Nonconvex optimization is becoming one of the most powerful workhorses of data science and artificial intelligence. Compared with convex optimization, it enjoys superior statistical accuracy, computational efficiency, and modeling flexibility in numerous modern settings. However, the empirical success of nonconvex optimization largely eludes the reach of classical statistical and optimization theory, which prohibits us from designing more efficient algorithms in a principled manner.

In this talk, I will illustrate how statistical thinking enables us to harness the power of nonconvex optimization. In specific, I will present an algorithmic framework for exploiting the latent geometry induced by the randomness of data. By integrating three new global exploration meta-algorithms — namely, homotopy continuation, tightening after relaxation, and noise regularization — with local search heuristics — such as the variants of gradient descent — this unified framework leads to new nonconvex optimization algorithms for a broad variety of challenging learning problems. In particular, these algorithms enjoy provably optimal statistical accuracy and computational efficiency, and moreover, lead to new scientific discoveries. Time permitting, I will discuss an interesting “more data, less computation” phenomenon, which arises from nonconvex optimization, but generalizes to even more algorithms.