Excess Optimism: How Biased is the Apparent Error of an Estimator Tuned by SURE?

Nearly all estimators in statistical prediction come with an associated tuning parameter, in one way or another. Common practice, given data, is to choose the tuning parameter value that minimizes a constructed estimate of the prediction error of the estimator; we focus on Stein’s unbiased risk estimator, or SURE (Stein, 1981; Efron, 1986), which forms an unbiased estimate of the prediction error by augmenting the observed training error with an estimate of the degrees of freedom of the estimator. Parameter tuning via SURE minimization has been advocated by many authors, in a wide variety of problem settings, and in general, it is natural to ask: what is the prediction error of the SURE-tuned estimator? An obvious strategy would be simply use the apparent error estimate as reported by SURE, i.e., the value of the SURE criterion at its minimum, to estimate the prediction error of the SURE-tuned estimator. But this is no longer unbiased; in fact, we would expect that the minimum of the SURE criterion is systematically biased downwards for the true prediction error. In this work, we define the excess optimism of the SURE-tuned estimator to be the amount of this downward bias in the SURE minimum. We argue that the following two properties motivate the study of excess optimism: (i) an unbiased estimate of excess optimism, added to the SURE criterion at its minimum, gives an unbiased estimate of the prediction error of the SURE-tuned estimator; (ii) excess optimism serves as an upper bound on the excess risk, i.e., the difference between the risk of the SUREtuned estimator and the oracle risk (where the oracle uses the best fixed tuning parameter choice). We study excess optimism in two common settings: shrinkage estimators and subset regression estimators. Our main results include a James-Stein-like property of the SURE-tuned shrinkage estimator, which is shown to dominate the MLE; and both upper and lower bounds on excess optimism for SURE-tuned subset regression. In the latter setting, when the collection of subsets is nested, our bounds are particularly tight, and reveal that in the case of no signal, the excess optimism is always in between 0 and 10 degrees of freedom, regardless of how many models are being selected from.