Gradient Boosted Mixed Models: Flexible Estimation of Mean and Variance Components for Clustered Data

Scalable Bayesian inference for generalized linear mixed models (GLMMs) provides uncertainty-aware analysis of correlated longitudinal data, but existing scalable approaches largely assume low-dimensional tabular predictors and do not directly accommodate high-dimensional modalities such as images and text. We address this limitation by learning one or more modality-specific neural encoders jointly with a GLMM objective, then performing variance-corrected stochasticgradient MCMC for the GLMM parameters conditional on the learned representation. This conditional-Bayes design combines supervised representation learning with posterior uncertainty quantification for population-level effects, subjectspecific heterogeneity, and modality-level random slopes. The resulting model preserves interpretable fixed and random effects for structured covariates and learned modalities while scaling gracefully to large longitudinal datasets. In simulation studies, our method recovers posterior means and variance estimates from full-data MCMC benchmarks after covariance correction. We further evaluate uncertainty through parameter-level interval coverage in simulations and predictive calibration on held-out data. Applications to glaucoma progression and adolescent mental health demonstrate that the framework allows nuanced assessment of the relative importance of each modality on both individual and population levels without sacrificing predictive performance.

We develop a statistical learning theory for gradient boosting applied to the estimation of covariate-dependent Generalized Pareto (GP) distributions in the context of Peaks-over-Threshold modeling. After an orthogonal reparametrization of the GP likelihood that diagonalizes its Fisher information matrix, we cast the estimation problem within the Empirical Risk Minimization (ERM) framework and derive non-asymptotic error bounds for the boosting estimator. Our analysis accounts for three distinct sources of error in the process: statistical fluctuations, the approximation bias inherent to the asymptotic nature of the GP model-controlled under second-order regular variation-and the approximation error associated with the finite number of boosting iterates, making explicit the resulting bias-variance trade-off. We illustrate the practical benefits of the reparametrization through simulations, showing that it significantly reduces gradient correlation during training and improves convergence stability. The methodology is applied to a medical malpractice insurance dataset from the Texas Department of Insurance, comprising over 18 000 closed claims. The gradient boosting approach yields a good fit for the tail of settlement cost distributions and reveals that the number of days to settlement is the dominant predictor of tail heaviness, consistent with earlier findings in the reserving literature.

Mixed-effects models are widely used to model data with complex grouping structures and high-cardinality categorical predictor variables. However, for high-dimensional crossed random effects, current standard computations relying on Cholesky decompositions can become prohibitively slow. In this work, we present Krylov subspace-based methods that address existing computational bottlenecks, and we analyze them both theoretically and empirically. In particular, we derive new results on the convergence and accuracy of the preconditioned stochastic Lanczos quadrature and conjugate gradient methods for mixed-effects models, and we develop scalable methods for calculating predictive variances. In experiments with simulated and real-world data, the proposed methods yield speedups of several orders of magnitude and are more computationally robust than Cholesky-based computations, while maintaining essentially the same accuracy.