LiD-GLM: Lipschitz-constrained Deep Generalized Linear Models

2026-08-17

AI Summary

This study addresses the trade-off between interpretability and flexibility in hybrid models, where neural networks often compromise model transparency. We propose a Generalized Linear Model augmentation framework based on Lipschitz-constrained invertible residual networks. By incorporating a controllable bias mechanism and posterior orthogonalization, the method achieves flexible nonlinear estimation and distribution correction while strictly preserving stochastic monotonicity and model identifiability. This approach establishes a semi-structured hybrid modeling paradigm that integrates high interpretability with flexibility, enabling user-defined trade-offs and quantifiable model bias. Consequently, it effectively mitigates the transparency bottleneck inherent in complex data modeling tasks.

Abstract

The combination of traditional statistical models and neural network (NN) components into semi-structured hybrid models is an intriguing approach to construct models that, ideally, combine traditional interpretability with the unprecedented flexibility of NNs. In order to preserve interpretability, it is usually necessary to restrict the NN components to prevent them from dominating the model. However, existing methods that enforce structural constraints on their NN components severely limit their models' flexibility; in contrast, methods that only enforce weak, indirect constraints lose meaningful interpretability. The method we propose therefore leverages invertible residual neural networks (i-ResNets) to equip generalized linear models with both nonlinear parameter estimation and a flexible correction of their distributional assumptions while always retaining stochastic monotonicity of the modeled distribution in the (formerly linear) predictor. The i-ResNets correspond to a controlled deviation from identity and by constraining their Lipschitz constant one can rigorously limit and quantify how far the hybrid model deviates from its traditional counterpart. This enables a user-specifiable compromise between flexibility and interpretability without limiting the structure of nonlinear and interaction effects that can be learned. Furthermore, we develop specific inherent interpretation techniques for our model and enforce model identifiability through an adapted post-hoc orthogonalization.

Problem

• Interpretability
• Flexibility
• Hybrid models
• Generalized Linear Models
• Structural constraints

Innovation

• Lipschitz-constrained i-ResNets
• Stochastic Monotonicity
• Interpretability-Flexibility Trade-off
• Model Identifiability
• Semi-structured Hybrid Models