Hessian Eigenspectra of More Realistic Nonlinear Models

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 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 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. • the (noisy) nonlinear factor model [22] , where y ∼ N (g(w T * x), σ 2 ) for some nonlinear linking function g : R → R and σ > 0; • the (noiseless) phase retrieval model [32] , with y = (w T * x) 2 , in which case we wish to reconstruct w * from its magnitude measurements; and • the single-layer NN model y = σ(w T * x), for some nonlinear activation function σ(t) such as the tanh-sigmoid σ(t) = tanh(t). For a given training set (x i , y i ) n i=1 of size n, the standard approach to obtain/recover the parameter w * ∈ R p is to solve the following optimization problem min w L(w) = min w 1 n n i=1 ℓ(y i , w T x i ), (3)