$i$. We ignore the special case of the input layer where errors do not need to be backpropagated. We also ignore the MAC operations needed to calculate the error at the top layer. Figure \[fig:traffic\_b\] illustrates the case when learning is done using errors generated by random local classifiers. As in standard backpropagation, $R^{i-1}\lvert A^{i-1} \rvert$ MAC operations are needed to calculate the activations of layer $i$. To calculate the local classifier output, $C \lvert A^i \rvert$ MAC operations are needed where $C$ is the number of classification classes. Note that the random classifier weights can be generated on the fly using a PRNG, and thus only require the PRNG seed (whose size can be 32 bits for 32-bit weights) to be stored. To backpropagate the local classifier error to obtain the error at layer $i$, an additional $C \lvert A^i \rvert$ MAC operations are needed and $R^{i-1}\lvert A^{i-1} \rvert$ MAC operations are needed to update the parameters of layer $i$, $P^i$, based on the layer’s error.

Table \[table:traffic\_macs\] summarizes the number of MAC operations and the memory read/write volume required by the two training methods. Learning using local errors has a decisive advantage when it comes to memory traffic 