Revenue maximization via machine learning with noisy data

1 Introduction . Revenue maximization in multi-item settings is one of the most important , long-standing open problems in mechanism design . In Bayesian Mechanism Design , there is a set of items for sale and an underlying distribution defining a set of agents ’ values for the items . A mechanism determines which buyers receive which items and what they pay . For decades , research in economics has assumed that the mechanism designer must know the exact distribution over buyers ’ values . An explosion of research [ e.g. , 1 , 4 , 19 , 20 , 25 , 26 , 28 , 30 , 31 , 33 , 37 , 40 , 44–48 , 51 ] has relaxed this strong assumption : instead , the distribution is unknown and the mechanism designer only has a training set of i.i.d . samples . Using the training set , the goal is to learn a mechanism with high expected revenue . Learning-based mechanism design is on the verge of taking over as the main tool for designing high-revenue mechanisms for selling items—a cornerstone of many modern enterprises . Motivated by recent literature on the brittleness of deep learning models in the face of imperceptible noise [ 32 , 41 , 52 ] , an important question is whether adversarial noise has the same effect in learningbased mechanism design . Learning methods deployed in real-world settings to design mechanisms 35th Conference on Neural Information Processing Systems ( NeurIPS 2021 ) . must be robust to minute noise in the data . We provide guarantees for learning mechanisms with high expected revenue in the face of an adversary that can add arbitrarily correlated noise to the training set ; we only require that the noise has bounded magnitude or is sub-Gaussian . In contrast , classic learning-theoretic results typically rely on the stronger assumption that the training samples are independent . 1.1 Summary of main contributions and overview of techniques . We set out to determine whether imperceptible adversarial noise in the training set can cause a catastrophic loss in the learned mechanism ’ s expected revenue . Our main contribution is a sensitivity analysis of revenue with respect to the noise ’ s magnitude , which answers this question in the negative . Our results apply to empirical revenue maximization , the canonical approach to learning-based mechanism design . To our knowledge , we are the first to study learning-based mechanism design under adversarial noise . We provide guarantees for optimizing revenue over several classic mechanism classes : second-price auctions with non-anonymous reserves under additive buyers , single-priced lottery mechanisms under unit-demand buyers , and item-pricing mechanisms under unit-demand buyers . These classes have been studied extensively [ e.g. , 6 , 18 , 24 , 46 , 51 ] and can be viewed as hypothesis classes , just as DNNs correspond to a particular ( brittle ) type of hypothesis class . In some settings , mechanisms from these classes have been shown to provide approximately optimal revenue [ 22 , 23 , 35 , 39 ] . The key challenge we face is that revenue functions are extremely sensitive to small perturbations of the buyers ’ values . In a second-price auction , for example , slightly shifting the highest bid from below to above the reserve price can cause an arbitrarily large drop in revenue , as illustrated in Figure 1 . The set of bidders whose bids are above their reserve prices can completely change when even an infinitesimal amount of noise is added , radically altering the analytical form of the revenue function . This is unlike most functions that we understand well from a learning-theoretic perspective , which generally are smooth , continuous , or—more broadly—exhibit a straightforward connection between parameters and output value . Despite the volatility of these revenue functions , we nonetheless are able to prove bounds on the revenue loss incurred by optimizing over a noisy training set . Our second main contribution is an application of our guarantees to multi-task mechanism design . To our knowledge , we are the first to study this problem . The existing literature on mechanism design via machine learning assumes that there is a single distribution defining the buyers ’ values . Often , however , the mechanism designer may be interested in designing high-revenue mechanisms for multiple related distributions . Each distribution thus defines a distinct learning task . The goal is to leverage the similarity between the distributions to learn a high-revenue mechanism per distribution . Multi-task learning has proven useful in fields such as Computer Vision and Natural Language Processing [ 53 ] , and we demonstrate its value in mechanism design as well . Our main technical insight is that we can transform training instances from one task into slightly noisy training instances for another task and make use of our near-optimality guarantees under noised samples . 1.2 Related research . Adversarial Machine Learning : A prominent line of research on machine learning in the presence of noise studies the case where some fraction of the training data may be noisy ( data poisoning ) . Research in this vein includes settings where every instance can be corrupted with probability η < 1 [ e.g. , 3 ] and where an η-fraction of the training data can be corrupted [ e.g. , 16 , 27 ] . Meanwhile , in our setting , every sample may be adversarially perturbed with probability 1 , not just an η-fraction , though we require that the perturbation be bounded . Another line of research , especially prominent in Computer Vision , studies the case where the test data can be adversarially perturbed within , for example , an ` p-ball [ 32 , 41 , 52 ] . We , on the other hand , are concerned with noisy training data and face the unique challenges imposed by mechanism design . Mechanism Design with Noised Distributions : While the robustness of mechanism design in the face of model uncertainty has been well studied ( see for instance work by Bergemann and Morris [ 14 ] and Bergemann and Schlag [ 15 ] ) , to our knowledge , there has been relatively less work done on mechanism design via machine learning in the particular setting where samples are not necessarily drawn i.i.d . from the true distribution . The paper most closely related to ours is that by Cai and Daskalakis [ 20 ] , who show that given access to a noisy distribution , it is possible to learn a mechanism with high expected revenue over the true distribution if the Kolmogorov distance between the noisy and true distributions are small . One might hope that the uniform distribution over the noisy training data could constitute the noisy distribution , but the Kolmogorov distance between this empirical distribution and the true distribution could be as large as 1 , so these results do not apply to our setting . Moreover , we note that Cai and Daskalakis [ 20 ] makes the stronger assumption that the buyers ’ values follow a product distribution , whereas our assumed noise model allows for arbitrarily correlated distributions . Multi- vs Single-item Mechanism Design : Further afield , Huang et al . [ 38 ] and Guo et al . [ 33 ] have also studied mechanism design under noisy settings , albeit under single-item settings . By contrast , we study a wider and different set of multi-item mechanism classes including multi-item lotteries and multi-item item-pricing mechanisms . The past few decades of research on revenue-maximizing multi-item mechanism design has demonstrated that results and intuition from the single-item setting do not always carry over to the multi-item setting . And we see that this is the case in our paper as well . For example , an item-pricing mechanism for a single item would have a straight-forward stability analysis ( similar to the analysis of a single-item anonymous second-price auction that we include for intuition in Appendix B.1 ) . With multiple items and unit-demand buyers , however , stability does not even hold . Therefore , intuition from the single-item case does not carry over to the multi-item case , and we must use completely different analysis techniques . Dispersion : The technique needed is that of dispersion ( section 3.4 ) . Our bounds improve based on how “ nice ” the distribution over buyers ’ values is , quantified by dispersion [ 5 ] . Dispersion has primarily been used to provide regret bounds [ 5 , 8 ] . Balcan et al . [ 7 ] also use dispersion in mechanism design , though for a very different problem : estimating how much utility an agent can gain by misreporting their value in a manipulable mechanism . The appendix summarizes additional related work on mechanism design with side information , multi-task learning , transfer learning , and approximate incentive compatibility . 2 Notation . There are n buyers and m items for sale . Each buyer i ∈ [ n ] has a value for each item j ∈ [ m ] , denoted as vij ≥ 0 . We analyze unit-demand and additive buyers . If buyer i is unit-demand , he is only interested in obtaining one item , so his value for a bundle b ⊆ [ m ] of goods is equal to the maximum value he has for any item in b , maxj∈b vij . If buyer i is additive , his value for a bundle b ⊆ [ m ] is ∑ j∈b vij . We use the notation vi = ( vi1 , . . . , vim ) to denote buyer i ’ s values for all m items and v = ( v1 , . . . , vn ) ∈ Rnm to denote all n buyers ’ values . When there is only one item , we use the notation v = ( v1 , . . . , vn ) ∈ Rn , where vi ∈ R is buyer i ’ s value for the item . We study mechanism classes parameterized by vectors r ∈ Rd for some d ( for example , the class of second-price auctions parameterized by non-anonymous reserves r ) . The mechanisms we analyze are dominant-strategy incentive compatible , so we assume the bids equal the buyers ’ true values . We denote the revenue of the mechanism defined by r when the buyers ’ values equal v by revr ( v ) . For simplicity of notation , we assume revr ( v ) is in [ 0 , 1 ] though our results can be extended to the case where revr ( v ) ∈ [ 0 , H ] for some H ( all bounds must simply be multiplied by H ) . 3 Learning under adversarial noise . In this section , we study mechanism design via machine learning when the data is corrupted . Let S = { v ( 1 ) , . . . , v ( L ) } ⊂ Rnm≥0 be a set of valuation vectors drawn from an unknown distribution D. Our learning algorithm receives a poisoned training set S ′ = { w ( 1 ) , . . . , w ( L ) } ⊂ Rnm≥0 . Bounded Noise Model : To model the imperceptible noise that corrupts the bids , we assume the bounded noise model that is commonly assumed in adversarial defense literature . That is , for some known > 0 , all samples ` ∈ [ L ] , all bidders i ∈ [ n ] , and all items j ∈ [ m ] , w ( ` ) ij ∈ [ v ( ` ) ij − , v ( ` ) ij ] , or more succinctly , v ( ` ) − ≤ w ( ` ) ≤ v ( ` ) . ( If we only know that ∥∥v ( ` ) −w ( ` ) ∥∥∞ ≤ , we can shift all bids in each vector w ( ` ) down by , in which case v ( ` ) − 2 ≤ w ( ` ) ≤ v ( ` ) . ) . Our goal is to use the noisy set S ′ to learn a mechanism with high expected revenue over D. We provide some more motivating factors for our choice of the noise model in the context of mechanism design . Besides its prevalence in adversarial learning literature , ( 1 ) Such noise may arise when one is using estimated bids to learn auction parameters . We give a concrete example in Section 4 where we study multi-task mechanism design . ( 2 ) The noise may result from bounded rationality on the part of the bidders , leading to small , seemingly-innocuous differences between the buyers ’ true values and reported values . ( 3 ) Lastly , we note that our results can immediately be extended to cover unbounded standard noise models such as sub-Gaussian noise . In this case , suppose that each element of each vector v ( i ) ∈ S is perturbed by subG ( σ2 ) to obtain the noisy vector w ( i ) . Then with probability 1 − δ , for all i ∈ [ L ] , ||w ( i ) − v ( i ) ||∞ ≤ σ √ 2 log 2Lnmδ . The noise need not be independent . Therefore , all of our results hold with ( high ) probability 1− 2δ over the draw of the true values S and the noise for = 2σ √ 2 log 2Lnmδ . Our Approach : We begin by fixing a mechanism class parameterized by vectors r ∈ Rd for some d ( for example , the class of second-price auctions parameterized by non-anonymous reserves r ) . Given the training set S ′ , the most widely used learning algorithm is empirical revenue maximization ( ERM ) , which returns the parameter setting r̂′ that maximizes average empirical revenue over S ′ . This algorithm will be the subject of our study . Throughout this section , we analyze the following key question : what is the difference between the expected revenue of the mechanism defined by r̂′ , Ev∼D [ revr̂′ ( v ) ] , and that of the optimal mechanism in the class , maxr∈Rd E [ revr ( v ) ] ? Another key aspect of learning besides optimality is sample complexity . In machine learning beyond the context of mechanism design , prior research [ e.g. , 49 ] has shown that a large increase in sample complexity is needed to handle noise with bounded ` ∞-norm . By contrast , under the same noise assumption , we show that empirical revenue maximization can achieve near-optimal revenue without significantly higher sample complexity . This exposes a notable contrast between the two learning tasks . We provide lower bounds showing that ERM ’ s loss has an optimal dependence on the noise .

Summary: This paper provides robustness analysis for several existing types of auctions when the empirical bids can be perturbed within an L1 ball. The authors proposed two conditions: p-stable and q-convergent, under which the expected revenue of the empirically optimal mechanism over the noisy samples is close to the expected revenue of the optimal mechanism in the class. The authors further show that several existing mechanisms satisfy these conditions to various degree.