Yes it's true that hyperparameters seem arbitrary, but that's just a consequence of the no-free-lunch theorem. Some models will always fit some problems better than others. There is no such thing as a perfect model. NNs turn out to be a really good prior for real world problems. I wrote about why that might be here: http://houshalter.tumblr.com/post/120134087595/approximating...
But in principle, that doesn't apply to things like layer numbers. In theory the best neural net has infinite layers of infinite size and infinite convolution, with a stride of 1. Because you can fit every other model into that, and as long as the parameters are properly regularized it can also avoid overfitting too. In the real world of course, we are limited to merely 50 layers, and need to cut corners with convolution sizes and such.
Likewise for training. The best algorithm for training is bayesian inference on the parameters. Since that is usually very impractical, we use approximations like maximum-likelihood or dropout.
It goes without saying that the reason we have 50 (vs. deeper) layer nets and strides, among other things, is not solely to reduce computational cost.
In order to do that you do need to use multiple layers. And the same is true for digital circuit, which NNs basically are. I'm sure there is mathematical theory and literature on the representation power of digital circuits.
There is a limit to what you can do with only one layer of circuits, and you can do more functions more efficiently with more layers. That is, taking the results of some operations, then doing more operations on those results. Composing functions. As opposed to just memorizing a lookup table, which is inefficient.
That's why multiple layers work better. It isn't some strange mystery.
>an infinite number of layers and convolutions makes no sense - I think you meant to use the word arbitrary
A better way to word it would be "as it approaches infinity" or "in the limit" or something. That is, the accuracy of the neural net should increase and only increase as you increase the number of layers and units (provided you have proper regularization/priors.) Since bigger models can emulate smaller models, but not vice versa.
The forward pass of a net is not theoretically interesting. It's the training of the net that has no theory. The training has nothing to do with digital circuits.
It goes without saying that you've handwaved some (perfectly fine) ideas about composing functions and such. And then claim "it isn't some strange mystery." That's my point. You've argued some ideas from intuition. There is little theoretical rigor around this, however.
Not with proper priors/regularization.
>You've argued some ideas from intuition. There is little theoretical rigor around this, however.
There's this paper which goes more into theoretical depth on the idea: http://arxiv.org/abs/1412.0233
So whenever you read about some 50-layer net trained with this architecture with this padding/stride/normalization, and you wonder how they came up with that, the answer is: some grad student sat there, thought about his past experiences and the papers he's read of architectures that have worked well, and then spent months trying a bunch of things.