Concept bottleneck model (CBM) is a ubiquitous method that can interpret neural networks using concepts. In CBM, concepts are inserted between the output layer and the last intermediate layer as observable values. This helps in understanding the reason behind the outputs generated by the neural networks: the weights corresponding to the concepts from the last hidden layer to the output layer. However, it has not yet been possible to understand the behavior of the generalization error in CBM since a neural network is a singular statistical model in general. When the model is singular, a one to one map from the parameters to probability distributions cannot be created. This non-identifiability makes it difficult to analyze the generalization performance. In this study, we mathematically clarify the Bayesian generalization error and free energy of CBM when its architecture is three-layered linear neural networks. We also consider a multitask problem where the neural network outputs not only the original output but also the concepts. The results show that CBM drastically changes the behavior of the parameter region and the Bayesian generalization error in three-layered linear neural networks as compared with the standard version, whereas the multitask formulation does not.

翻译：概念瓶颈模型（CBM）是一种能够利用概念解释神经网络的通用方法。在CBM中，概念作为可观测值被插入输出层与最后一个中间层之间。这有助于理解神经网络生成输出背后的原因：即从最后一个隐藏层到输出层中对应于概念的权重。然而，由于神经网络通常是一种奇异统计模型，目前尚无法理解CBM中泛化误差的行为。当模型是奇异的时，无法建立从参数到概率分布的一一映射。这种非可识别性使得分析泛化性能变得困难。在本研究中，我们从数学上阐明了当CBM架构为三层线性神经网络时其贝叶斯泛化误差和自由能。我们还考虑了一种多任务问题，其中神经网络不仅输出原始输出，还输出概念。结果表明，与标准版本相比，CBM极大地改变了三层线性神经网络中参数区域和贝叶斯泛化误差的行为，而多任务形式则未产生这种影响。