Similar to showing a problem falls in NP, you can reduce the problem down to another problem in NP and be done with it.
There's a number of problems with svms; complexity for training and inference scales with the amount of training data, which is pretty sad panda for complex problems.
Extremely spicy/cynical take: it's not cool to say "you all should go look at all these possible applications" when the thrust is the paper is to prop up the relevance of an obsolete approach. You gotta do the actual work to close the gap if you still want your PhD to be worth something...
That said, I haven't read the paper terribly closely, and am always happy to be proven wrong!
What are the leading theories for why this seems to be the case? Less nodes to capture and direct decisions?
Is this true though, or does network architecture only matter in terms of efficiency? This is non rhetorical, I really don't know much about deep learning. :) I guess i'm asking if with enough data and compute, is architecture still relevant?
https://papers.nips.cc/paper/2018/file/a41b3bb3e6b050b6c9067...
In effect, there's a big gap between an existence proof and actually workable models, and the tricks of the trade do quite a lot to close the gap. (And there are almost certainly more tricks that we're still not aware of! I'm still amazed at how late in the game batch normalization was discovered.)
OTOH, so long as you're using the basic tricks of the trade, IME architecture doesn't really matter much. Our recent kaggle competition for birdsong identification was a great example of this: pretty much everyone reported that the difference between five or so 'best practices' feature extraction architectures (various permutations of resnet/efficientnet) was negligible.
Can we do the same with neural networks? Is there a formalization of why 'skip connections' (which I know nothing about) are better, why transformers are more efficient than recurrance, etc?
Is it useful to talk about their complexity or universal properties or size (and I realize this is muddled up a bit by the fact that hardware architecture can sometimes trump efficiency).
It would be fantastic to have better measures of problem complexity. My thinking at this point is that huge parameter size makes it easier to search for a solution, but once we've found one, there should be interesting ways to simplify the function you've found. Recall above: there are many equivalent models with the same loss when the learning slows down... Some of these equivalent models have lots of zeros in them. We find that often you can prune 90% of the weights and still have a perfectly good model. Eventually you hit a wall, where it gets hard to prune more without large drops in model quality; this /might/ correspond to the actual problem complexity somehow, but the pruned model you happened to find may not actually be the best, and there may be better methods we haven't discovered yet.
For many non-deep machine learning models like support vector machines (SVMs), "find the function in my class that fits the data best" can be posed as a convex optimization problem. So the magical algorithm actually exists. In this setting, the PAC analysis is the natural thing to study. (Typically, we find that you need more samples if your function class is more expressive, which agrees with the observation that deep learning requires huge data sets.)
In deep learning, the magical algorithm doesn't exist. The optimization problem is nonconvex, but everyone uses stochastic gradient descent (SGD), an algorithm that is only guaranteed to find the global optimum on convex problems. Theory suggests that SGD will often converge on a local optimum that is significantly worse than the global optimum. However, in practice this doesn't happen much! If the network is big enough and all the algorithm hyperparameters are tuned well, and you run deep the deep learning algorithm with different random seeds, the result will be about equally good every time.
ML theory people working in deep learning tend to focus on this phenomenon: why does SGD usually find good local optima? This is totally different from the PAC analysis, and the analogy with computational complexity is less crisp.