  By learning a sequence of tasks continually, an agent in continual learning
(CL) can improve the learning performance of both a new task and `old' tasks by
leveraging the forward knowledge transfer and the backward knowledge transfer,
respectively. However, most existing CL methods focus on addressing
catastrophic forgetting in neural networks by minimizing the modification of
the learnt model for old tasks. This inevitably limits the backward knowledge
transfer from the new task to the old tasks, because judicious model updates
could possibly improve the learning performance of the old tasks as well. To
tackle this problem, we first theoretically analyze the conditions under which
updating the learnt model of old tasks could be beneficial for CL and also lead
to backward knowledge transfer, based on the gradient projection onto the input
subspaces of old tasks. Building on the theoretical analysis, we next develop a
ContinUal learning method with Backward knowlEdge tRansfer (CUBER), for a fixed
capacity neural network without data replay. In particular, CUBER first
characterizes the task correlation to identify the positively correlated old
tasks in a layer-wise manner, and then selectively modifies the learnt model of
the old tasks when learning the new task. Experimental studies show that CUBER
can even achieve positive backward knowledge transfer on several existing CL
benchmarks for the first time without data replay, where the related baselines
still suffer from catastrophic forgetting (negative backward knowledge
transfer). The superior performance of CUBER on the backward knowledge transfer
also leads to higher accuracy accordingly.
