I have not dug into the math in the paper yet, but the surprising thing from my current perspective is: backprop is basically for supervised learning, while Hebbian learning is basically for unsupervised learning. In particular, Hebbian learning has been touted as an (inefficient but biologically plausible) algorithm for PCA. How can you chain a bunch of PCAs together and get gradient descent?

Aside from that, here's what I understood from the paper so far.

• By predictive coding, they basically mean: take the structure of the computation graph (eg, the structure of the NN) and interpret it as a gaussian bayes net instead.
• Calculating learning using only local information follows, therefore, from the general fact that bayes nets let you efficiently compute gradient descent (and some other rules, such as the EM algorithm) using only local information, so you don't have to perform automatic differentiation on the whole network.
• So the local computation of gradient descent isn't surprising: it's standard for graphical models, it's just unusual for NNs. This is one reason why graphical models might be a better model of the brain than artificial neural networks.
• The contribution of this paper is the nice correspondence between Gaussian bayes nets and NN backprop. I'm not really sure this should be exciting. It's not like it's useful for anything. If we were really excited about local learning rules, well, we already had some. 

Maybe the tremendous success of backprop lends some fresh credibility to bayes nets due to this correspondence. IE, maybe we are supposed to make an inference like: "I know backprop on NNs can be super effective, so I draw the lesson that learning for bayes nets (at least Gaussian bayes nets) can also be super effective, at a 100x slowdown." But this should have already been plausible, I claim. The machine learning community didn't really put bayes nets and NNs side by side and find bayes nets horribly lacking in learning capacity. Rather, I think the 100x slowdown was the primary motivator: bayes nets eliminate the need for an extra automatic differentiation step, but at the cost of a more expensive inference algorithm.

In particular, someone might take this as evidence that the brain uses Gaussian networks in particular, because we now know Gaussian approximates backprop, and we know backprop is super effective. I think this would be a mistaken inference: I don't think this provides much evidence that Gaussian bayes nets are especially intelligent compared to other Bayes nets.

On the other hand, the simplicity of the math for the Gaussian case does provide some evidence. It seems more plausible that the brain uses Gaussian bayes nets than, say, particle filters.

On page 8 at the end of section 4.1:

Due to the need to iterate the vs until convergence, the predictive coding network had roughly a 100x greater computational cost than the backprop network.

This seems to imply that artificial NNs are 100x more computationally efficient (at the cost of not being able to grow and probably lower fault tolerance etc.). Still, I'm updating to simulating a brain requiring much less CPU than the neurons in the brain would indicate. 

If they set ${\eta }_v$ to 1 they converge in a single backward pass${}^1$, since they then calculate precisely backprop. Setting ${\eta }_v$ to less than that and perhaps mixing up the pass order merely obfuscates and delays this process, but converges because any neuron without incorrect children has nowhere to go but towards correctness. And the entire convergence is for a single input! After which they manually do a gradient step on the weights as usual.

[Preliminary edit: I think this was partly wrong. Replicating...]

It's neat that you can treat activations and parameters by the same update rule, but then you should actually do it. Every "tick", replace the input and label and have every neuron update its parameters and data in lockstep, where every neuron can only look at its neighbors. Of course, this only has a chance of working if the inputs and labels come from a continuous stream, as they would if the input were the output of another network. They also notice the possibility of continuous data. And then one could see how its performance degrades as one speeds up the poor brain's environment :).

${}^1$: Which has to be in backward order and ${ϵ}_i\leftarrow v_i-{\hat{v}}_i$ has to be done once more after the v update line.

I think there's another big implication of this:

Current neural nets seem to require more data (e.g. games of Starcraft) to reach the same level of performance as a human adult. There are different hypotheses as to why this is:

--Human brains are bigger (bigger NN's require less data, it seems?)

--Human brains have a ton of pre-training from which they can acquire good habits and models which generalize to new tasks/situations (we call this "childhood" and "education"). GPT-3 etc. show that something similar works for neural nets, maybe we just need to do more of this with bigger brains.

--Human brains have tons of instincts built in (priors?)

--Human brains have special architectures, various modules that interact in various ways (priors?)

--Human brains don't use Backprop; maybe they have some sort of even-better algorithm

I've heard all of these hypotheses seriously maintained by various people. If this is true it rules out the last one.

You guys will probably find this Slate Star Codex post interesting:

https://slatestarcodex.com/2017/09/05/book-review-surfing-uncertainty/

Scott summarizes the Predictive Processing theory, explains it in a very accessible way (no math required), and uses it to explain a whole bunch of mental phenomena (attention, imagination, motor behavior, autism, schizophrenia, etc.)

Can someone ELI5/TLDR this paper for me, explain in a way more accessible to a non-technical person?

- How does backprop work if the information can't flow backwards?
- In Scotts post, he says that when lower-level sense data contradicts high-level predictions, high-level layers can override lower-level predictions without you noticing it. But if low-level sensed data has high confidence/precision - the higher levels notice it and you experience "surprise". Which one of those is equivalent to the backdrop error? Is it low-level predictions being overridden, or high-level layers noticing the surprise, or something else, like changing the connections between neurons to train the network and learn from the error somehow?

Is it known whether predictive coding is easier to train than backprop? Local learning rules seem like they would be more parallelizable.

Every Model Learned by Gradient Descent Is Approximately a Kernel Machine seems relevant to this discussion:

We show, however, that deep networks learned by the standard gradient descent algorithm are in fact mathematically approximately equivalent to kernel machines, a learning method that simply memorizes the data and uses it directly for prediction via a similarity function (the kernel). This greatly enhances the interpretability of deep network weights, by elucidating that they are effectively a superposition of the training examples. The network architecture incorporates knowledge of the target function into the kernel.

Thanks for sharing!

Two comments:

• There seem to be a couple of sign errors in the manuscript. (Probably worth reaching out to the authors directly)
• Their predictive coding algorithm holds the vhat values fixed during convergence, which actually implies a somewhat different network topology than the more traditional one shown in your figure.

I'm confused. The bottom diagram seems to also involve bidirectional flow of information.

There is no relationship between neurons and the "neurons" of an ANN. It's just a naming mishap at this point.