Estimating the expected output of wide random MLPs more efficiently than sampling

TL;DR

This paper introduces a novel method to estimate the expected output of wide random MLPs over Gaussian inputs efficiently, avoiding sampling and reducing computational costs.

Contribution

The authors develop a sampling-free estimator using cumulants and Hermite expansions, improving efficiency for wide networks and rare event probability estimation.

Findings

Estimator achieves target mean squared error with fewer FLOPs than Monte Carlo sampling.

Method performs well in estimating probabilities of rare events.

Approach can be used for model training to reduce tail risks.

Abstract

By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.