{"title": "Sample Complexity of Automated Mechanism Design", "book": "Advances in Neural Information Processing Systems", "page_first": 2083, "page_last": 2091, "abstract": "The design of revenue-maximizing combinatorial auctions, i.e. multi item auctions over bundles of goods, is one of the most fundamental problems in computational economics, unsolved even for two bidders and two items for sale. In the traditional economic models, it is assumed that the bidders' valuations are drawn from an underlying distribution and that the auction designer has perfect knowledge of this distribution. Despite this strong and oftentimes unrealistic assumption, it is remarkable that the revenue-maximizing combinatorial auction remains unknown. In recent years, automated mechanism design has emerged as one of the most practical and promising approaches to designing high-revenue combinatorial auctions. The most scalable automated mechanism design algorithms take as input samples from the bidders' valuation distribution and then search for a high-revenue auction in a rich auction class. In this work, we provide the first sample complexity analysis for the standard hierarchy of deterministic combinatorial auction classes used in automated mechanism design. In particular, we provide tight sample complexity bounds on the number of samples needed to guarantee that the empirical revenue of the designed mechanism on the samples is close to its expected revenue on the underlying, unknown distribution over bidder valuations, for each of the auction classes in the hierarchy. In addition to helping set automated mechanism design on firm foundations, our results also push the boundaries of learning theory. In particular, the hypothesis functions used in our contexts are defined through multi stage combinatorial optimization procedures, rather than simple decision boundaries, as are common in machine learning.", "full_text": "Sample Complexity of Automated Mechanism Design\n\nMaria-Florina Balcan, Tuomas Sandholm, Ellen Vitercik\n\n{ninamf,sandholm,vitercik}@cs.cmu.edu\n\nSchool of Computer Science\nCarnegie Mellon University\n\nPittsburgh, PA 15213\n\nAbstract\n\nThe design of revenue-maximizing combinatorial auctions, i.e. multi-item auctions\nover bundles of goods, is one of the most fundamental problems in computational\neconomics, unsolved even for two bidders and two items for sale. In the traditional\neconomic models, it is assumed that the bidders\u2019 valuations are drawn from an\nunderlying distribution and that the auction designer has perfect knowledge of\nthis distribution. Despite this strong and oftentimes unrealistic assumption, it is\nremarkable that the revenue-maximizing combinatorial auction remains unknown.\nIn recent years, automated mechanism design has emerged as one of the most prac-\ntical and promising approaches to designing high-revenue combinatorial auctions.\nThe most scalable automated mechanism design algorithms take as input samples\nfrom the bidders\u2019 valuation distribution and then search for a high-revenue auction\nin a rich auction class. In this work, we provide the \ufb01rst sample complexity analysis\nfor the standard hierarchy of deterministic combinatorial auction classes used in\nautomated mechanism design. In particular, we provide tight sample complexity\nbounds on the number of samples needed to guarantee that the empirical revenue\nof the designed mechanism on the samples is close to its expected revenue on the\nunderlying, unknown distribution over bidder valuations, for each of the auction\nclasses in the hierarchy. In addition to helping set automated mechanism design on\n\ufb01rm foundations, our results also push the boundaries of learning theory. In partic-\nular, the hypothesis functions used in our contexts are de\ufb01ned through multi-stage\ncombinatorial optimization procedures, rather than simple decision boundaries, as\nare common in machine learning.\n\n1\n\nIntroduction\n\nMulti-item, multi-bidder auctions have been studied extensively in economics, operations research,\nand computer science. In a combinatorial auction (CA), the bidders may submit bids on bundles of\ngoods, rather than on individual items alone, and thereby they may fully express their complex valua-\ntion functions. Notably, these functions may be non-additive due to the presence of complementary\nor substitutable goods for sale. There are many important and practical applications of CAs, ranging\nfrom the US government\u2019s wireless spectrum license auctions to sourcing auctions, through which\ncompanies coordinate the procurement and distribution of equipment, materials and supplies.\nOne of the most important and tantalizing open questions in computational economics is the design\nof optimal auctions, that is, auctions that maximize the seller\u2019s expected. In the standard economic\nmodel, it is assumed that the bidders\u2019 valuations are drawn from an underlying distribution and that\nthe mechanism designer has perfect information about this distribution. Astonishingly, even with this\nstrong assumption, the optimal CA design problem is unsolved even for auctions with just two distinct\nitems for sale and two bidders. A monumental advance in the study of optimal auction design was\nthe characterization of the optimal 1-item auction [Myerson, 1981]. However, the problem becomes\n\n30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.\n\n\fsigni\ufb01cantly more challenging with multiple items for sale. In particular, Conitzer and Sandholm\nproved that the problem of \ufb01nding a revenue-maximizing deterministic CA is NP-complete [Conitzer\nand Sandholm, 2004]. We note here that it is well-known that randomization can increase revenue in\nCAs, but we focus on deterministic CAs in this work because in many applications, randomization is\nnot palatable and very few, if any, randomized CAs are used in practice.\nIn recent years, a novel approach known as automated mechanism design (AMD) has been adopted\nto attack the revenue-maximizing auction design problem [Conitzer and Sandholm, 2002, Sandholm,\n2003]. In the most scalable strand of AMD, algorithms have been developed which take samples\nfrom the bidders\u2019 valuation distributions as input, optimize over a rich class of auctions, and return an\nauction which is high-performing over the sample [Likhodedov and Sandholm, 2004, 2005, Sandholm\nand Likhodedov, 2015]. AMD algorithms have yielded deterministic mechanisms with the highest\nknown revenues in the contexts used for empirical evaluations [Sandholm and Likhodedov, 2015].\nThis approach relaxes the unrealistic assumption that the mechanism designer has perfect information\nabout the bidders\u2019 valuation distribution.\nHowever, until now, there was no formal characterization of the number of samples required to\nguarantee that the empirical revenue of the designed mechanism on the samples is close to its\nexpected revenue on the underlying, unknown distribution over bidder valuations. In this paper,\nwe provide that missing link. We present tight sample complexity guarantees over an extensive\nhierarchy of expressive CA families. These are the most commonly used auction families in AMD.\nThe classes in the hierarchy are based on the classic VCG mechanism, which is a generalization of\nthe well-known second-price, or Vickrey, single-item auction. The auctions we consider achieve\nsigni\ufb01cantly higher revenue than the VCG baseline by weighting bidders (multiplicatively increasing\nall of their bids) and boosting outcomes (additively increasing the liklihood that a particular outcome\nwill be the result of the auction).\nA major strength of our results is their applicability to any algorithm that determines the optimal\nauction over the sample, a nearly optimal approximation, or any other black box procedure. Therefore,\nthey apply to any automated mechanism design algorithm, optimal or not. One of the key challenges\nin deriving these general sample complexity bounds is that to do so, we must develop deep insights\ninto how changes to the auction parameters (the bidder weights and allocation boosts) effect the\noutcome of the auction (who wins which items and how much each bidder pays) and thereby the\nrevenue of the auction. In our context, we show that the functions which determine the outcome of an\nauction are highly complex, consisting of multi-stage optimization procedures.\nTherefore, the function classes we consider are much more challenging than those commonly found\nin machine learning contexts. Typically, for well-understood classes of functions used in machine\nlearning, such as linear separators or other smooth curves in Euclidean spaces, there is a simple\nmapping from the parameters of a speci\ufb01c hypothesis to its prediction on a given example and a\nclose connection between the distance in the parameter space between two parameter vectors and the\ndistance in function space between their associated hypotheses. Roughly speaking, it is necessary to\nunderstand this connection in order to determine how many signi\ufb01cantly different hypotheses there\nare over the full range of parameters. In our context, due to the inherent complexity of the classes we\nconsider, connecting the parameter space to the space of revenue functions requires a much more\ndelicate analysis. The key technical part of our work involves understanding this connection from a\nlearning theoretic perspective. For the more general classes in the hierarchy, we use Rademacher\ncomplexity to derive our bounds, and for the auction classes with more combinatorial structure, we\nexploit that structure to prove pseudo-dimension bounds. This work is both of practical importance\nsince we \ufb01ll a fundamental gap in AMD, and of learning theoretical interest, as our sample complexity\nanalysis requires a deep understanding of the structure of the revenue function classes we consider.\nRelated Work. In prior research, the sample complexity of revenue maximization has been studied\nprimarily in the single-item or the more general single-dimensional settings [Elkind, 2007, Cole and\nRoughgarden, 2014, Huang et al., 2015, Medina and Mohri, 2014, Morgenstern and Roughgarden,\n2015, Roughgarden and Schrijvers, 2016, Devanur et al., 2016], as well as some multi-dimensional\nsettings which are reducible to the single-bidder setting [Morgenstern and Roughgarden, 2016]. In\ncontrast, the combinatorial settings that we study are much more complex since the revenue functions\nconsist of multi-stage optimization procedures that cannot be reduced to a single-bidder setting. The\ncomplexity intrinsic to the multi-item setting is explored in [Dughmi et al., 2014], who show that\n\n2\n\n\ffor a single unit-demand bidder, when the bidder\u2019s values for the items may be correlated, \u2126(2m)\nsamples are required to determine a constant-factor approximation to the optimal auction.\nLearning theory tools such as pseudo-dimension and Rademacher complexity were used to prove\nstrong guarantees in [Medina and Mohri, 2014, Morgenstern and Roughgarden, 2015, 2016], which\nanalyze piecewise linear revenue functions and show that few samples are needed to learn over the\nrevenue function classes in question. In a similar direction, bounds on the sample complexity of\nwelfare-optimal item pricings have been developed [Feldman et al., 2015, Hsu et al., 2016]. Earlier\nwork of Balcan et al. [2008] addressed sample complexity results for revenue maximization in\nunrestricted supply settings. In that context, the revenue function decomposes additively among\nbidders and does not apply to our combinatorial setting.\nDespite the inherent complexity of designing high-revenue CAs, Morgenstern and Roughgarden\nuse linear separability as a tool to prove that certain simple classes of multi-parameter auctions\nhave small sample complexity. The auctions they study are sequential auctions with item and grand\nbundle pricings, as well as second-price item auctions with item reserve prices [Morgenstern and\nRoughgarden, 2016].\nIn the item pricing auctions, the bidders show up one at a time and the\nseller offers each item that remains at some price. Each buyer then chooses the subset of goods\nthat maximizes her utility. In the grand bundle pricing auctions, the bidders are each offered the\ngrand bundle in some \ufb01xed order, and the \ufb01rst bidder to have a value greater than the price buys it.\nThey show that bounding the sample complexity of these sequential auctions can be reduced to the\nsingle-buyer setting.\nIn contrast, the auctions we study are more versatile than item pricing auctions, as they give the\nmechanism designer many more degrees of freedom than the number of items. This level of\nexpressiveness allows the designer to increase competition between bidders, much like Myerson\u2019s\noptimal auction, and thus boost revenue.\nIt is easy to construct examples where even simple\nAMAs achieve signi\ufb01cantly greater revenue than sequential auctions with item and grand bundle\nprices. Moreover, even the simpler auction classes we consider pose a unique challenge because the\nparameters de\ufb01ning the auctions in\ufb02uence the multi-stage allocation procedure and resulting revenue\nin non-intuitive ways. This is unlike item and grand bundle pricing auctions, as well as second-price\nitem auctions, which are simple by design. Our function classes therefore require us to understand\nthe speci\ufb01c form of the weighted VCG payment rule and its interaction with the parameter space.\nThus, our context and techniques diverge from those in [Morgenstern and Roughgarden, 2016].\nFinally, there is a wealth of work on characterizing the optimal CA for restricted settings and designing\nmechanisms which achieve high, if not optimal revenue in speci\ufb01c contexts. Due to space constraints,\nin Section A of the supplementary materials, we describe these results as well as what is known\ntheoretically about the classes in the hierarchy of deterministic CAs we study.\n\n2 Preliminaries, notation, and the combinatorial auction hierarchy\n\nIn the following section, we explain the basic mechanism design problem, \ufb01x notation, and then\ndescribe the hierarchy of combinatorial auction families we study.\nMechanism Design Preliminaries. We consider the problem of selling m heterogeneous goods to\nn bidders. This means that there are 2m different bundles of goods, B = {b1, . . . , b2m}. Each bidder\ni \u2208 [n] is associated with a set-wise valuation function over the bundles, vi : B \u2192 R. We assume\nthat the bidders\u2019 valuations are drawn from a distribution D.\nEvery auction is de\ufb01ned by an allocation function and a payment function. The allocation function\ndetermines which bidders receive which items based on their bids and the payment function determines\nhow much the bidders need to pay based on their bids and the allocation. It is up to the mechanism\ndesigner to determine which allocation and payment functions should be used. In our context, the\ntwo functions are \ufb01xed based on the samples from D before the bidders submit their bids.\nEach auction family that we consider has a design based on the classic Vickrey-Clarke-Groves\nmechanism (VCG). The VCG mechanism, which we describe below, is the canonical strategy-proof\nmechanism, which means that every bidder\u2019s dominant strategy is to bid truthfully. In other words,\nfor every Bidder i, no matter the bids made by the other bidders, Bidder i maximizes her expected\nutility (her value for her allocation minus the price she pays) by bidding her true value. Therefore, we\ndescribe the VCG mechanism assuming that the bids equal the bidders\u2019 true valuations.\n\n3\n\n\fj\n\nj(cid:54)=i vj\n\nj\n\nj\n\nj(cid:54)=i vj\n\n(cid:2)vj\n\n1, . . . , b\u2217\n\n(cid:0)b\u2212i\n\n(cid:0)b\u2212i\n(cid:1)(cid:105)\n\n1 , . . . , b\u2212i\nj(cid:54)=i\n\n(cid:1) be the disjoint\n(cid:0)b\u2212i\n(cid:1)(cid:3) =\n(cid:0)b\u2217\n(cid:1) \u2212 vj\n\ni ). Meanwhile, let(cid:0)b\u2212i\n(cid:1). Then Bidder i must pay(cid:80)\n\nThe VCG mechanism allocates the items such that the social welfare of the bidders, that is, the\nsum of each bidder\u2019s value for the items she wins, is maximized. Intuitively, each winning bid-\nder then pays her bid minus a \u201crebate\u201d equal to the increase in welfare attributable to Bidder i\u2019s\npresence in the auction. This form of the payment function is crucial to ensuring that the auction\nis strategy-proof. More concretely, the allocation of the VCG mechanism is the disjoint set of\nsubsets (b\u2217\n\nn) \u2286 B that maximizes(cid:80) vi (b\u2217\nset of subsets that maximizes(cid:80)\ni ) \u2212(cid:104)(cid:80) vj\n(cid:1) \u2212(cid:80)\n(cid:0)b\u2217\n\nj\nvi (b\u2217\n. In the special case where there is one item for sale, the\nVCG mechanism is known as the second price, or Vickrey, auction, where the highest bidder wins\nthe item and pays the second highest bid. We note that every auction in the classes we study is\nstrategy-proof, so we may assume that the bids equal the bidders\u2019 valuations.\nNotation. We study auctions with n bidders and m items. We refer to the bundle of all m items\nas the grand bundle. In total, there are (n + 1)m possible allocations, which we denote as the\noi,j = b(cid:96) \u2208 B denotes the bundle of items allocated to Bidder j in allocation (cid:126)oi. We use the\nnotation (cid:126)v1 = (v1 (b1) , . . . , v1 (b2m)) and (cid:126)v = ((cid:126)v1, . . . , (cid:126)vn) to denote a vector of bidder valuation\nfunctions. We say that revA((cid:126)v) is the revenue of an auction A on the valuation vector (cid:126)v. Denoting\nthe payment of any one bidder under auction A given valuation vector (cid:126)v as pi,A ((cid:126)v), we have that\ni=1 pi,A ((cid:126)v). Finally, U is an upper bound on the revenue achievable for any auction\n\nvectors O =(cid:8)(cid:126)o1, . . . , (cid:126)o(n+1)m\nrevA((cid:126)v) =(cid:80)n\n\n(cid:9) . Each allocation vector (cid:126)oi can be written as (oi,1, . . . , oi,n), where\n\nn\n\nj\n\nover the support of the bidders\u2019 valuation distribution.\n\n(cid:111)\n\n(cid:111)\n\n(cid:110)(cid:80)\n\nAuction Classes. We now give formal de\ufb01nitions of the CA families in the hierarchy we study. See\nFigure 1 for the hierarchical organization of the auction classes, together with the papers which\nintroduced each family.\nAf\ufb01ne maximizer auctions (AMAs).\n\nbidder (w1, . . . , wn) \u2282 R>0 and boosts per allocation (cid:0)\u03bb ((cid:126)o1) , . . . , \u03bb(cid:0)(cid:126)o(n+1)m\n(cid:0)w1, . . . , wn, \u03bb ((cid:126)o1) , . . . , \u03bb(cid:0)(cid:126)o(n+1)m\n\n(cid:1)(cid:1) \u2282 R.\n(cid:1)(cid:1). To simplify notation, we write \u03bbi = \u03bb ((cid:126)oi) interchange-\n\nAn auction A uniquely corresponds to a set of these parameters, so we write A =\n\nAn AMA A is de\ufb01ned by a set of weights per\n\n(cid:96)(cid:54)=j w(cid:96)v(cid:96) (o\u2217\n\n(cid:96)(cid:54)=j w(cid:96)v(cid:96) (oi,(cid:96)) + \u03bb ((cid:126)oi)\n\nj=1 wjvj (oi,j) + \u03bb ((cid:126)oi)\n\n(cid:105)\n(cid:96) ) \u2212 \u03bb ((cid:126)o\u2217)\n\n(cid:110)(cid:80)n\n(cid:104)(cid:80)\n(cid:96)(cid:54)=j w(cid:96)v(cid:96) (o\u2212j,(cid:96)) + \u03bb ((cid:126)o\u2212j) \u2212(cid:80)\n\nably. These parameters allow the mechanism designer to multiplicatively boost any bidder\u2019s bids\nby their corresponding weight and to increase the likelihood that any one allocation is returned\nas the output of an auction. More concretely, the allocation (cid:126)o\u2217 of an AMA A is the one which\nmaximizes the weighted social welfare, i.e. (cid:126)o\u2217 = argmax(cid:126)oi\u2208O\n. The\npayment function of A has the same form as the VCG payment rule, with the parameters fac-\nIn particular, for all j \u2208 [n], the\ntored in to ensure that the auction remains strategy-proof.\npayments are pj,A ((cid:126)v) = 1\n, where\nwj\n(cid:126)o\u2212j = argmax(cid:126)oi\u2208O\nWe assume that Hw \u2264 wi \u2264 Hw, \u03bbi \u2264 H\u03bb, and vi (b(cid:96)) \u2264 Hv for some Hw, Hw, H\u03bb, Hv \u2208 R\u22650.\nIt is typical to assume an upper bound (here, Hv) on the bidders\u2019 valuation for any bundle. This is\nrelated to the fact that an upper bound on a target function\u2019s range is always assumed in standard\nmachine learning sample complexity bounds. Intuitively, generalizability depends on how much any\none sample can skew the empirical average of a hypothesis, or in this case, auction. The bounds on\nthe AMA parameters are closely related to the bound on the bidders\u2019 valuations Hv. For example, it\nis a simple exercise to see that we need not search for a lambda value which is greater than Hv.\nVirtual valuation combinatorial auctions (VVCAs). VVCAs are a subset of AMAs. The de\ufb01ning\ni=1 \u03bbi ((cid:126)oj)\nwhere \u03bbi ((cid:126)oj) = ci,b for all allocations (cid:126)oj that give Bidder i exactly bundle b \u2208 B.\n\u03bb-auctions. \u03bb-auctions are the subclass of AMAs where wi = 1 for all i \u2208 [n].\nMixed bundling auctions (MBAs). The class of MBAs is parameterized by a constant c \u2265 0 which\ncan be seen as a discount for any bidder who receives the grand bundle. Formally, the c-MBA is the\n\u03bb-auction with \u03bb((cid:126)o) = c if some bidder receives the grand bundle in allocation (cid:126)o and 0 otherwise.\nMixed bundling auctions with reserve prices (MBARPs). MBARPs are identical to MBAs though\nwith reserve prices. In a single-item VCG auction (i.e. second price auction) with a reserve price, the\n\ncharacteristic of a VVCA is that each \u03bb ((cid:126)oj) is split into n terms such that \u03bb ((cid:126)oj) =(cid:80)n\n\n.\n\n4\n\n\fitem is only sold if the highest bidder\u2019s bid exceeds the reserve price, and the winner must pay the\nmaximum of the second highest bid and the reserve price. We describe how this intuition generalizes\nto MBAs in Section 3.\nGeneralization bounds. In order to derive sample complexity bounds which apply to any algorithm\nthat determines the optimal auction over the sample, a nearly optimal approximation, or any other\nblack-box procedure, we derive uniform convergence sample complexity bounds with respect to the\nauction classes we examine. Formally, we de\ufb01ne the sample complexity of uniform convergence over\nan auction class A as follows.\nDe\ufb01nition 1 (Sample complexity of uniform convergence over A). We say that N (\u0001, \u03b4,A) is the\na sample of size N \u2265 N (\u0001, \u03b4,A) drawn at random from D, with probability at least 1 \u2212 \u03b4, for all\nauctions A \u2208 A,\n\nsample complexity of uniform convergence over A if for any \u0001, \u03b4 \u2208 (0, 1), if S =(cid:8)(cid:126)v1, . . . , (cid:126)vN(cid:9) is\n\n(cid:12)(cid:12)(cid:12) \u2264 \u0001.\n(cid:0)(cid:126)vi(cid:1) \u2212 E(cid:126)v\u223cD [revA((cid:126)v)]\n\n(cid:80)N\n\ni=1 revA\n\n(cid:12)(cid:12)(cid:12) 1\n\nN\n\n3 Sample complexity bounds over the hierarchy of auction classes\n\nIn this section, we provide an overview of our sample complexity guarantees over the hierarchy of\nauction classes we consider (Section 3.1 and 3.2). We show that more structured classes require\ndrastically fewer samples to learn over. We conclude with a note about sample complexity guarantees\nfor algorithms that \ufb01nd an approximately optimal mechanism over a sample, as opposed to the\noptimal mechanism. All omitted proofs are presented in full in the supplementary material.\n\n3.1 The sample complexity of AMA, VVCA, and \u03bb-auction revenue maximization\n\nWe begin by analyzing the most general families in the CA hierarchy \u2014 AMAs, VVCAs, and\n\u03bb-auctions \u2014 proving a general upper bound and class-speci\ufb01c lower bounds.\nTheorem 1. The sample complexity of uniform convergence over the classes of n-bidder, m-item\n. Moreover, for \u03bb-Auctions,\n\nAMAs, VVCAs, and \u03bb-Auctions is N = (cid:101)O\n\nm(cid:0)U + nm/2(cid:1) /\u0001(cid:3)2(cid:17)\n\n(cid:16)(cid:2)U nm\u221a\n\nN = \u2126 (nm) and for VVCAs, N = \u2126 (2m).\n\nWe derive the upper bound by analyzing the Rademacher complexity of the class of n-bidder, m-item\nAMA revenue functions. For a family of functions G and a \ufb01nite sample S = {x1, . . . , xN} of size\n\u03c3 = (\u03c31, . . . , \u03c3N ), with \u03c3is independent uniform random variables taking values in {\u22121, 1}. The\n\nN, the empirical Rademacher complexity is de\ufb01ned as (cid:98)RS (G) = E\u03c3[supg\u2208G\nRademacher complexity of G is de\ufb01ned as RN (G) = ES\u223cDN [(cid:98)RS (G)].\n\n(cid:80) \u03c3ig(xi)], where\n\n1\nN\n\nThe AMA revenue function, de\ufb01ned in Section 2, can be summarized as a multi-stage optimization\nprocedure: determine the weighted-optimal allocation and then compute the n different payments,\neach of which requires a separate optimization procedure. Luckily, we are able to decompose the\nrevenue functions into small components, each of which is easier to analyze on its own, and then\ncombine our results to prove the following theorem about this class of revenue functions as a whole.\nTheorem 2. Let F be the set of n-bidder, m-item AMA revenue functions revA such that A =\n\n(cid:1) , Hw \u2264 |wi| \u2264 Hw,|\u03bbi| \u2264 H\u03bb. Then\n\n(cid:0)w1, . . . , wn, \u03bb1, . . . , \u03bb(n+1)m\n\nRN (F) = O\n\nnm+2 (HwHv + H\u03bb)\n\nHw\n\nm log n\n\nN\n\nn \u02c6Hv (nHw + H\u03bb)\n\nHw\n\nwhere \u02c6Hv = max{Hv, 1}.\n\nProof sketch. First, we describe how we split each revenue function into smaller, easier to analyze\natoms, which together allow us to bound the Rademacher complexity of the class of AMA revenue\nfunctions. To this end, it is well-known (e.g. [Mohri et al., 2012]) that if every function f in a class\nF can be written as the summation of two functions g and h from classes G and H, respectively, then\nRN (F) \u2264 RN (G) + RN (H). Therefore, we split each revenue function into n + 1 components\nsuch that the sum of these components equals the revenue function.\n\n5\n\n(cid:32)\n\n(cid:114)\n\n(cid:32)\n\n+(cid:112)nm log N\n\n(cid:33)(cid:33)\n\n,\n\n\fAf\ufb01ne maximizer auctions [Roberts, 1979]\n\n\u222a\n\n\u222a\n\nVirtual valuation CAs [Likhodedov and Sandholm, 2004]\n\n\u222a\n\n\u03bb-auctions [Jehiel et al., 2007]\n\n\u222a\n\nMixed bundling auctions with reserve prices [Tang and Sandholm, 2012]\n\n\u222a\n\nMixed bundling auctions [Jehiel et al., 2007]\n\nFigure 1: The hierarchy of deterministic CA families. Generality increases upward in the hierarchy.\n\nA = argmax(cid:126)oi\u2208O\n\nWith this objective in mind, let (cid:126)o\u2217\nA((cid:126)v) be the outcome of the AMA A on the bidding instance (cid:126)v,\ni.e. (cid:126)o\u2217\nand let \u03c6A,\u2212j((cid:126)v) be the weighted social welfare\nof the welfare-maximizing outcome without Bidder j\u2019s participation. In other words, \u03c6A,\u2212j((cid:126)v) =\nmax(cid:126)oi\u2208O\n\nj=1 wjvj (oi,j) + \u03bbi\n\n. Then we can write\n\n(cid:110)(cid:80)\n\n(cid:96)(cid:54)=j w(cid:96)v(cid:96) (oi,(cid:96)) + \u03bbi\n\n(cid:111)\n\n\uf8f6\uf8f8 1(cid:126)oi=(cid:126)o\u2217\n\uf8f6\uf8f8 1(cid:126)oi=(cid:126)o\u2217\n\n(cid:110)(cid:80)n\n\nn(cid:88)\n\nj=1\n\n(cid:111)\n\uf8eb\uf8ed n(cid:88)\n(cid:88)\n\nj=1\n\n1\nwj\n\n(cid:88)\n\n(cid:96)(cid:54)=j\n\n1\nwj\n\ni=1\n\n(n+1)m(cid:88)\n\uf8eb\uf8ed n(cid:88)\n\n(n+1)m(cid:88)\n\nrevA((cid:126)v) =\n\n\u03c6A,\u2212j((cid:126)v) \u2212\n\n1\nwj\n\nw(cid:96)v(cid:96)(oi,(cid:96)) + \u03bbi\n\nA((cid:126)v).\n\nWe can now split revA into n + 1 simpler functions: revA,j((cid:126)v) = 1\nwj\n\n\u03c6A,\u2212j((cid:126)v) for j \u2208 [n] and\n\nrevA,n+1((cid:126)v) = \u2212\n\ni=1\n\nj=1\n\n(cid:96)(cid:54)=j\n\nA((cid:126)v),\n\nw(cid:96)v(cid:96) (oi,(cid:96)) + \u03bbi\n\nso revA((cid:126)v) =(cid:80)n+1\n(cid:8)revA,j | (cid:0)w1, . . . , wn, \u03bb1, . . . , \u03bb(n+1)m\n(cid:1) , Hw \u2264 |wi| \u2264 Hw,|\u03bbi| \u2264 H\u03bb\nof all n-bidder, m-item AMA revenue functions, then RN (F) \u2264(cid:80)n+1\n\nj=1 revA,j((cid:126)v). Intuitively, for j \u2208 [n], revA,j is a weighted version of what the\nsocial welfare would be if Bidder j had not participated in the auction, whereas revA,n+1((cid:126)v) measures\nthe amount of revenue subtracted to ensure that the resulting auction is strategy-proof.\nAs to be expected, bounding the Rademacher complexity of each smaller class of functions Lj =\nthan bounding the Rademacher complexity the class of revenue functions itself, and if F is the set\nj=1 RN (Lj). In Lemma 2 and\nLemma 3 of Section B.1 in the supplementary materials, we obtain bounds on RN (Lj) for j \u2208 [n+1]\nwhich lead us to our bound on RN (F).\n\n(cid:9) for j \u2208 [n+1] is simpler\n\n3.2 The sample complexity of MBA revenue maximization\n\nFortunately, these negative sample complexity results are not the end of the story. We do achieve\npolynomial sample complexity upper bounds for the important classes of mixed bundling auctions\n(MBAs) and mixed bundling auctions with reserve prices (MBARPs). We derive these sample\ncomplexity bounds by analyzing the pseudo-dimensions of these classes of auctions. In this section,\nwe present our results in increasing complexity, beginning with the class of n-bidder, m-item MBAs,\nwhich we show has a pseudo-dimension of 2. We build on the proof of this result to show that the\n\nclass of n-bidder, m-item MBARPs has a pseudo-dimension of O(cid:0)m3 log n(cid:1).\n\nWe note that when we analyze the class of MBARPs, we assume additive reserve prices, rather than\nbundle reserve prices. In other words, each item has its own reserve price, and the reserve price of a\nbundle is the sum of its components\u2019 reserve prices, as opposed to each bundle having its own reserve\nprice. We have good reason to make this restriction; in Section C.1, we prove that an exponential\nnumber of samples are required to learn over the class of MBARPs with bundle reserve prices.\nBefore we prove our sample complexity results, we \ufb01x some notation. For any c-MBA, let revc ((cid:126)v)\nbe its revenue on (cid:126)v, which is determined in the exact same way as the general AMA revenue function\nwith the \u03bb terms set as described in Section 2. Similarly, let rev(cid:126)v(cid:96)(c) be the revenue of the c-MBA\non (cid:126)v(cid:96) as a function of c. We will use the following result regarding the structure of revc ((cid:126)v) in order\nto derive our pseudo-dimension results. The proof is in Section C of the supplementary materials.\n\n6\n\n\fFigure 2: Example of rev(cid:126)v(cid:96)(c).\n\nLemma 1. There exists c\u2217 \u2208 [0,\u221e) such that rev(cid:126)v(c) is non-decreasing on the interval [0, c\u2217] and\nnon-increasing on the interval (c\u2217,\u221e).\nThe form of rev(cid:126)v(c) as described in Lemma 1 is depicted in Figure 2. The full proof of the following\npseudo-dimension bound can be found in Section C of the supplementary materials.\nTheorem 3. The pseudo-dimension of the class of n-bidder, m-item MBAs is 2.\n\nProof sketch. First, we recall what we must show in order to prove that the pseudo-dimension of this\nclass is 2 (for more on pseudo-dimension, see, for example, [Mohri et al., 2012]). The proof structure\nis similar to those involved in VC dimension derivations. To begin with, we must provide a set of\nThis means that there exist two targets z1, z2 \u2208 R with the property that for any T \u2286 S, there exists\na cT \u2208 C such that if (cid:126)vi \u2208 T , then revcT\nwords, S can be labeled in every possible way by MBA revenue functions (whether or not revc\nis greater than its target zj). We must also prove that no set of three valuation vectors is shatterable.\n\ntwo valuation vectors S =(cid:8)(cid:126)v1, (cid:126)v2(cid:9) that can be shattered by the class of MBA revenue functions.\n(cid:0)(cid:126)vi(cid:1) > zi. In other\n(cid:0)(cid:126)vj(cid:1)\nOur construction of the set S =(cid:8)(cid:126)v1, (cid:126)v2(cid:9) that can be shattered by the set of MBAs can be found in\n\n(cid:0)(cid:126)vi(cid:1) \u2264 zi and if (cid:126)vi (cid:54)\u2208 T , then revcT\n\nthe full proof of this theorem in Section C of the supplementary materials. We now show that no set\nof size N \u2265 3 can be shattered by the class of MBAs. Fix one sample (cid:126)vi \u2208 S and consider rev(cid:126)vi (c).\ni \u2208 [0,\u221e), such that rev(cid:126)vi(c) is non-decreasing on\nFrom Lemma 1, we know that there exists c\u2217\ni ,\u221e). Therefore, there exist two thresholds\nthe interval [0, c\u2217\ni ] and non-increasing on the interval (c\u2217\ni \u2208 [0, c\u2217\ni \u2208 (c\u2217\ni ), above\ni ] and t2\nt1\nits threshold for c \u2208 (t1\ni ,\u221e). Now, merge these thresholds\nfor all N samples on the real line and consider the interval (t1, t2) between two adjacent thresholds.\nThe binary labeling of the samples in S on this interval is \ufb01xed. In other words, for any sample\n(cid:126)vj \u2208 S, rev(cid:126)vj (c) is either at least zj or strictly less than zj for all c \u2208 (t1, t2). There are at most\n2N + 1 intervals between adjacent thresholds, so at most 2N + 1 different binary labelings of S.\nSince we assumed S is shatterable, it must be that 2N \u2264 2N + 1, so N \u2264 2.\n\ni ,\u221e) \u222a {\u221e} such that rev(cid:126)vi(c) is below its threshold for c \u2208 [0, t1\ni , t2\n\ni ), and below its threshold for c \u2208 (t2\n\nThis result allows us to prove the following sample complexity guarantee.\nTheorem 4. The sample complexity of uniform convergence over the class of n-bidder, m-item MBAs\nis N = O\n\n(U/\u0001)2 (log(U/\u0001) + log(1/\u03b4))\n\n.\n\n(cid:17)\n\n(cid:16)\n\nMixed bundling auctions with reserve prices (MBARPs). MBARPs are a variation on MBAs,\nwith the addition of reserve prices. Reserve prices in the single-item case, as described in Section 2,\ncan be generalized to the multi-item case as follows. We enlarge the set of agents to include the seller,\nwho is now Bidder 0 and whose valuation for a set of items is the set\u2019s reserve price. Working in this\nexpanded set of agents, the bidder weights are all 1 and the \u03bb terms are the same as in the standard\nMBA setup. Importantly, the seller makes no payments, no matter her allocation. More formally, given\ni=0 vi (oi) + \u03bb ((cid:126)o) .\nFor each i \u2208 {1, . . . , n}, Bidder i\u2019s payment is\n\na vector of valuation functions (cid:126)v, the MBARP allocation is (cid:126)o\u2217 = argmax(cid:126)o\u2208O(cid:80)n\n(cid:0)o\u2217\n\nvj (o\u2212i,j) + \u03bb ((cid:126)o\u2212i) \u2212 (cid:88)\n\n(cid:1) \u2212 \u03bb ((cid:126)o\u2217) ,\n\n(cid:88)\n\npA,i((cid:126)v) =\n\nvj\n\nj\n\nj\u2208{0,...,n}\\{i}\n\nj\u2208{0,...,n}\\{i}\n\nwhere\n\n(cid:126)o\u2212i = argmax\n\n(cid:126)o\u2208O\n\nvj (oj) + \u03bb ((cid:126)o) .\n\n(cid:88)\n\nj\u2208{0,...,n}\\{i}\n\n7\n\n\fthe ith good.\n\nAs mentioned, we restrict our attention to item-speci\ufb01c reserve prices. In this case, the the reserve\nprice of a bundle is the sum of the reserve prices of the items in the bundle.\nEach MBARP is therefore parameterized by m + 1 values (c, r1, . . . , rm), where ri\nis the reserve price for\nFor a \ufb01xed valuation function vector (cid:126)v =\n(v1 (b1) , . . . , v1 (b2m) , . . . , vn (b1) , . . . , vn (b2m)), we can analyze the MBARP revenue function\non (cid:126)v as a mapping rev(cid:126)v : Rm+1 \u2192 R, where rev(cid:126)v (c, r1, . . . , rm) is the revenue of the MBARP\nparameterized by (c, r1, . . . , rm) on (cid:126)v.\nTheorem 5. The psuedo-dimension of the class of n-bidder, m-item MBARPs with item-speci\ufb01c\n\nreserve prices is O(cid:0)m3 log n(cid:1).\nProof sketch. Let S =(cid:8)(cid:126)v1, . . . , (cid:126)vN(cid:9) of size N be a set of n-bidder valuation function samples that\n\ncan be shattered by a set C of 2N MBARPs. This means that there exist N targets z1, . . . , zN such\nthat each MBARP in C induces a binary labeling of the samples (cid:126)vj in S (whether the revenue of the\nMBARP on (cid:126)vj is greater than or less than zj). Since S is shatterable, we can thus label S in every\npossible way using MBARPs in C.\nThis proof is similar to the proof of Theorem 3, where we split the real line into a set of intervals\nI such that for any I \u2208 I, the binary labeling of S by the c-MBA revenue function was \ufb01xed for\nall c \u2208 I. In the case of MBARPs, however, the domain is Rm+1, so we cannot split the domain\ninto intervals in the same way. Instead, we show that we can split the domain into cells such that\nthe binary labeling of S by the MBARP revenue function is a \ufb01xed linear function as we range over\n\nparameters in a single cell. In this way, we show that N = O(cid:0)m3 log n(cid:1).\n\nThis is enough to prove the following guarantee.\nTheorem 6. The sample complexity of uniform convergence over the class of n-bidder, m-item\nMBARPs with item-speci\ufb01c reserve prices is N = O\n.\n\n(U/\u0001)2(cid:0)m3 log n log (U/\u0001) + log (1/\u03b4)(cid:1)(cid:17)\n\n(cid:16)\n\n3.3 Sample complexity bounds for approximation algorithms\nIt may not always be computationally feasible to solve for the best auction over S for the given\nauction family. Rather, we may only be able to determine an auction A that has average revenue\nover S that is within a (1 + \u03b1) multiplicative factor of the revenue-maximizing auction over S\nwithin the family. Nonetheless, in Theorem 11 of the supplementary materials, we prove that with\nslightly more samples, we can ensure that the expected revenue of A is close to being with a (1 + \u03b1)\nmultiplicative factor of the expected revenue of the optimal auction within the family with respect to\nthe real distribution D. We prove a similar bound for an additive factor approximation as well.\n\n4 Conclusion\n\nIn this paper, we proved strong bounds on the sample complexity of uniform convergence for the\nwell-studied and standard auction families that constitute the hierarchy of deterministic combinatorial\nauctions. We thereby answered a crucial question in the study of (automated) mechanism design:\nhow to relate the performance of the mechanisms in the search space over the input samples to\ntheir expectation over the underlying\u2014unknown\u2014distribution. Speci\ufb01cally, for a \ufb01xed class of\nauctions, we determine the sample complexity necessary to ensure that with high probability, for\nany auction in that class, the average revenue over the sample is close to the expected revenue with\nrespect to the underlying, unknown distribution over bidders\u2019 valuations. Our bounds apply to any\nalgorithm that \ufb01nds an optimal or approximately optimal auction over an input sample, and therefore\nto any automated mechanism design algorithm. Moreover, our results and analyses are of interest\nfrom a learning theoretic perspective because the function classes which make up the hierarchy of\ndeterministic combinatorial auctions diverge signi\ufb01cantly from well-understood hypothesis classes\ntypically found in machine learning.\nAcknowledgments. This work was supported in part by NSF grants CCF-1535967, CCF-1451177,\nCCF-1422910, IIS-1618714, IIS-1617590, IIS-1320620, IIS-1546752, ARO award W911NF-16-\n1-0061, a Sloan Research Fellowship, a Microsoft Research Faculty Fellowship, an NSF Graduate\nResearch Fellowship, and a Microsoft Research Women\u2019s Fellowship.\n\n8\n\n\fReferences\nMaria-Florina Balcan, Avrim Blum, Jason Hartline, and Yishay Mansour. Reducing mechanism design to\n\nalgorithm design via machine learning. 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