network trained using local errors, sign-concordant local feedback weights with independent and random magnitudes were used. []{data-label="fig:mnist_results"}](mnist_signconcordant "fig:"){width="\textwidth"} \[fig:mnist\_concordant\]

[lp[35mm]{}p[35mm]{}p[35mm]{}]{} & Local error learning (symmetric feedback weights) & Local error learning (sign-concordant feedback weights) & Local error learning (trainable local classifier)\
Test error & & &\
\
& Learning using feedback alignment & 2-layer network trained using backprop & 3-layer network trained using backprop\
Test error & $1.70 \pm 0.087 \%$ & $1.26 \pm 0.068 \%$ & $1.27 \pm 0.050 \%$\

\[tab:mnist\]

Next, to lessen concern of biological implausibility of exact symmetry in feedforward and feedback weights, we relaxed the weight symmetry requirement in the local error loops and initialized the error feedback weights ${\bf K}^i$ randomly and independently of ${\bf M}{^i}$, except we then modified the sign of the weights in ${\bf K}^i$ so that $sign({\bf K}^i) = sign({\bf M}{^i}^T)$. The signs of the feedback weights in the local error loops thus match the signs of the feedforward weights (both are fixed and have independent magnitudes). This is the ’sign-concordant feedback weights’ case shown in Fig. \[fig:mnist\_concordant\]. Performance deteriorates slightly in this case compared to symmetric feedforward and feedback local classifier weights. When we relax the symmetry requirement further 