classifiers in the network [@Szegedy_etal14; @Lee_etal15]. These auxiliary objectives provide additional sources of error to deeper layers. Training, however, involves error signals that must propagate backwards from the top layer. Standard backpropagation is biologically unrealistic for several reasons: the need to buffer network states until errors arrive from the top layer; weight symmetry in the forward and backward passes; and the need to precisely interleave the forward and backward passes. Several biologically-motivated learning mechanisms have been proposed to explain how circuits in the brain are able to learn complex, hierarchical representations. One broad class of these proposals is based on contrastive learning in energy-based models [@Seung_etal03; @Bengio_Fischer15; @Scellier_Bengio17]. In these models, the network is trained to minimize the discrepancy between its equilibrium points when running freely and when observables clamp the values of some units in the network. Weight symmetry is required, though: each synaptic connection from one neuron to another assumes a matching synaptic connection of identical strength in the reverse direction. In [@Lillicrap_etal16; @Baldi_etal16], weight symmetry is avoided by using an independent set of fixed random weights to backpropagate errors between the network layers. However, like standard backpropagation, the error signals are non-local. Instead of backpropagating errors layer 