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A General Theory of Sample Complexity for Multi-Item Profit Maximization
The design of profit-maximizing multi-item mechanisms is a notoriously
challenging problem with tremendous real-world impact. The mechanism designer's
goal is to field a mechanism with high expected profit on the distribution over
buyers' values. Unfortunately, if the set of mechanisms he optimizes over is
complex, a mechanism may have high empirical profit over a small set of samples
but low expected profit. This raises the question, how many samples are
sufficient to ensure that the empirically optimal mechanism is nearly optimal
in expectation? We uncover structure shared by a myriad of pricing, auction,
and lottery mechanisms that allows us to prove strong sample complexity bounds:
for any set of buyers' values, profit is a piecewise linear function of the
mechanism's parameters. We prove new bounds for mechanism classes not yet
studied in the sample-based mechanism design literature and match or improve
over the best known guarantees for many classes. The profit functions we study
are significantly different from well-understood functions in machine learning,
so our analysis requires a sharp understanding of the interplay between
mechanism parameters and buyer values. We strengthen our main results with
data-dependent bounds when the distribution over buyers' values is
"well-behaved." Finally, we investigate a fundamental tradeoff in sample-based
mechanism design: complex mechanisms often have higher profit than simple
mechanisms, but more samples are required to ensure that empirical and expected
profit are close. We provide techniques for optimizing this tradeoff
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