Block-wise Primal-dual Algorithms for Large-scale Doubly Penalized ANOVA Modeling

TL;DR

This paper introduces block-wise primal-dual algorithms, including stochastic variants, for efficient large-scale doubly penalized ANOVA modeling, effectively handling hierarchical total variation and empirical norm penalties.

Contribution

The paper develops novel primal-dual algorithms tailored for large-scale DPAM with HTV and empirical norm penalties, including stochastic methods for improved scalability.

Findings

Stochastic primal-dual algorithms outperform batch versions in large-scale tests.

Algorithms effectively handle combined HTV and empirical norm penalties.

Numerical experiments validate the efficiency and accuracy of the proposed methods.

Abstract

For multivariate nonparametric regression, doubly penalized ANOVA modeling (DPAM) has recently been proposed, using hierarchical total variations (HTVs) and empirical norms as penalties on the component functions such as main effects and multi-way interactions in a functional ANOVA decomposition of the underlying regression function. The two penalties play complementary roles: the HTV penalty promotes sparsity in the selection of basis functions within each component function, whereas the empirical-norm penalty promotes sparsity in the selection of component functions. We adopt backfitting or block minimization for training DPAM, and develop two suitable primal-dual algorithms, including both batch and stochastic versions, for updating each component function in single-block optimization. Existing applications of primal-dual algorithms are intractable in our setting with both HTV and empirical-norm penalties. Through extensive numerical experiments, we demonstrate the validity and advantage of our stochastic primal-dual algorithms, compared with their batch versions and a previous active-set algorithm, in large-scale scenarios.