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Most high-dimensional nonconvex optimization problems cannot be solved to optimality. However, deep neural networks have a benign geometry that allows stochastic optimization methods find acceptable solutions. There are, nevertheless, many open questions concerning the optimization process, including trade-offs between parallelism and the predictive ability of solutions, as well as the choice of a metric with the right statistical properties. In this talk we discuss classical and new optimization methods in the light of these observations, and conclude with some open questions.