Smoothness and Monotonicity Constraints for Neural Networks Using ICEnet

33 Pages Posted: 21 May 2023

See all articles by Ronald Richman

Ronald Richman

Old Mutual Insure; University of the Witwatersrand

Mario V. Wuthrich

RiskLab, ETH Zurich

Date Written: May 15, 2023


Deep neural networks have become an important tool for use in actuarial tasks, due to the significant gains in accuracy provided by these techniques compared to traditional methods, but also due to the close connection of these models to the Generalized Linear Models (GLMs) currently used in industry. Whereas constraining GLM parameters relating to insurance risk factors to be smooth or exhibit monotonicity is trivial, methods to incorporate such constraints into deep neural networks have not yet been developed. This is a barrier for the adoption of neural networks in insurance practice since actuaries often impose these constraints for commercial or statistical reasons. In this work, we present a novel method for enforcing constraints within deep neural network models, and we show how these models can be trained. Moreover, we provide example applications using real-world datasets. We call our proposed method ICEnet to emphasize the close link of our proposal to the individual conditional expectation (ICE) model interpretability technique.

Keywords: Smoothing, Whittaker–Henderson Smoothing, Graduation, Monotonicity, Deep Neural Networks, Constrained Likelihood, Individual Conditional Expectation

JEL Classification: G22

Suggested Citation

Richman, Ronald and Wuthrich, Mario V., Smoothness and Monotonicity Constraints for Neural Networks Using ICEnet (May 15, 2023). Available at SSRN: or

Ronald Richman (Contact Author)

Old Mutual Insure ( email )

Wanooka Place
St Andrews Road
Johannesburg, 2192
South Africa

University of the Witwatersrand ( email )

Mario V. Wuthrich

RiskLab, ETH Zurich ( email )

Department of Mathematics
Ramistrasse 101
Zurich, 8092

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