nit: a divergent training run is somewhat separate from the question of local minima. The latter observes the loss curve stabilize and asks 'is the _best_ (loss-minimizing) set of parameters for this loss/data/model/hypers/seed/etc?'. The former observes the loss curve explode, and asks for another 10k GPU hours to try again with a different choice of loss/data/model/... .

They're definitely related in some instances e.g. 'flat' vs 'steep' local minima with comparable losses.