Submultiplicative Glivenko-Cantelli and Uniform Convergence of Revenues
This work addresses the problem of revenue learning in economics and algorithmic game theory, providing theoretical guarantees for uniform convergence, but it is incremental as it builds on classic statistical theorems.
The authors derived a variant of the Glivenko-Cantelli Theorem with tighter convergence bounds for extreme values and applied it to revenue learning, establishing sample-complexity bounds based on the kth moment of valuations and a zero-one law for uniform convergence depending on the finiteness of the first moment.
In this work we derive a variant of the classic Glivenko-Cantelli Theorem, which asserts uniform convergence of the empirical Cumulative Distribution Function (CDF) to the CDF of the underlying distribution. Our variant allows for tighter convergence bounds for extreme values of the CDF. We apply our bound in the context of revenue learning, which is a well-studied problem in economics and algorithmic game theory. We derive sample-complexity bounds on the uniform convergence rate of the empirical revenues to the true revenues, assuming a bound on the $k$th moment of the valuations, for any (possibly fractional) $k>1$. For uniform convergence in the limit, we give a complete characterization and a zero-one law: if the first moment of the valuations is finite, then uniform convergence almost surely occurs; conversely, if the first moment is infinite, then uniform convergence almost never occurs.