Robust Learning of Optimal Auctions

√ α ) of the optimal revenue when the distributions are regular . We prove that these upper bounds can not be further improved , by providing matching lower bounds . Lastly , we derive sample complexity upper bounds for learning a near-optimal auction for both MHR and regular distributions . 1 Introduction . Optimal auctions play a crucial role in economic theory , with a wide range of applications across various industries , public sectors , and online platforms [ Myerson , 1981 , Bykowsky et al. , 2000 , Roth and Ockenfels , 2002 , Klemperer , 2002 , Milgrom and Milgrom , 2004 , Lahaie et al. , 2007 ] . In such auctions , pricing mechanisms need to be determined by the auction designer so as to satisfy various desired goals , such as revenue maximization and incentive compatibility . Often this determination is made based on information about the buyers that is assumed to be available a priori . For example , in a standard valuation model , each bidder has a valuation over the available items , and if the sellers knows the distribution of these valuations , they could design an optimal auction which maximizes the revenue . Arguably the fundamental difficulty in the design of optimal auctions is that real valuations are private and unknown to the auction designer . Consider specifically the problem of selling one item to multiple buyers . Suppose that we model the buyers ’ valuations as arising as independent draws from buyer-specific prior distributions . In this scenario , what is the optimal mechanism in terms of the expected revenue ? This problem was solved by Myerson [ 1981 ] through a characterization of virtual value functions . In particular , we can define a virtual value function of each buyer based on their 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . prior distributions . An optimal auction then lets the buyer with the largest non-negative virtual value win the item , and charges the winner a price that equals the threshold value above which she wins.1 Unfortunately , there is a further fundamental challenge in deploying these theoretical results in practice , which is that in real-world settings the auction designer may not even know the prior distributions on valuations . Instead , what the designer might hope for is that there is a stream of previous transactions , or some other relevant auxiliary data , that is helpful in inferring the buyers ’ private distributions . This perspective has motivated an active recent literature learning optimal auctions from samples [ Cole and Roughgarden , 2014 , Devanur et al. , 2016 , Morgenstern and Roughgarden , 2015 , 2016 , Syrgkanis , 2017 , Dudík et al. , 2017 , Gonczarowski and Nisan , 2017 , Huang et al. , 2018 , Roughgarden and Schrijvers , 2016 , Balcan et al. , 2018 , Guo et al. , 2019 , Roughgarden and Wang , 2019 , Gonczarowski and Weinberg , 2021 ] . In this line of work , the central question is : suppose we are only able to access the prior distributions in the form of independent samples , how many samples are sufficient and necessary for finding an approximately optimal auction ? While this merging of mechanism design and learning theory is appealing , a further concern arises . Given the potentially adversarial setting of auction design , do we really believe that the data that we observe are drawn in accord with our assumptions ? More concretely , is the learning of optimal auctions robust to adversarial corruptions of the samples ? This problem is arguably at the core of what it means to learn an optimal auction . It is a challenging problem ; indeed , as we show in Counterexample 1 in Section 4 , auction designs that are optimal in the absence of corruptions can become arbitrarily bad even if a small portion of the samples are corrupted . Building on earlier work by Cai and Daskalakis [ 2017 ] and Brustle et al . [ 2020 ] , we tackle a key open problem—what is the best approximation to the optimal revenue for arbitrary levels of corruption for distributions with unbounded support ? And what is the mechanism that achieves it ? In summary , in this work we explore the problem of the robust learning of optimal auctions , where the samples of bidders ’ valuations are subject to corruption and their support is unbounded . In particular , we consider having access to samples that are drawn from some distribution D̃ which is within a Kolmogorov-Smirnov ( KS ) distance α of the true distribution D∗ . Denote OPT as the maximum revenue we can achieve under the true valuation distributions . Our goal is to design mechanisms that are guaranteed to achieve a revenue of at least ( 1− ρ ( α ) ) ·OPT for the smallest possible error ρ ( α ) and with the use of a minimal number of samples . 1.1 Our results . We study the problem of learning revenue-optimal multi-bidder auctions from samples when the samples of bidders ’ valuations can be adversarially corrupted or drawn from distributions that are adversarially perturbed . We summarize our main results as follows : 1 . We derive tight upper bounds on the revenue we can obtain with a corrupted distribution under a population model . For distributions with monotone hazard rate ( MHR ) , and with total corruption α , we obtain an approximation ratio of 1−O ( α ) compared to the optimal revenue under the true distribution ( see Theorem 3.6 ) . For regular valuation distributions , where for total corruption α , we get an approximation ratio of 1−O ( √ α ) ( see Theorem 3.8 ) . 2 . To achieve these upper bounds , we propose a new theoretical algorithm for the population model ( see Algorithm 1 ) that , given only an “ approximate distribution ” for the bidder ’ s valuation , can learn a mechanism whose revenue is nearly optimal simultaneously for all “ true distributions ” that are α-close to the given distribution in Kolmogorov-Smirnov distance . The proposed algorithm operates beyond the setting of bounded distributions that have been studied in prior works ; indeed , they apply to general unbounded MHR and regular distributions . 3 . We further show that these upper bounds under the population model can not be further improved ( up to constant log factors ) , by providing matching lower bounds for both the MHR and regular distributions ( see Theorem 3.7 and Theorem 3.9 ) . 4 . Lastly , we derive sample complexity upper bounds for learning a near-optimal auction for both MHR and regular distributions with multiple bidders ( Theorem 4.3 and Theorem 4.4 ) , and propose a practical algorithm ( see Algorithm 2 ) which takes samples as input . We also 1More generally , the optimal auction picks the winner based on the virtual value after an “ ironing ” procedure . provide accompanying sample complexity lower bounds ( Theorem 4.5 ) , and demonstrate a small gap relative to the corresponding upper bounds . 1.2 Related work . Designing revenue optimal auctions is a classic problem in economic theory that has attracted much research attention . We survey the most closely related work in two main areas . Learning optimal auctions from samples . Recent work has explored settings of learning approximately optimal auction from samples , both for single-item auctions [ Cole and Roughgarden , 2014 ] , and multi-item auctions [ Balcan et al. , 2018 , 2016 , Morgenstern and Roughgarden , 2015 , Syrgkanis , 2017 ] . Most recently , Guo et al . [ 2019 ] provide a complete set of sample complexity bounds for single-item auctions , by deriving matching upper and lower bounds up to a poly-logarithmic factor . While these approaches have obtained fruitful results on the sample complexity of learning optimal auctions , a key assumption that is commonly made in this work is that the samples are independently and identically drawn from the bidders ’ valuation distributions , with the goal of learning an auction which maximizes the expected revenue on the underlying , unknown distribution over bidder valuations . A major difference in our work is that we consider that the samples can suffer from potential corruptions , which is a significantly more challenging setting . Robustness of learning optimal auctions . Our paradigm on the robust learning of optimal auctions is closely related to recent work that considers the learning of auctions from mismatched distributions or corrupted samples . Cai and Daskalakis [ 2017 ] consider a multi-item auction setting , where there is a given “ approximate distribution , ” and the goal is to compute an auction whose revenue is approximately optimal simultaneously for all “ true distributions '' that are close to the given one . They provide an algorithm that achieves a poly-α additive loss compared to the true optimal revenue . More recently , Brustle et al . [ 2020 ] consider learning multi-item auctions where bidders ’ valuations are drawn from correlated distributions that can be captured by Markov random fields . However , they make a key simplifying assumption—that the bidders ’ valuation for the items lie in some bounded interval . Our results , by contrast , apply to the general setting of unbounded valuation distributions , a setting that requires new theoretical machinery . To the best of our knowledge , our work constitutes the first analysis of the learnability of single-item optimal auctions from corrupted samples for unbounded distributions . Organization . In Section 2 , we provide background on auction models and formally state our problem . Section 3 contains our main theoretical statements for the population model . We propose an algorithm that achieves optimal theoretical upper bounds , by providing matching lower bounds . Section 4 contains our main results on learning with finite samples . We provide a practical algorithm that takes samples from the corrupted distribution , and provides sample complexity upper and lower bounds for both the regular and MHR distributions cases . We conclude in Section 5 . 2 Preliminaries . We begin by formally defining the setting we study for robust learning of optimal auctions , which includes the revenue objective and the general classes of valuation distributions that we consider . 2.1 Auction models . Single-bidder setting . Consider one item for sale to one bidder . The bidder has a private valuation v ∈ R+ for this item . We assume that v is a random variable distributed according to the distribution D∗ , with support R+ , cumulative distribution function F , and probability density function f . It is well known that the optimal auction in this setting is a reserve price auction , such that the task for the seller is to compute a reserve price p that optimizes revenue [ Myerson , 1981 ] . We assume that the bidder has a quasi-linear utility that is equal to u ( p ) = v − p if she decides to buy the item and u ( p ) = 0 otherwise . The seller aims to set p such that her expected revenue—i.e. , the received payment—is maximized . We consider the setting where both v and D∗ are unknown to the seller . However , the seller can access i.i.d . samples that are drawn from a distribution D̃ , which is α-close to D with regard to the Kolmogorov distance : Definition 2.1 . ( Kolmogorov-Smirnov distance ) For probability measures µ and ν on R , define dk ( µ , ν ) = sup x∈R |µ ( ( −∞ , x ) ) − ν ( ( −∞ , x ) ) | . It is well known that dk ( µ , ν ) 6 dTV ( µ , ν ) , where dTV denotes the total variation ( TV ) distance between µ and ν . The closeness of D̃ to D∗ is thus formalized as follows : dk ( D∗ , D̃ ) 6 α , for some α > 0 . Multi-bidder setting . Consider one item for sale to n bidders . Each bidder has a private valuation , vi ∈ R+ , where vi is independently drawn from the corresponding prior distribution D∗i . Thus , the valuations v = ( v1 , v2 , · · · , vn ) follow a product distribution D∗ = D∗1 × · · · × D∗n . Each bidder submits a bid bi > 0 . Denote all the bids as b = ( b1 , · · · , bn ) . A mechanism in this setting consists of two rules : the allocation rule x ( b ) that takes the bids b and outputs the probability xi ( b ) that each bidder i will receive the item , and the payment rule p ( b ) that takes the bids b and outputs the payment of bidder i . Bidder i ’ s utility is then ui ( b ) = vi · xi ( b ) − pi ( b ) . The goal of the seller is to find a mechanism that maximizes the expected revenue E [ ∑ i∈ [ n ] pi ( b ) ] , where the expectation is over v ∼ D∗ , under the following Dominant Strategy Incentive Compatibility ( DSIC ) and the Individual Rationality ( IR ) constraints : ui ( vi , b−i ) > ui ( bi , b−i ) for all vi , bi ∈ R+ and all b−i ∈ Rn−1+ ( DSIC ) ui ( vi , b−i ) > 0 for all vi ∈ R+ and all b−i ∈ Rn−1+ . ( IR ) We consider the setting in which the valuations and the prior distributions are unknown to the seller . Instead , the seller has access to a finite number of i.i.d . samples drawn from the product distribution D̃ = D̃1 × · · · × D̃n , where each D̃i satisfies dk ( D∗i , D̃i ) 6 αi , for some αi > 0 , ∀i ∈ [ n ] . Revenue objective . Letting D , D′ be product or single bidder distributions as described above , we define MD to be the mechanism that achieves the optimal revenue for the value distributions D and OPT ( D ) its expected revenue . Let also Rev ( MD , D′ ) be the expected revenue of the mechanism MD when applied to a setting where the values are drawn with respect to D′ .

Summary: The paper studies the problem of designing approximately revenue-optimal auctions using corrupted / inaccurate priors, where inaccuracy is measured by the Kolmogorov-Smirnov distance, i.e., the maximum difference in the two CDFs.  The authors consider the multi-bidder single-item setting, and focus on the cases of regular and MHR prior distributions.  Notably, they do not assume bounded support.  The main results are almost matching (multiplicative) upper and lower bounds on the maximum revenue achievable given inaccurate priors with bounded KS distances to the real ones.  The authors further apply their results to sample-based settings, and derive similar upper bounds with polynomial number of samples from the inaccurate priors.