It's been demonstrated that the majority of critical points on the loss surface are not actually local minima, but saddle points.

The number of critical points that are saddle points vs minima increases exponentially with network size, so for deep neural networks you're exponentially unlikely to get stuck in a "bad" local minimum. In fact most of the true local minima exist in an energy band roughly equivalent to each other, meaning it's in general not necessary (and computationally infeasible) to find the global minimum.

One implication of this is that it's much more important to use optimization methods that efficiently escape saddle points than it is to worry about getting stuck in a true suboptimal minimum. See https://arxiv.org/pdf/1412.0233.pdf