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Two Routes to Scalable Credit Assignment without Weight Symmetry This paper looks at five different learning rules, three purely local, and two nonlocal, to see if they can work as well as backprop in training a deep convolutional net on ImageNet. The local learning networks all feature forward weights W and backward weights B; the forward weights (+ nonlinearities) pass the information to lead to a classification; the backward weights pass the error, which is used to locally adjust the forward weights. Hence, each fake neuron has locally the forward activation, the backward error (or loss gradient), the forward weight, backward weight, and Hebbian terms thereof (e.g the outer product of the inout vectors for both forward and backward passes). From these available variables, they construct the local learning rules:
Each of these serves as a "regularizer term" on the feedback weights, which governs their learning dynamics. In the case of backprop, the backward weights B are just the instantaneous transpose of the forward weights W. A good local learning rule approximates this transpose progressively. They show that, with proper hyperparameter setting, this does indeed work nearly as well as backprop when training a ResNet18 network. But, hyperparameter settings don't translate to other network topologies. To allow this, they add in nonlocal learning rules:
In "Symmetric Alignment", the Self and Decay rules are employed. This is similar to backprop (the backward weights will track the forward ones) with L2 regularization, which is not new. It performs very similarly to backprop. In "Activation Alignment", Amp and Sparse rules are employed. I assume this is supposed to be more biologically plausible  the Hebbian term can track the forward weights, while the Sparse rule regularizes and stabilizes the learning, such that overall dynamics allow the gradient to flow even if W and B aren't transposes of each other. Surprisingly, they find that Symmetric Alignment to be more robust to the injection of Gaussian noise during training than backprop. Both SA and AA achieve similar accuracies on the ResNet benchmark. The authors then go on to explain the plausibility of nonlocal but approximate learning rules with Regression discontinuity design ala Spiking allows neurons to estimate their causal effect. This is a decent paper,reasonably well written. They thought trough what variables are available to affect learning, and parameterized five combinations that work. Could they have done the full matrix of combinations, optimizing just they same as the metaparameters? Perhaps, but that would be even more work ... Regarding the desire to reconcile backprop and biology, this paper does not bring us much (if at all) closer. Biological neural networks have specific and local uses for error; even invoking 'error' has limited explanatory power on activity. Learning and firing dynamics, of course of course. Is the brain then just an overbearing mess of details and overlapping rules? Yes probably but that doesn't mean that we human's can't find something simpler that works. The algorithms in this paper, for example, are well described by a bit of linear algebra, and yet they are performant.  
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PMID17946450[0] An Autonomous, broadband, multichannel neural recording system for freely behaving primates
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