Hessian Eigenspectra of More Realistic Nonlinear Models

TL;DR

This paper uses advanced random matrix theory techniques to precisely analyze the Hessian eigenspectra of broad nonlinear models, revealing diverse spectral behaviors and advancing understanding of complex machine learning models.

Contribution

It provides a rigorous characterization of Hessian eigenspectra for nonlinear models without simplifying assumptions, connecting spectral features to data and model properties.

Findings

Hessian spectra can have bounded or unbounded support.

Spectral behavior varies with data, response models, and loss functions.

Analysis explains features observed in complex machine learning models.

Abstract

Given an optimization problem, the Hessian matrix and its eigenspectrum can be used in many ways, ranging from designing more efficient second-order algorithms to performing model analysis and regression diagnostics. When nonlinear models and non-convex problems are considered, strong simplifying assumptions are often made to make Hessian spectral analysis more tractable. This leads to the question of how relevant the conclusions of such analyses are for more realistic nonlinear models. In this paper, we exploit deterministic equivalent techniques from random matrix theory to make a \emph{precise} characterization of the Hessian eigenspectra for a broad family of nonlinear models, including models that generalize the classical generalized linear models, without relying on strong simplifying assumptions used previously. We show that, depending on the data properties, the nonlinear response model, and the loss function, the Hessian can have \emph{qualitatively} different spectral behaviors: of bounded or unbounded support, with single- or multi-bulk, and with isolated eigenvalues on the left- or right-hand side of the bulk. By focusing on such a simple but nontrivial nonlinear model, our analysis takes a step forward to unveil the theoretical origin of many visually striking features observed in more complex machine learning models.