Covariate-Guided Clusterwise Linear Regression for Generalization to Unseen Data

摘要

In many tabular regression tasks, the relationships between covariates and response can often be approximated as linear only within localized regions of the input space; a single global linear model therefore fails to capture these local relationships. Conventional Clusterwise Linear Regression (CLR) mitigates this issue by learning $K$ local regressors. However, existing algorithms either optimize latent binary indicators, (i) providing no explicit rule for assigning an $\textit{unseen}$ covariate vector to a cluster at test time, or rely on heuristic mixture of experts approaches, (ii) lacking convergence guarantees. To address these limitations, we propose $\textit{covariate-guided}$ CLR, an end-to-end framework that jointly learns an assignment function and $K$ linear regressors within a single gradient-based optimization loop. During training, a proxy network iteratively predicts coefficient vectors for inputs, and hard vector quantization assigns samples to their nearest codebook regressors. This alternating minimization procedure yields monotone descent of the empirical risk, converges under mild assumptions, and enjoys a PAC-style excess-risk bound. By treating the covariate data from all clusters as a single concatenated design matrix, we derive an $F$-test statistic from a nested linear model, quantitatively characterizing the effective model complexity. As $K$ varies, our method spans the spectrum from a single global linear model to instance-wise fits. Experimental results show that our method exactly reconstructs synthetic piecewise-linear surfaces, achieves accuracy comparable to strong black-box models on standard tabular benchmarks, and consistently outperforms existing CLR and mixture-of-experts approaches.