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We introduce a new sample complexity measure, which we refer to assplit-sample growth rate. For any hypothesis $H$ and for any sample $S$ of size$m$, the split-sample growth rate $\hat{\tau}_H(m)$ counts how many differenthypotheses can empirical risk minimization output on any sub-sample of $S$ ofsize $m/2$. We show that the expected generalization error is upper bounded by$O\left(\sqrt{\frac{\log(\hat{\tau}_H(2m))}{m}}\right)$. Our result is enabledby a strengthening of the Rademacher complexity analysis of the expectedgeneralization error. We show that this sample complexity measure, greatlysimplifies the analysis of the sample complexity of optimal auction design, formany auction classes studied in the literature. Their sample complexity can bederived solely by noticing that in these auction classes, ERM on any sample orsub-sample will pick parameters that are equal to one of the points in thesample.

A Sample Complexity Measure with Applications to Learning Optimal Auctions