Uncertainty in Neural Networks

This project contains different implementation and evaluations of approaches to model uncertainty in neural networks.

• Bootstrapping-Method from Osband et al.
• (Monte Carlo)Dropout-Method by Gal
• Combined method (heteroscedastic aleatoric + epistemic) from Kendal & Gal
• Mixture Density Networks as used by Choi et al.

Those models are evaluated on 2D data for function approximation. Specifically there is a dataset having six points at (-1,1) , (-1,-1), (0,1), (0,-1), (1,1), (1,-1) which shows problems with the Dropout and Combined Method. And a ''x + sin(x)'' function with added noise.

Additionally the evaluation is done on MNIST data, for which I crafted adversarial attacks to evaluate the effectiveness of the Uncertainty methods.

Methods

Bootstrapping

The idea is to have a network with k distinct heads that share a network. The dataset is masked, so that every head only sees a subset of all data. Predictive variance and mean can then be gained by averaging over the prediction of every head.

https://arxiv.org/pdf/1602.04621v3.pdf

Monte-Carlo Dropout

Using dropout during training and test time which is approximate variational inference. Mean and variance is gained by doing stochastic forward passes (MC Dropout) and averaging over the outputs. This model can't differentiate between aleatoric and epistemic uncertainty.

http://mlg.eng.cam.ac.uk/yarin/thesis/thesis.pdf

Combined (Aleatoric + Epistemic Uncertainty) Method

Using Monte-Carlo Dropout and a modified loss function, you can get heteroscedastic aleatoric and epistemic uncertainty separated and also combine them.

https://arxiv.org/pdf/1703.04977.pdf

Mixture Density Networks

The last layer(s) are replaced by a layer that output Gaussian Mixture Models.

https://arxiv.org/pdf/1709.02249.pdf

Results

Results can be found in results or generated with

• python evaluation/boostrap_evaluation.py
• python evaluation/combined_evaluation.py
• python evaluation/dropout_evaluation.py
• python evaluation/mixture_evaluation.py

or all at once with python evaluation/evaluate_all.py

TODO

• Working with higher dimensional data (MNIST)
• Analyze influence of adversarial attacks

Problems

• When running on the command-line, you might have to set $PYTHONPATH to the root dir: export PYTHONPATH=.