Faithful Heteroscedastic Regression with Neural Networks

TL;DR

This paper introduces a simple optimization modification for neural heteroscedastic regression that ensures accurate mean estimates and well-calibrated variance, improving over existing methods without complex surrogates or Bayesian approaches.

Contribution

It proposes a novel, theoretically justified optimization technique for heteroscedastic neural regression that guarantees mean accuracy and enhances variance calibration.

Findings

Mean estimates match those of homoscedastic models.

Variance calibration is significantly improved.

Method recovers underlying heteroscedastic noise variance.

Abstract

Heteroscedastic regression models a Gaussian variable's mean and variance as a function of covariates. Parametric methods that employ neural networks for these parameter maps can capture complex relationships in the data. Yet, optimizing network parameters via log likelihood gradients can yield suboptimal mean and uncalibrated variance estimates. Current solutions side-step this optimization problem with surrogate objectives or Bayesian treatments. Instead, we make two simple modifications to optimization. Notably, their combination produces a heteroscedastic model with mean estimates that are provably as accurate as those from its homoscedastic counterpart (i.e.~fitting the mean under squared error loss). For a wide variety of network and task complexities, we find that mean estimates from existing heteroscedastic solutions can be significantly less accurate than those from an equivalently expressive mean-only model. Our approach provably retains the accuracy of an equally flexible mean-only model while also offering best-in-class variance calibration. Lastly, we show how to leverage our method to recover the underlying heteroscedastic noise variance.