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We identify a class of over-parameterized deep neural networks with standardactivation functions and cross-entropy loss which provably have no bad localvalley, in the sense that from any point in parameter space there exists acontinuous path on which the cross-entropy loss is non-increasing and getsarbitrarily close to zero. This implies that these networks have no sub-optimalstrict local minima.

On the loss landscape of a class of deep neural networks with no bad local valleys