  Most high-dimensional estimation and prediction methods propose to minimize a
cost function (empirical risk) that is written as a sum of losses associated to
each data point. In this paper we focus on the case of non-convex losses, which
is practically important but still poorly understood. Classical empirical
process theory implies uniform convergence of the empirical risk to the
population risk. While uniform convergence implies consistency of the resulting
M-estimator, it does not ensure that the latter can be computed efficiently.
  In order to capture the complexity of computing M-estimators, we propose to
study the landscape of the empirical risk, namely its stationary points and
their properties. We establish uniform convergence of the gradient and Hessian
of the empirical risk to their population counterparts, as soon as the number
of samples becomes larger than the number of unknown parameters (modulo
logarithmic factors). Consequently, good properties of the population risk can
be carried to the empirical risk, and we can establish one-to-one
correspondence of their stationary points. We demonstrate that in several
problems such as non-convex binary classification, robust regression, and
Gaussian mixture model, this result implies a complete characterization of the
landscape of the empirical risk, and of the convergence properties of descent
algorithms.
  We extend our analysis to the very high-dimensional setting in which the
number of parameters exceeds the number of samples, and provide a
characterization of the empirical risk landscape under a nearly
information-theoretically minimal condition. Namely, if the number of samples
exceeds the sparsity of the unknown parameters vector (modulo logarithmic
factors), then a suitable uniform convergence result takes place. We apply this
result to non-convex binary classification and robust regression in very
high-dimension.
