friendly_aixi

1) It's definitely not the first. Other methods have universal guarantees of some form or other with well quantified rates of convergence, e.g. k-NN would be the best known example.

2) There are some restrictions on the class of density functions it can model, so arbitrarily weird is a bit strong, but the model class is very general.

3) The weights needed to model any function in this class although finite, can be arbitrarily large. The regret of a single neuron has a dependence on the diameter of the convex set your weights reside in, so there is a nasty constant of sorts in there, and this will also carry over when you analyse the regret of a complete network. With such a general statement, it's unavoidable sadly.

4) The universality result on its own is just a nice property. See it as the first stepping stone to a more meaningful analysis. What you really want is for the model class to grow as you add more neurons, using weights within a realistic range, and that the method performs well in practice on some problems people care about -- we provide empirical evidence that the capacity grows on par with deep relu networks with our capacity experiments, and show a bunch of results where the method works, but we don't have a theoretical characterisation of the class of density functions it can model well (i.e. if the function has some nice structural property, then a network of reasonable size is guaranteed to learn a good approximation quickly). Such a result would be extraordinary in my eyes. Because the network is composed of simple and well understood building blocks, I am optimistic that such an analysis will be possible in the future.

1) Indeed, GLNs don't learn features... but I would claim they do learn some notion of an intermediate representation, it's just different from the DL mainstream -- in particular its closely related to the inverse Radon transform in medical imaging.

2) Inputs which are similar in terms of cosine similarity will map to similar (data dependent) products of weight matrices, and thus behave similarly, which of course can affect performance in both good and bad ways. With the results we show on permuted MNIST, its well... just not particularly likely that they will interfere. This is a good thing -- why should completely different data distributions interfere with one another? The point is the method is resiliant to catastrophic forgetting when the cosine similarity between data items from different tasks is small. This highlights the different kind of inductive bias a halfspace gated GLN has compared to a Deep ReLu network.

3) Re bird example, that's slightly unfair. I am sure one could easily make use of the pre-trained resnet to provide informative features to a GLN -- it's early days for this method, hybrid systems haven't been investigated, so I don't know whether it would work better than current SOTA methods for image classification. But I would be pretty confident that some simple combination would work better than chopping the head off a pretrained network and fitting an SVM on top. This is all speculation on my part though. :)

The result here is stronger, in the sense that typical NN universality results are statements with respect to just capacity (and not how you optimise them). Here, the result holds with respect to both capacity + a choice of suitable no regret online convex optimisation algorithm (e.g. online gradient descent). Of course, this is just one desirable property of a general purpose learning algorithm.

Convolution is a linear operation; in the case of images, you can view it as a multiplication with a doubly block circulant matrix. I can't see any barriers to hybrid approaches here, though it seems difficult to avoid using backpropagation for credit assignment within the convolutional layers.

Re: significance, how about their application in regression: https://arxiv.org/abs/2006.05964 ? Or in contextual bandits: https://arxiv.org/abs/2002.11611 ?

Disclaimer: I am one of the authors (Joel).