Stochastic gradient methods and risk of neural nets Under many situations it is currently provable that we can minimize the risk of neural nets using stochastic gradient based algorithms. For example : https://arxiv.org/abs/1811.03804, https://arxiv.org/abs/1811.04918 https://arxiv.org/abs/1811.03962, https://arxiv.org/abs/1810.12065, https://arxiv.org/pdf/1810.02054.pdf, https://arxiv.org/abs/1705.04591 and dozens more!

Is the focus on stochastic gradient based methods for this question purely motivated by the fact that this is pretty much the only thing used in practice? 

Or is there a fundamental/complexity-theoretic reason why there cant be an algorithm (or its improbably hard to find an algorithm) that does not use stochastic gradients (is maybe deterministic!) and yet minimizes the risk of neural nets as fast as one can using SGD-like methods?
The question has changed somewhat in the comments, so I'll address its new version: "Given a class of algorithms $A$ and an $\epsilon &gt;0$ and a loss class $L$ and a data distribution $D$, one cannot use algorithms of type $A$ to find a member of $L$ whose generalization error is below $\epsilon$ unless running time is $f(\epsilon)$"... .

One such lower bound I'm aware of is for the case of Statistical Query based algorithms. See, for example, the paper by Yang, "New lower bounds for statistical query learning",
https://www.sciencedirect.com/science/article/pii/S0022000004001291
where, in particular,
it is shown that "any SQ-based algorithm needs running time $\Omega(2^n)$".