The Annals of Statistics

The landscape of empirical risk for nonconvex losses

Song Mei, Yu Bai, and Andrea Montanari

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Most high-dimensional estimation methods propose to minimize a cost function (empirical risk) that is a sum of losses associated to each data point (each example). In this paper, we focus on the case of nonconvex losses. Classical empirical process theory implies uniform convergence of the empirical (or sample) risk to the population risk. While under additional assumptions, 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 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 are able to establish one-to-one correspondence of their stationary points. We demonstrate that in several problems such as nonconvex 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 provides 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 parameters vector (modulo logarithmic factors), then a suitable uniform convergence result holds. We apply this result to nonconvex binary classification and robust regression in very high-dimension.

Article information

Ann. Statist., Volume 46, Number 6A (2018), 2747-2774.

Received: January 2017
Revised: August 2017
First available in Project Euclid: 7 September 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62F10: Point estimation 62J02: General nonlinear regression
Secondary: 62H30: Classification and discrimination; cluster analysis [See also 68T10, 91C20]

Nonconvex optimization empirical risk minimization landscape uniform convergence


Mei, Song; Bai, Yu; Montanari, Andrea. The landscape of empirical risk for nonconvex losses. Ann. Statist. 46 (2018), no. 6A, 2747--2774. doi:10.1214/17-AOS1637.

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Supplemental materials

  • Supplement: Proofs and simulations. The supplement provides some technical background lemmas and gives all the proofs of the theorems, and additional numerical simulations.