Adaptive Monotone Shrinkage for Regression

Zhuang Ma, Dean Foster, Robert Stine
Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, PMLR R12:499-508, 2014.

Abstract

We develop an adaptive monotone shrinkage es- timator for regression models with the following characteristics: i) dense coefficients with small but important effects; ii) a priori ordering that in- dicates the probable predictive importance of the features. We capture both properties with an em- pirical Bayes estimator that shrinks coefficients monotonically with respect to their anticipated importance. This estimator can be rapidly com- puted 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 in- formation. We further observe that the estima- tor also minimizes Stein’s unbiased risk estimate. Along with our key result that the estimator mim- ics the oracle Bayes rule under an order assump- tion, we also prove that the estimator is robust. Even without the order assumption, our estima- tor mimics the best performance of a large family of estimators that includes the least squares es- timator, 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.

Cite this Paper

BibTeX
@InProceedings{pmlr-vR12-ma14a, title = {Adaptive Monotone Shrinkage for Regression}, author = {Ma, Zhuang and Foster, Dean and Stine, Robert}, booktitle = {Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence}, pages = {499--508}, year = {2014}, editor = {Zhang, Nevin L. and Tian, Jin}, volume = {R12}, series = {Proceedings of Machine Learning Research}, month = {23--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/r12/main/assets/ma14a/ma14a.pdf}, url = {https://proceedings.mlr.press/r12/ma14a.html}, abstract = {We develop an adaptive monotone shrinkage es- timator for regression models with the following characteristics: i) dense coefficients with small but important effects; ii) a priori ordering that in- dicates the probable predictive importance of the features. We capture both properties with an em- pirical Bayes estimator that shrinks coefficients monotonically with respect to their anticipated importance. This estimator can be rapidly com- puted 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 in- formation. We further observe that the estima- tor also minimizes Stein’s unbiased risk estimate. Along with our key result that the estimator mim- ics the oracle Bayes rule under an order assump- tion, we also prove that the estimator is robust. Even without the order assumption, our estima- tor mimics the best performance of a large family of estimators that includes the least squares es- timator, 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.}, note = {Reissued by PMLR on 04 October 2026.} }
Endnote
%0 Conference Paper %T Adaptive Monotone Shrinkage for Regression %A Zhuang Ma %A Dean Foster %A Robert Stine %B Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence %C Proceedings of Machine Learning Research %D 2014 %E Nevin L. Zhang %E Jin Tian %F pmlr-vR12-ma14a %I PMLR %P 499--508 %U https://proceedings.mlr.press/r12/ma14a.html %V R12 %X We develop an adaptive monotone shrinkage es- timator for regression models with the following characteristics: i) dense coefficients with small but important effects; ii) a priori ordering that in- dicates the probable predictive importance of the features. We capture both properties with an em- pirical Bayes estimator that shrinks coefficients monotonically with respect to their anticipated importance. This estimator can be rapidly com- puted 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 in- formation. We further observe that the estima- tor also minimizes Stein’s unbiased risk estimate. Along with our key result that the estimator mim- ics the oracle Bayes rule under an order assump- tion, we also prove that the estimator is robust. Even without the order assumption, our estima- tor mimics the best performance of a large family of estimators that includes the least squares es- timator, 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. %Z Reissued by PMLR on 04 October 2026.
APA
Ma, Z., Foster, D. & Stine, R.. (2014). Adaptive Monotone Shrinkage for Regression. Proceedings of the 30th Conference on Uncertainty in Artificial Intelligence, in Proceedings of Machine Learning Research R12:499-508 Available from https://proceedings.mlr.press/r12/ma14a.html. Reissued by PMLR on 04 October 2026.

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