Adaptive Monotone Shrinkage for Regression

TL;DR

This paper introduces an adaptive monotone shrinkage estimator for regression that effectively leverages feature importance ordering, demonstrating competitive performance and robustness through theoretical guarantees and empirical validation.

Contribution

It proposes a novel empirical Bayes estimator that enforces monotone shrinkage based on feature importance, with rapid computation and strong theoretical properties.

Findings

Estimator is competitive with Bayesian methods sharing the prior

Minimizes Stein's unbiased risk estimate

Mimics oracle Bayes rule under order assumption

Abstract

We develop an adaptive monotone shrinkage estimator for regression models with the following characteristics: i) dense coefficients with small but important effects; ii) a priori ordering that indicates the probable predictive importance of the features. We capture both properties with an empirical Bayes estimator that shrinks coefficients monotonically with respect to their anticipated importance. This estimator can be rapidly computed using a version of Pool-Adjacent-Violators algorithm. We show that the proposed monotone shrinkage approach is competitive with the class of all Bayesian estimators that share the prior information. We further observe that the estimator also minimizes Stein's unbiased risk estimate. Along with our key result that the estimator mimics the oracle Bayes rule under an order assumption, we also prove that the estimator is robust. Even without the order assumption, our estimator mimics the best performance of a large family of estimators that includes the least squares estimator, constant-$\lambda$ ridge estimator, James-Stein estimator, etc. All the theoretical results are non-asymptotic. Simulation results and data analysis from a model for text processing are provided to support the theory.