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For nonconvex optimization in machine learning, this article proves thatevery local minimum achieves the globally optimal value of the perturbablegradient basis model at any differentiable point. As a result, nonconvexmachine learning is theoretically as supported as convex machine learning witha handcrafted basis in terms of the loss at differentiable local minima, exceptin the case when a preference is given to the handcrafted basis over theperturbable gradient basis. The proofs of these results are derived under mildassumptions. Accordingly, the proven results are directly applicable to manymachine learning models, including practical deep neural networks, without anymodification of practical methods. Furthermore, as special cases of our generalresults, this article improves or complements several state-of-the-arttheoretical results on deep neural networks, deep residual networks, andoverparameterized deep neural networks with a unified proof technique and novelgeometric insights. A special case of our results also contributes to thetheoretical foundation of representation learning.

Every Local Minimum Value is the Global Minimum Value of Induced Model in Non-convex Machine Learning