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We analyze a new robust method for the reconstruction of probabilitydistributions of observed data in the presence of output outliers. It is basedon a so-called gradient conjugate prior (GCP) network which outputs theparameters of a prior. By rigorously studying the dynamics of the GCP learningprocess, we derive an explicit formula for correcting the obtained variance ofthe marginal distribution and removing the bias caused by outliers in thetraining set. Assuming a Gaussian (input-dependent) ground truth distributioncontaminated with a proportion $\varepsilon$ of outliers, we show that thefitted mean is in a $c e^{-1/\varepsilon}$-neighborhood of the ground truthmean and the corrected variance is in a $b\varepsilon$-neighborhood of theground truth variance, whereas the uncorrected variance of the marginaldistribution can even be infinite. We explicitly find $b$ as a function of theoutput of the GCP network, without a priori knowledge of the outliers (possiblyinput-dependent) distribution. Experiments with synthetic and real-world datasets indicate that the GCP network fitted with a standard optimizer outperformsother robust methods for regression.

Robustness Against Outliers For Deep Neural Networks By Gradient Conjugate Priors