ParamBoost: Gradient Boosted Piecewise Cubic Polynomials

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.

Tabular applications often require inspectable prediction rules and stable behavior when records are incomplete. We propose FlagGAM, a rule-basis framework that separates feature-level rule construction from prediction. A Flag Core Module converts numerical and categorical variables into sparse, human-readable univariate bases: threshold flags, category-level flags, tail-deviation bases, and categorical step functions. A default additive head combines these bases as a restricted GAM-style predictor, while the retained sparse rule-basis matrix supports mixed-type classification and regression, feature-specific weighting, and optional flexible heads. On clean benchmarks, additive FlagGAM stays close to modern additive and rule-based baselines on classification and improves over global linear modeling on regression, while remaining less flexible than tree-based predictors. Its clearest advantage appears under deployment-time perturbations: across three classification datasets, FlagGAM has the smallest mean AUROC degradation under missingness and numerical noise. Flexible heads improve absolute accuracy and approach strong tree-based baselines, but should be interpreted as nonlinear predictors over learned rule bases. These results support FlagGAM as a constrained additive rule-basis model for applications that need readable rules and stable behavior with incomplete inputs.

We introduce Gradient Boosted Mixed Models (GBMixed), a framework which extends boosting to clustered data by jointly modeling the mean and variance components in a linear mixed model via likelihood-based gradients. GBMixed estimates a nonparametric fixed effects function characterizing the overall mean of the response, while also allowing the random effects covariance matrix along with the residual variance to depend on covariates in a flexible manner. We demonstrate how GBMixed facilitates covariate-dependent random effect predictions, and subsequently point predictions and prediction intervals for individual treatment effects, that can adapt between population-level and cluster-level information. Simulations and applications to two real-world datasets demonstrate that GBMixed can accurately recover complex nonlinear fixed effect functions and covariate-dependent covariances in a linear mixed model, while also improving point and probabilistic predictive performance compared with several existing approaches such as parametric linear mixed models, Natural Gradient Boosting, and Gaussian Process Boosting.