Upper Bound of Bayesian Generalization Error in Partial Concept
  Bottleneck Model (CBM): Partial CBM outperforms naive CBM
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).