  Backpropagation through time (BPTT) is the de facto standard for training
recurrent neural networks (RNNs), but it is non-causal and non-local. Real-time
recurrent learning is a causal alternative, but it is highly inefficient.
Recently, e-prop was proposed as a causal, local, and efficient practical
alternative to these algorithms, providing an approximation of the exact
gradient by radically pruning the recurrent dependencies carried over time.
Here, we derive RTRL from BPTT using a detailed notation bringing intuition and
clarification to how they are connected. Furthermore, we frame e-prop within in
the picture, formalising what it approximates. Finally, we derive a family of
algorithms of which e-prop is a special case.
