A core element of microeconomics and game theory is that consumers have valuation functions over bundles of goods and that these valuation functions drive their purchases. In particular, the value assigned to a bundle need not be the sum of values on the individual items but rather is often a more complex function of how the items relate. The literature considers a hierarchy of valuation classes that includes subadditive, XOS (i.e. fractionally subadditive), submodular, and OXS valuations. Typically it is assumed that these valuations are known to the center or that they come from a known distribution. Two recent lines of work, by Goemans et al. (SODA 2009) and by Balcan and Harvey (STOC 2011), have considered a more realistic setting in which valuations are learned from data, focusing specifically on submodular functions. In this paper we consider the approximate learnability of valuation functions at all levels in the hierarchy. We first study their learnability in the distributional learning (PAC-style) setting due to Balcan and Harvey (STOC 2011). We provide nearly tight lower and upper bounds ofΘ(n 1/2 ) on the approximation factor for learning XOS and subadditive valuations, both important classes that are strictly more general than submodular valuations. Interestingly, we show that theΘ(n 1/2 ) lower bound can be circumvented for XOS functions of polynomial complexity; we provide an algorithm for learning the class of XOS valuations with a representation of polynomial size to within an O(n ǫ ) approximation factor in running time n O(1/ε) for any ǫ > 0. We also establish learnability and hardness results for subclasses of the class of submodular valuations, i.e. gross substitutes valuations and interesting subclasses of OXS valuations. In proving our results for the distributional learning setting, we provide novel structural results for all these classes of valuations. We show the implications of these results for the learning everywhere with value queries model, considered by Goemans et al. (SODA 2009). Finally, we also introduce a more realistic variation of these models for economic settings, in which information on the value of a bundle S of goods can only be inferred based on whether S is purchased or not at a specific price. We provide lower and upper bounds for learning both in the distributional setting and with value queries.
Subsequently, Balcan et al. gave a distribution-independent PMAC learning algorithm for XOS functions that achieves factorÕ( √ n) approximation and showed that this it is essentially optimal REF .
Learning Valuation Functions
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