Estimating the Maximum Expected Value: An Analysis of (Nested) Cross Validation and the Maximum Sample Average
This addresses a fundamental statistical estimation problem for researchers and practitioners in machine learning and statistics, but it is incremental as it builds on existing methods without introducing a new paradigm.
The paper tackles the problem of estimating the maximum expected value of a set of random variables, analyzing the bias and variance of common estimators like maximum sample average and cross-validation, and shows that cross-validation can reduce variance but risks large bias, with performance being highly problem-dependent.
We investigate the accuracy of the two most common estimators for the maximum expected value of a general set of random variables: a generalization of the maximum sample average, and cross validation. No unbiased estimator exists and we show that it is non-trivial to select a good estimator without knowledge about the distributions of the random variables. We investigate and bound the bias and variance of the aforementioned estimators and prove consistency. The variance of cross validation can be significantly reduced, but not without risking a large bias. The bias and variance of different variants of cross validation are shown to be very problem-dependent, and a wrong choice can lead to very inaccurate estimates.