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Loss functions with non-isolated minima have emerged in several machinelearning problems, creating a gap between theory and practice. In this paper,we formulate a new type of local convexity condition that is suitable todescribe the behavior of loss functions near non-isolated minima. We show thatsuch condition is general enough to encompass many existing conditions. Inaddition we study the local convergence of the SGD under this mild condition byadopting the notion of stochastic stability. The corresponding concentrationinequalities from the convergence analysis help to interpret the empiricalobservation from some practical training results.

A Local Convergence Theory for the Stochastic Gradient Descent Method in Non-Convex Optimization With Non-isolated Local Minima