Adding to sibling post from phao, Optimizers like ADAM essentially do a diagonal approximation to Newton's method, which gives some, though certainly not all, of the benefit of Newton.

There is the seminal paper "Deep Learning via Hessian-free Optimization" [1] that applies Newton's method (Hessian free because of a clever trick of solving Hx=-g iteratively without explicitly constructing the Hessian, and getting an approximation without full O(n^3)) - but n^2 or n^3 is just too large in practice when the diagonal approximations work just fine.

[1] https://www.cs.toronto.edu/~jmartens/docs/Deep_HessianFree.p...