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Scalable Krylov Subspace Methods for Generalized Mixed Effects Models with Crossed Random Effects
arXiv:2505.09552v2 Announce Type: replace-cross
Abstract: Mixed-effects models are widely used to model data with hierarchical grouping structures and high-cardinality categorical predictor variables. However, for high-dimensional crossed random effects, sparse Cholesky decompositions, the current standard approach, can become prohibitively slow. In this work, we present Krylov subspace-based methods that address these computational bottlenecks and 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 by factors of up to about 10,000 and are numerically more stable than Cholesky-based computations as implemented in state-of-the-art packages such as lme4 and glmmTMB. Our methodology is available in the open-source C++ software library GPBoost, with accompanying high-level Python and R packages.
Abstract: Mixed-effects models are widely used to model data with hierarchical grouping structures and high-cardinality categorical predictor variables. However, for high-dimensional crossed random effects, sparse Cholesky decompositions, the current standard approach, can become prohibitively slow. In this work, we present Krylov subspace-based methods that address these computational bottlenecks and 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 by factors of up to about 10,000 and are numerically more stable than Cholesky-based computations as implemented in state-of-the-art packages such as lme4 and glmmTMB. Our methodology is available in the open-source C++ software library GPBoost, with accompanying high-level Python and R packages.
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