Naoki Hayashi

Naoki Hayashi, Yoshihide Sawada

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.

Naoki Hayashi, Yoshihide Sawada

Concept Bottleneck Model (CBM) is a methods for explaining neural networks. In CBM, concepts which correspond to reasons of outputs are inserted in the last intermediate layer as observed values. It is expected that we can interpret the relationship between the output and concept similar to linear regression. However, this interpretation requires observing all concepts and decreases the generalization performance of neural networks. Partial CBM (PCBM), which uses partially observed concepts, has been devised to resolve these difficulties. Although some numerical experiments suggest that the generalization performance of PCBMs is almost as high as that of the original neural networks, the theoretical behavior of its generalization error has not been yet clarified since PCBM is singular statistical model. In this paper, we reveal the Bayesian generalization error in PCBM with a three-layered and linear architecture. The result indcates that the structure of partially observed concepts decreases the Bayesian generalization error compared with that of CBM (full-observed concepts).