known as the density power divergence) which we call `rRNet'. It applies to a broad class of regression NNs, including models with non-smooth activation functions and error densities, and recovers the classical maximum likelihood learning as a special case. The rRNet is implemented via an alternating optimization scheme, for which we establish convergence guarantees to stationary points under mild, verifiable conditions. The (local) robustness of rRNet is theoretically characterized through the influence functions of both the parameter estimates and the resulting rRNet predictor, which are shown to be bounded for suitable choices of the tuning parameter $β$, depending on the error density. We further prove that rRNet attains the optimal 50\% asymptotic breakdown point at the assumed model for all $β\in(0, 1]$, providing a strong global robustness guarantee that is largely absent for existing NN learning methods. Our theoretical results are complemented by simulation experiments and real-data analyses, illustrating practical advantages of rRNet over existing approaches in both function approximation problems and prediction tasks with noisy observations.

翻译：回归神经网络通常通过最小化均方预测误差进行训练，而该方法对异常值和数据污染极为敏感。现有的回归神经网络鲁棒训练方法往往适用范围有限，且主要依赖经验验证，仅少数方法提供部分理论保证。本文提出一种基于β-散度的回归神经网络鲁棒学习新框架，我们称之为"rRNet"。该框架适用于广泛的回归神经网络类别，包括具有非光滑激活函数和误差密度的模型，并能以经典最大似然学习作为特例进行还原。rRNet通过交替优化方案实现，我们在温和且可验证的条件下建立了其收敛至稳定点的理论保证。rRNet的（局部）鲁棒性通过参数估计值和所得预测器的影响函数进行理论刻画，证明在适当选择调谐参数β（取决于误差密度）时具有有界性。我们进一步证明，在假设模型下，rRNet对所有β∈(0, 1]均能达到最优的50%渐近崩溃点，这为现有神经网络学习方法普遍缺失的全局鲁棒性提供了有力保证。通过仿真实验和实际数据分析，我们的理论结果得到进一步验证，表明rRNet在含噪声观测的函数逼近问题和预测任务中均优于现有方法。