Quantifying Uncertainty In Wide Two-Layer Neural Networks: On The Law Of The Limiting Fluctuation Process
Authors: Arnaud Descours, Arnaud Guillin, Geoffrey Lacour, Manon Michel, Boris Nectoux, Paul Stos
Organizations: LSAF · LMBP · MaIAGE · UCA, LMBP
Abstract
Uncertainty quantification in neural networks prediction is a main issue for usual applications. Our approach seeks at reducing computation costs by directly evaluating uncertainty using PDE's information on the asymptotic variance, rather than the deep ensemble method which may be seen as a Monte Carlo estimation of the prediction, requiring the training of multiple networks. We thus study the law of the limiting process describing the random fluctuations around the mean-field limit of wide two-layer neural networks trained by stochastic gradient descent in a weak-noise regime. Building on a recent trajectorial central limit theorem, in which this limit is characterized as the weak solution of a linear stochastic evolution equation, we identify its law explicitly. More precisely, we show that it is a centered Gaussian process in the dual of a weighted Sobolev space, and we derive a closed covariance representation for the finite-dimensional distributions obtained by testing it against smooth functions. This covariance is expressed through the solution of a backward transport equation with a nonlocal source term, whose coefficients are driven by the mean-field trajectory. As a consequence, by testing against the activation function at a fixed input, we obtain an expression for the limiting variance of the corresponding network-output fluctuations. We illustrate this result numerically on a one-dimensional regression example.
We establish central and non-central limit theorems for sequences of functionals of the Gaussian output of an infinitely-wide random neural network on the d-dimensional sphere . We show that the asymptotic behaviour of these functionals as the depth of the network increases depends crucially on the fixed points of the covariance function, resulting in three distinct limiting regimes: convergence to the same functional of a limiting Gaussian field, convergence to a Gaussian distribution, convergence to a distribution in the Qth Wiener chaos. Our proofs exploit tools that are now classical (Hermite expansions, Diagram Formula, Stein-Malliavin techniques), but also ideas which have never been used in similar contexts: in particular, the asymptotic behaviour is determined by the fixed-point structure of the iterative operator associated with the covariance, whose nature and stability governs the different limiting regimes.
Simmaco Di Lillo, Leonardo Maini, Domenico Marinucci
We consider the infinite-width limit of a fully connected deep neural network with general weights, and we prove quantitative general bounds on the 2-Wasserstein distance between the network and its infinite-width Gaussian limit, under appropriate regularity assumptions on the activation function. Our main tool is a Lindeberg principle for Deep Neural Networks, which we use to successively replace the weights on each layer by Gaussian random variables.
Filippo Giovagnini, Sotirios Kotitsas, Marco Romito
We study the infinite-width Gaussian-process limit of random neural networks through the lens of tensor programs, and we provide a quantitative convergence theory in Wasserstein distance. Our main result gives explicit finite-width error bounds, of order inverse square-root of the widths between finite-network executions and their Gaussian-process limits. The framework is architecture-agnostic and covers feed-forward models together with weight-sharing schemes relevant for recurrent and transformer-type architectures.