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.
much to chew on here, it really does seem like a very interesting class of models. from the papers it sounds like in practice clipping weights to a small set works okay, so the constant factors shouldn't be too bad.
i may have to sit down and try to implement these...
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.