Benign landscapes provide a geometric explanation for the success of iterative methods on nonconvex problems arising in data science. The landscape is benign if every local minimizer is global and every other critical point is a strict saddle. For methods that avoid strict saddles, convergence to a critical point therefore implies convergence to a global minimizer. For smooth stochastic optimization, a common strategy for establishing such landscapes proceeds in two steps: $(i)$ establish benign geometry for the population objective---which is often easier to analyze---and $(ii)$ transfer this geometry to the empirical objective using concentration of gradients and Hessians. Recent work has extended saddle-point avoidance guarantees to weakly convex nonsmooth optimization through the notion of active strict saddles. We show, however, that benign population geometry need not transfer to empirical objectives in this setting. Specifically, for robust phase retrieval with Gaussian measurements---a well-studied and highly structured problem---the population objective is benign. Nevertheless, the empirical objective has sharp spurious local minima with probability bounded away from zero as the sample size tends to infinity. This asymptotic probability lower bound is independent of the dimension.
Mateo Díaz, Nicolás García Trillos, Daniel López-Castaño, Congwei Yang
Dimension-reduction methods such as PCA identify a subspace capturing most of the variation in the data, but not a distinguished basis within it. Selecting a basis in which the samples are sparsely represented is a recurring task in unsupervised learning, addressed by methods in factor analysis, dictionary learning, and subspace clustering. We pose this task as Wasserstein approximation of the data distribution by measures supported on the axes of an orthogonal frame, which, in turn, reduces to a nonsmooth optimization problem over the orthogonal group $\mathrm{O}(d)$. We show that, under mild assumptions, both empirical risk minimizers and a locally initialized stochastic gradient method recover the population-optimal frame at the near-parametric rate $\widetilde{O}_d(n^{-1/2})$, where $n$ is the sample size. The main difficulty is that, although the population objective is smooth, the empirical objective is not-this mismatch precludes standard covering arguments. We overcome this issue by replacing Lipschitz continuity of the empirical gradient with a coarsely Lipschitz bound, a tool that may be of independent interest. Numerical experiments support our theoretical findings.