This repository contains the code release for the paper "On the Pitfalls of Heteroscedastic Uncertainty Estimation with Probabilistic Neural Networks", published at ICLR 2022.

This work was done by Maximilian Seitzer, Arash Tavakoli, Dimitrije Antic, Georg Martius at the Autonomous Learning Group, Max-Planck Institute for Intelligent Systems in Tübingen.

If you make use of our work, please use the citation information below.

Capturing aleatoric uncertainty is a critical part of many machine learning systems. In deep learning, a common approach to this end is to train a neural network to estimate the parameters of a heteroscedastic Gaussian distribution by maximizing the logarithm of the likelihood function under the observed data. In this work, we examine this approach and identify potential hazards associated with the use of log-likelihood in conjunction with gradient-based optimizers. First, we present a synthetic example illustrating how this approach can lead to very poor but stable parameter estimates. Second, we identify the culprit to be the log-likelihood loss, along with certain conditions that exacerbate the issue. Third, we present an alternative formulation, termed β-NLL, in which each data point's contribution to the loss is weighted by the β-exponentiated variance estimate. We show that using an appropriate β largely mitigates the issue in our illustrative example. Fourth, we evaluate this approach on a range of domains and tasks and show that it achieves considerable improvements and performs more robustly concerning hyperparameters, both in predictive RMSE and log-likelihood criteria.

Here is a ready-to-use snippet for our beta-NLL loss such that you don't have to go dig through the code:

def beta_nll_loss(mean, variance, target, beta=0.5):
    """Compute beta-NLL loss
    
    :param mean: Predicted mean of shape B x D
    :param variance: Predicted variance of shape B x D
    :param target: Target of shape B x D
    :param beta: Parameter from range [0, 1] controlling relative 
        weighting between data points, where `0` corresponds to 
        high weight on low error points and `1` to an equal weighting.
    :returns: Loss per batch element of shape B
    """
    loss = 0.5 * ((target - mean) ** 2 / variance + variance.log())

    if beta > 0:
        loss = loss * (variance.detach() ** beta)
    
    return loss.sum(axis=-1)

We use conda to manage the environment. Use the following command to recreate the environment:

conda env create -f environment.yml

We only use standard packages: pytorch, torchvision, numpy, scipy, sklearn, pandas, matplotlib. Activate the environment with conda activate prob-nns.

The datasets must reside in the folder data/ by default.

• UCI datasets are already included in data/UCI_Datasets
• MNIST and Fashion-MNIST are handled by torchvision
• ObjectSlide and FPP coming soon

Use the following commands to reproduce the experiments.

./train.sh <dataset> <method> <beta>

where <dataset> can be one of

• Sine (sinusoidal without heteroscedastic noise)
• SineDetlefsen (sinusoidal with heteroscedastic noise)
• ObjectSlide (dynamics models)
• FPP (dynamics models)
• MNIST (generative modeling)
• Fashion-MNIST (generative modeling)

and <method> can be one of likelihood, mse, moment_matching. For likelihood, you can also specify the value for <beta> (default 0).

To reproduce the UCI results, use

./train_uci.sh <dataset> <method> <beta>

where <dataset> can be one of carbon,concrete,housing,energy,kin8m,naval,power,protein,superconductivity,wine-red,wine-white,yacht.

The results will be put into the folder ./logs by default.

For the depth estimation experiments, see the depth_estimation folder.

Please use the following citation if you make use of our work:

@inproceedings{Seitzer2022PitfallsOfUncertainty,
  title = {On the Pitfalls of Heteroscedastic Uncertainty Estimation with Probabilistic Neural Networks},
  author = {Seitzer, Maximilian and Tavakoli, Arash and Antic, Dimitrije and Martius, Georg},
  booktitle = {International Conference on Learning Representations},
  month = apr,
  year = {2022},
  url = {https://openreview.net/forum?id=aPOpXlnV1T},
  month_numeric = {4}
}

This implementation is licensed under the MIT license.

The dynamics dataset ObjectSlide originally stems from our previous work Causal Influence Detection for Improving Efficiency in Reinforcement Learning, which you can find here: https://github.com/martius-lab/cid-in-rl.

Code for loading UCI datasets adapted from https://github.com/yaringal/DropoutUncertaintyExps/ by Yarin Gal under CC BY-NC 4.0 license.

Code for variational variance and for downloading UCI datasets adapted from https://github.com/astirn/variational-variance by Andrew Stirn under MIT license.

Code for depth estimation with AdaBins was adapted from https://github.com/shariqfarooq123/AdaBins by Shariq Farooq Bhat, licensed under GPL v3 license.