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Learning in Auctions: Regret is Hard, Envy is Easy

A line of recent work provides welfare guarantees of simple combinatorial auction formats, such as selling m items via simultaneous second price auctions (SiSPAs) (Christodoulou et al. 2008, Bhawalkar and Roughgarden 2011, Feldman et al. 2013). These guarantees hold even when the auctions are repeatedly executed and players use no-regret learning algorithms. Unfortunately, off-the-shelf no-regret algorithms for these auctions are computationally inefficient as the number of actions is exponential. We show that this obstacle is insurmountable: there are no polynomial-time no-regret algorithms for SiSPAs, unless RP$\supseteq$ NP, even when the bidders are unit-demand. Our lower bound raises the question of how good outcomes polynomially-bounded bidders may discover in such auctions. To answer this question, we propose a novel concept of learning in auctions, termed "no-envy learning." This notion is founded upon Walrasian equilibrium, and we show that it is both efficiently implementable and results in approximately optimal welfare, even when the bidders have fractionally subadditive (XOS) valuations (assuming demand oracles) or coverage valuations (without demand oracles). No-envy learning outcomes are a relaxation of no-regret outcomes, which maintain their approximate welfare optimality while endowing them with computational tractability. Our results extend to other auction formats that have been studied in the literature via the smoothness paradigm. Our results for XOS valuations are enabled by a novel Follow-The-Perturbed-Leader algorithm for settings where the number of experts is infinite, and the payoff function of the learner is non-linear. This algorithm has applications outside of auction settings, such as in security games. Our result for coverage valuations is based on a novel use of convex rounding schemes and a reduction to online convex optimization.

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