as it requires drastically less read and write operations compared to standard backpropagation. The reduction in the number of MAC operations is less unequivocal as it depends on the number of classification classes, $C$, and the fanout of the neurons in the network, $R^i$. Learning using local error reduces the MAC operations count if $L\times C < 0.5\sum_i  R^i$. This condition is easily satisfied when the number of classes is small and it was satisfied by all the networks presented in this paper.

  Training method               Memory read(words)                                                      Memory write (words)                                                  MAC operations
  ----------------------------- ----------------------------------------------------------------------- --------------------------------------------------------------------- ----------------------------------------------------
  Standard backpropagation      $N_{e}N_{b} \sum\limits_i (2\lvert P^i \rvert  + \lvert A^i \rvert )$   $N_{e}N_{b} \sum\limits_i (\lvert P^i \rvert  + \lvert A^i \rvert)$   $N_{e}N_{b} \sum\limits_i 3R^{i}\lvert A^i \rvert$
  Learning using local errors   $N_e N_b \sum\limits_i \lvert P^i \rvert $                              $N_e N_b \sum\limits_i \lvert P^i \rvert$                             $N_eN_b\sum\limits_i(2R^i + 2C)\lvert A^i \rvert$

  : Memory traffic and number of MAC operations for different learning methods[]{data-label="table:traffic_macs"}

Results {#sec:results}
=======

MNIST
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We first validate the performance of our training approach on the MNIST hand-written digit recognition task. We used the standard split of 50,000/10,000/10,000 examples for training/validation/testing respectively. The validation set was added to the training set after choosing the 