  We show that deep networks can be trained using Hebbian updates yielding
similar performance to ordinary back-propagation on challenging image datasets.
To overcome the unrealistic symmetry in connections between layers, implicit in
back-propagation, the feedback weights are separate from the feedforward
weights. The feedback weights are also updated with a local rule, the same as
the feedforward weights - a weight is updated solely based on the product of
activity of the units it connects. With fixed feedback weights as proposed in
Lillicrap et. al (2016) performance degrades quickly as the depth of the
network increases. If the feedforward and feedback weights are initialized with
the same values, as proposed in Zipser and Rumelhart (1990), they remain the
same throughout training thus precisely implementing back-propagation. We show
that even when the weights are initialized differently and at random, and the
algorithm is no longer performing back-propagation, performance is comparable
on challenging datasets. We also propose a cost function whose derivative can
be represented as a local Hebbian update on the last layer. Convolutional
layers are updated with tied weights across space, which is not biologically
plausible. We show that similar performance is achieved with untied layers,
also known as locally connected layers, corresponding to the connectivity
implied by the convolutional layers, but where weights are untied and updated
separately. In the linear case we show theoretically that the convergence of
the error to zero is accelerated by the update of the feedback weights.
