What question did this study set out to answer?

The aim is to develop robust learning methods for deep neural networks that can effectively handle outliers.

June 6, 2026Open Access

Density power divergence based learning of deep neural network models with application to outlier detection

Key Points

The aim is to develop robust learning methods for deep neural networks that can effectively handle outliers.
Adopt minimum density power divergence framework for flexible trade-off between robustness and efficiency.
Extend approach to univariate time series settings with a novel loss function.
Integrate outlier detection using standardized residuals and tail probability estimation.
Achieved robust estimation with down-weighting of observations with large residuals during training.
Demonstrated reliable outlier detection in simulated environments.
Results highlight the effectiveness of the data-driven tuning parameter selection strategy.

Abstract

This paper studies robust learning methods for deep neural networks in the presence of outliers.While conventional training based on mean squared error (MSE) is optimal under normality assumptions, it is highly sensitive to anomalous observations commonly encountered in real-world data.To address this limitation, we adopt the minimum density power divergence framework, which enables a flexible trade-off between robustness and statistical efficiency through a tuning parameter.This paper extends the framework to univariate time series settings and shows that the resulting loss function down-weight the contribution of observations with large residuals to the gradient of model parameters during training.In addition, we integrate an outlier detection procedure based on standardized residuals and tail probability estimation.A data-driven strategy for selecting the tuning parameter is also provided.Simulation studies demonstrate the effectiveness of the proposed method in achieving robust estimation and reliable outlier detection.

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