The Annals of Statistics

Heuristics of instability and stabilization in model selection

Leo Breiman

Full-text: Open access


In model selection, usually a "best" predictor is chosen from a collection ${\hat{\mu}(\cdot, s)}$ of predictors where $\hat{\mu}(\cdot, s)$ is the minimum least-squares predictor in a collection $\mathsf{U}_s$ of predictors. Here s is a complexity parameter; that is, the smaller s, the lower dimensional/smoother the models in $\mathsf{U}_s$.

If $\mathsf{L}$ is the data used to derive the sequence ${\hat{\mu}(\cdot, s)}$, the procedure is called unstable if a small change in $\mathsf{L}$ can cause large changes in ${\hat{\mu}(\cdot, s)}$. With a crystal ball, one could pick the predictor in ${\hat{\mu}(\cdot, s)}$ having minimum prediction error. Without prescience, one uses test sets, cross-validation and so forth. The difference in prediction error between the crystal ball selection and the statistician's choice we call predictive loss. For an unstable procedure the predictive loss is large. This is shown by some analytics in a simple case and by simulation results in a more complex comparison of four different linear regression methods. Unstable procedures can be stabilized by perturbing the data, getting a new predictor sequence ${\hat{\mu'}(\cdot, s)}$ and then averaging over many such predictor sequences.

Article information

Ann. Statist. Volume 24, Number 6 (1996), 2350-2383.

First available: 16 September 2002

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Digital Object Identifier

Zentralblatt MATH identifier

Primary: 62H99: None of the above, but in this section

Regression subset selection predictive loss cross-validation prediction error


Breiman, Leo. Heuristics of instability and stabilization in model selection. The Annals of Statistics 24 (1996), no. 6, 2350--2383. doi:10.1214/aos/1032181158.

Export citation


  • BREIMAN, L. 1992. The little bootstrap and other methods for dimensionality selection in regression: x-fixed prediction error. J. Amer. Statist. Assoc. 87 738 754. Z.
  • BREIMAN, L. 1995. Better subset selection using the non-negative garotte. Technometrics 37 373 384. Z.
  • BREIMAN, L. 1996a. Stacked regressions. Machine Learning 24 41 64. Z.
  • BREIMAN, L. 1996b. Bagging predictors. Machine Learning 26 123 140. Z.
  • BREIMAN, L. 1996c. Bias, variance and arcing classifiers. Report 460, Dept. Statistics, Univ. California. Z.
  • BREIMAN, L. and SPECTOR, P. 1992. Submodel selection and evaluation in regression. The random X case. Internat. Statist. Rev. 60 291 319. Z.
  • WOLPERT, D. 1992. Stacked generalization. Neural Networks 5 241 259.