The Landscape of Empirical Risk for Non-convex Losses (1607.06534v3)

Abstract: 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.

• The paper demonstrates that under suitable sample conditions, the gradients and Hessians of empirical risks uniformly converge to those of the population risk.
• The paper shows that stationary points of the empirical risk correspond to those of the population risk, providing valuable insights for optimization.
• The paper applies its framework to nonconvex classification, robust regression, and Gaussian mixtures, extending analysis to high-dimensional regimes.

The Landscape of Empirical Risk for Non-Convex Losses

The paper investigates the empirical risk landscape for high-dimensional estimation problems involving non-convex loss functions— a topic of significant practical importance but still not thoroughly understood within empirical risk minimization frameworks. It presents a theoretical framework and demonstrates that despite the inherent challenges, valuable insights into the computation of M-estimators can be obtained by focusing on the stationary points and associated properties of the empirical risk function.

Classical empirical process theory provides assurances for the uniform convergence of empirical risk to population risk. However, convergence alone does not suffice for computational practicality due to the potential complexity of non-convex landscapes, which can have numerous local minima. The authors propose examining the landscape of empirical risk through its gradient and Hessian, extending classical analysis. They provide conditions under which these derivatives converge uniformly to their population counterparts.

Key Contributions:

1. Uniform Convergence of Derivatives: The paper establishes conditions under which the gradients and Hessians of empirical risks converge uniformly to those of the population risks. As soon as the number of samples, $n$, surpasses the number of parameters, $p$ (modulo logarithmic factors), good properties of population risk are shown to hold for empirical risks as well.
2. Stationary Points and Correspondence with Population Risk: When the gradient and Hessian of empirical risk match those of the population risk, stationary points can be correlated. This result implies significant insight into correspondence, where stationary points (minima, saddles) properly align with the stationary points of the population risk.
3. Analyzing Non-Convex Applications: Three quintessential cases are dissected: non-convex binary classification, robust regression, and Gaussian mixtures, showing how this framework helps delineate the complex landscape of the empirical risk surface and optimizes descent algorithms' convergence.
4. Extension to Very High-Dimensional Settings: The work extends analysis into settings where $p \gg n$, conditional on the sparsity of unknown parameters. This setup leads to a nearly information-theoretically minimal result concerning the uniform convergence of risk landscapes even in these less favorable "p > n" regimes.

Implications and Future Directions:

This paper's framework significantly impacts both theoretical aspects and practical algorithm implementations for high-dimensional machine learning tasks using non-convex loss functions. The results provide a basis to design algorithms that utilize the provable smoothness of empirical risk landscapes for efficient optimization, even in scenarios characterized by very high-dimensional feature spaces often found in genomics and signal processing.

Future research may explore further model-specific conditions where empirical landscapes demonstrate particularly desirable properties. Additionally, expanding this framework to multi-modal or highly irregular datasets promises significant advances in AI's ability to deduce meaningful parameters from complex data.

Consistency and Local Minima:

The results underscore that many non-convex optimization challenges can still be globally reconciled under established conditions of local consistency in derivatives' convergence. The theoretical backing encourages the exploration of more refined algorithms ensuring faster convergence to effectively global optimal solutions.

With promising results, the work paves the way for extensive, application-specific empirical and theoretical studies to fully exploit the rich landscape of empirical risk minimization using non-convex losses in various domains of artificial intelligence and machine learning.