Deep learning practitioners are aware that they're using non-convex optimization, stochastic gradient descent, and so on. In fact, those topics are core to modern deep learning research and explicitly acknowledged in most published research: LSTMs were invented to solve the vanishing gradient problem, Restricted Boltzmann Machines were used as a pre-training step to avoid local minima, and optimizers like ADAM have explicit guarantees about things like convergence.
You may know all this stuff already--not sure, based on your comment above.
If not, here are a few example papers from mainstream AI researchers explicitly talking about deep learning as an optimization problem or function approximator: [1] Why Does Unsupervised Pre-training Help Deep Learning?http://www.jmlr.org/papers/volume11/erhan10a/erhan10a.pdf [2] Multilayer feedforward networks are universal approximators: http://www.sciencedirect.com/science/article/pii/08936080899... [3] Deep Learning, Nature. http://www.cs.toronto.edu/~hinton/absps/NatureDeepReview.pdf [4] Adam: A Method for Stochastic Optimization. https://arxiv.org/abs/1412.6980