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.
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