We study the sample complexity of learning revenue-optimal multi-item auctions. We obtain the first set of positive results that go beyond the standard but unrealistic setting of item-independence. In particular, we consider settings where bidders' valuations are drawn from correlated distributions that can be captured by Markov Random Fields or Bayesian Networks -- two of the most prominent graphical models. We establish parametrized sample complexity bounds for learning an up-to-ε optimal mechanism in both models, which scale polynomially in the size of the model, i.e. the number of items and bidders, and only exponential in the natural complexity measure of the model, namely either the largest in-degree (for Bayesian Networks) or the size of the largest hyper-edge (for Markov Random Fields).
We obtain our learnability results through a novel and modular framework that involves first proving a robustness theorem. We show that, given only "approximate distributions" for bidder valuations, we can learn a mechanism whose revenue is nearly optimal simultaneously for all "true distributions" that are close to the ones we were given in Prokhorov distance. Thus, to learn a good mechanism, it suffices to learn approximate distributions. When item values are independent, learning in Prokhorov distance is immediate, hence our framework directly implies the main result of Gonczarowski and Weinberg. When item values are sampled from more general graphical models, we combine our robustness theorem with novel sample complexity results for learning Markov Random Fields or Bayesian Networks in Prokhorov distance, which may be of independent interest. Finally, in the single-item case, our robustness result can be strengthened to hold under an even weaker distribution distance, the Levy distance.
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Learning Multi-Item Auctions with (or without) Samples
We provide algorithms that learn simple auctions whose revenue is approximately optimal in multi-item multi-bidder settings, for a wide range of valuations including unit-demand, additive, constrained additive, XOS, and subadditive. We obtain our learning results in two settings. The first is the commonly studied setting where sample access to the bidders' distributions over valuations is given, for both regular distributions and arbitrary distributions with bounded support. Our algorithms require polynomially many samples in the number of items and bidders. The second is a more general max-min learning setting that we introduce, where we are given "approximate distributions," and we seek to compute an auction whose revenue is approximately optimal simultaneously for all "true distributions" that are close to the given ones. These results are more general in that they imply the sample-based results, and are also applicable in settings where we have no sample access to the underlying distributions but have estimated them indirectly via market research or by observation of previously run, potentially non-truthful auctions.
Our results hold for valuation distributions satisfying the standard (and necessary) independence-across-items property. They also generalize and improve upon recent works, which have provided algorithms that learn approximately optimal auctions in more restricted settings with additive, subadditive and unit-demand valuations using sample access to distributions. We generalize these results to the complete unit-demand, additive, and XOS setting, to i.i.d. subadditive bidders, and to the max-min setting.
Our results are enabled by new uniform convergence bounds for hypotheses classes under product measures. Our bounds result in exponential savings in sample complexity compared to bounds derived by bounding the VC dimension, and are of independent interest.
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- Award ID(s):
- 1650733
- NSF-PAR ID:
- 10086306
- Date Published:
- Journal Name:
- 58th IEEE Symposium on Foundations of Computer Science (FOCS)
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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