LiD-GLM: Lipschitz-constrained Deep Generalized Linear Models

We formally introduce a class of models inspired by renormalization group (RG) theory, built on additive hierarchical expansions analogous to those appearing in functional ANOVA and mixed-effects models. Like ReLU convolutional neural networks, they are almost everywhere locally linear; unlike ReLU networks, their partition structure is explicit, interpretable, and easy to modify or constrain. In these models, one defines a multidimensional lattice partition of the input space and uses it to scaffold variations in regression parameters. Each dimension of the lattice corresponds to an attribute by which the statistics of the problem may vary. The parameters are themselves expressed in the form of an expansion, where each term captures variations relative to a lower (coarser) interaction scale. These models admit multiple equivalent interpretations: as piecewise GLMs, as hierarchical mixed-effects regressions, or as regression trees with structured parameter sharing. Since RG motivates the design of these models, we use techniques from statistical physics -- specifically replica analysis -- to study their generalization properties. Specifically, we analyze the behavior of the Watanabe-Akaike Information Criterion (WAIC) as a proxy for generalization loss. This analysis yields two practical results: (i) guidance on the lattice design as a function of dataset size and predictor dimensionality; and (ii) a principled scaling law for the regularization prior when adding higher-order terms to the expansion so that one can increase model complexity without an expected increase in generalization loss. We evaluate the methodology on public datasets and find performance competitive against both blackbox methods and other intrinsically interpretable approaches.

Interpretable Mesomorphic Neural Networks (IMNs) offer a promising framework that combines the predictive power of deep neural networks with the interpretability of linear models. However, the original formulation lacks safeguards to ensure that the learned interpretations are in fact reliable. In particular, the network is free to concentrate all explanatory variance into a single weight of the linear output layer, achieving strong predictive performance while producing interpretations that are largely meaningless. Paradoxically, the L1 penalty proposed to encourage sparse solutions exacerbates this problem by further incentivizing such degenerate configurations. To address this vulnerability, we introduce Local Fidelity Regularization (LFR), a novel penalty term that prevents degenerate weight collapse by aligning the linear output weights with local data variations. This structural constraint guarantees faithful explanations and substantially improves the reliability of model interpretations. Furthermore, empirical evaluations across the OpenML benchmark suite demonstrate that LFR does not compromise accuracy for explainability; rather, it achieved improved AUROC over the unregularized IMN. By yielding results highly competitive with state-of-the-art black-box models, LFR provides the dual benefit of reliable interpretability and superior predictive performance. Source code and usage instructions are available at https://github.com/hugohammer/LFR-IMN.git.

Generalized linear mixed-effects models (GLMMs) are widely used to analyze grouped and hierarchical data. In a GLMM, each response is assumed to follow an exponential-family distribution where the natural parameter is given by a linear function of observed covariates and a latent group-specific random effect. Since exact marginalization over the random effects is typically intractable, model parameters are estimated by maximizing an approximate marginal likelihood. In this paper, we replace the linear function with neural networks. The result is a more flexible model, the neural generalized mixed-effects model (NGMM), which captures complex relationships between covariates and responses. To fit NGMM to data, we introduce an efficient optimization procedure that maximizes the approximate marginal likelihood and is differentiable with respect to network parameters. We show that the approximation error of our objective decays at a Gaussian-tail rate in a user-chosen parameter. On synthetic data, NGMM improves over GLMMs when covariate-response relationships are nonlinear, and on real-world datasets it outperforms prior methods. Finally, we analyze a large dataset of student proficiency to demonstrate how NGMM can be extended to more complex latent-variable models.