Quantifying the complexity of feed-forward neural networks (FFNNs) remains challenging due to their nonlinear, hierarchical structure and numerous parameters. We apply generalized degrees of freedom (GDF) to measure model complexity in FFNNs with binary outcomes, adapting the algorithm for discrete responses. We compare GDF with both the effective number of parameters derived via log-likelihood cross-validation and the null degrees of freedom of Landsittel et al. Through simulation studies and a real data analysis, we demonstrate that GDF provides a robust assessment of model complexity for neural network models, as it depends only on the sensitivity of fitted values to perturbations in the observed responses rather than on assumptions about the likelihood. In contrast, cross-validation-based estimates of model complexity and the null degrees of freedom rely on the correctness of the assumed likelihood and may exhibit substantial variability. We find that GDF, cross-validation-based measures, and null degrees of freedom yield similar assessments of model complexity only when the fitted model adequately represents the data-generating mechanism. These findings highlight GDF as a stable and broadly applicable measure of model complexity for neural networks in statistical modeling.

翻译：量化前馈神经网络（FFNNs）的复杂度仍然具有挑战性，这源于其非线性、层次化的结构以及大量的参数。我们应用广义自由度（GDF）来测量具有二元结果的前馈神经网络的模型复杂度，并针对离散响应调整了算法。我们将GDF与通过对数似然交叉验证得到的有效参数数量以及Landsittel等人提出的零自由度进行了比较。通过模拟研究和真实数据分析，我们证明GDF为神经网络模型提供了稳健的模型复杂度评估，因为它仅依赖于拟合值对观测响应扰动的敏感性，而非关于似然函数的假设。相比之下，基于交叉验证的模型复杂度估计和零自由度依赖于所假设的似然函数的正确性，并可能表现出显著的变异性。我们发现，只有当拟合模型充分代表数据生成机制时，GDF、基于交叉验证的度量以及零自由度才会对模型复杂度给出相似的评估。这些发现凸显了GDF作为统计建模中神经网络模型复杂度的一种稳定且广泛适用的度量方法。