  When a computational system continuously learns from an ever-changing
environment, it rapidly forgets its past experiences. This phenomenon is called
catastrophic forgetting. While a line of studies has been proposed with respect
to avoiding catastrophic forgetting, most of the methods are based on intuitive
insights into the phenomenon, and their performances have been evaluated by
numerical experiments using benchmark datasets. Therefore, in this study, we
provide the theoretical framework for analyzing catastrophic forgetting by
using teacher-student learning. Teacher-student learning is a framework in
which we introduce two neural networks: one neural network is a target function
in supervised learning, and the other is a learning neural network. To analyze
continual learning in the teacher-student framework, we introduce the
similarity of the input distribution and the input-output relationship of the
target functions as the similarity of tasks. In this theoretical framework, we
also provide a qualitative understanding of how a single-layer linear learning
neural network forgets tasks. Based on the analysis, we find that the network
can avoid catastrophic forgetting when the similarity among input distributions
is small and that of the input-output relationship of the target functions is
large. The analysis also suggests that a system often exhibits a characteristic
phenomenon called overshoot, which means that even if the learning network has
once undergone catastrophic forgetting, it is possible that the network may
perform reasonably well after further learning of the current task.
