trainable parameters. This had no effect on accuracy as shown in Table \[tab:mnist\].

[0.48]{} ![() MNIST test set errors obtained from three networks: a 3-layer network trained using local errors and symmetric local feedback weights (${\bf K}^i = {\bf M}{^i}^T$ ) where the errors for the three random local classifiers are shown, a network with two hidden layers trained using standard backpropagation, and a network with three hidden layers trained using standard backpropagation. The networks were trained for 100 epochs. Each line is the average of 20 training trials. () Same as () except that in the network trained using local errors, sign-concordant local feedback weights with independent and random magnitudes were used. []{data-label="fig:mnist_results"}](mnist_symmetric "fig:"){width="\textwidth"} \[fig:mnist\_symmetric\]

[0.48]{} ![() MNIST test set errors obtained from three networks: a 3-layer network trained using local errors and symmetric local feedback weights (${\bf K}^i = {\bf M}{^i}^T$ ) where the errors for the three random local classifiers are shown, a network with two hidden layers trained using standard backpropagation, and a network with three hidden layers trained using standard backpropagation. The networks were trained for 100 epochs. Each line is the average of 20 training trials. () Same as () except that in the 