Every Local Minimum is a Global Minimum of an Induced Model

For non-convex optimization in machine learning, this paper proves that every local minimum achieves the global optimality of the perturbable gradient basis model at any differentiable point. As a result, non-convex machine learning is theoretically as supported as convex machine learning with a hand-crafted basis in terms of the loss at differentiable local minima, except in the case when a preference is given to the hand-crafted basis over the perturbable gradient basis. The proofs of these results are derived under mild assumptions. Accordingly, the proven results are directly applicable to many machine learning models, including practical deep neural networks, without any modification of practical methods. Furthermore, as special cases of our general results, this paper improves or complements several state-of-the-art theoretical results in the literature with a simple and unified proof technique.