Estimating the expected output of wide random MLPs more efficiently than sampling
This work addresses the computational bottleneck of estimating expected losses in MLPs at initialization, offering a more efficient alternative to sampling for practitioners concerned with tail risks.
The authors propose a method to estimate the expected output of wide random MLPs over Gaussian inputs without sampling, using cumulants and Hermite expansions. Their estimator achieves a target MSE with substantially fewer FLOPs than Monte Carlo sampling, especially for rare events.
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.