Benefits of Permutation-Equivariance in Auction Mechanisms
Anonymous Author(s)
Abstract
Designing an incentive-compatible auction mechanism that maximizes the auctioneer’s revenue while minimizes the bidders’ ex-post regret is an importantyet intricate problem in economics.
Remarkable progresses have been achievedthrough learning the optimal auction mechanism by neural networks.
In this paper,we consider the popular additive valuation and symmetric valuation setting; i.e. ,the valuation for a set of items is defined as the sum of all items’ valuations in theset, and the valuation distribution is invariant when the bidders and/or the itemsare permutated.
We prove that permutation-equivariant neural networks have significant advantages: the permutation-equivariance decreases the expected ex-postregret, improves the model generalizability, while maintains the expected revenueinvariant.
This implies that the permutation-equivariance helps approach the theoretically optimal dominant strategy incentive compatible condition, and reducesthe required sample complexity for desired generalization.
Extensive experimentsfully support our theory.
To our best knowledge, this is the first work towardsunderstanding the benefits of permutation-equivariance in auction mechanisms.
Code will be released publicly.
Introduction
Optimal auction design [26] has wide applications in economics, including online advertising [16],resource allocation [15], and supply chain [4].
In an auction, every bidder has a private valuationprofile over all items, and accordingly, submits her bid profile.
An auctioneer collects the bids fromall bidders and determines a feasible item allocation to the bidders as well as the prices the biddersneed to pay.
Consequently, every bidder receives her utility.
From the auctioneer’s perspective, theoptimal auction mechanism is required to maximize her revenue , defined as the sum of all bidders’payments.
From the aspect of the bidders, the optimal auction mechanism needs to incentivize everybidder to bid according to their truthful valuation profiles (truthful bidding).
This is summarized asthe dominant strategy incentive compatible (DSIC) condition; i.e. , the truthful bidding is always thedominant strategy for every bidder [22].
The optimal auction mechanism can be approximated via neural networks [12, 25, 10].
The “approximation error”, or the “distance” to the DSIC condition, is usually measured by the ex-post regret ,defined as the gap between the bidder’s utility of truthful bidding and the utility when her bid profileis only the best to herself (selfish bidding), while the bid profiles of all other bidders are fixed in bothcases [12].
When a bidder’s ex-post regret is always equal to 0 , the truthful bidding is her dominantstrategy.
Therefore, the optimal auction design can be modeled as a linear programming problem,where the object is to maximize the expected revenue subject to the expected ex-post regret being
Another major consideration in learning the optimal mechanism is the generalizability tounseen data, usually measured by the generalization bound , i.e. , the upper bound of the gap betweenthe expected revenue/ex-post regret and their empirical counterparts on the training data [12].
In this paper, we consider the popular setting of additive valuation and symmetric valuation [12,25, 10].
The additive valuation condition defines the valuation for a set of items as the sum of thevaluations for all items in this set.
The symmetric valuation condition assumes the joint distributionof all bidders’ valuation profiles to be invariant when bidders and/or items are permutated.
Thissetting covers many applications in practice.
For example, when items are independent, the additivevaluation condition holds.
Moreover, if the auction is anonymous or the order of the items is notprior-known, the symmetric valuation condition holds.
We demonstrate that permutation-equivariant models have significant advantages in learning theoptimal auction mechanism as follows.
(1) We prove that the permutation-equivariance in auctionmechanisms decreases the expected ex-post regret while maintaining the expected revenue invariant.
Conversely, and equivalently, the permutation-equivariance promises a larger expected revenue,when the expected ex-post regret is fixed.
(2) We show that the permutation-equivariance of auctionmechanisms reduces the required sample complexity for a desirable generalizability.
We prove thatthe l ∞ , 1 -distance between any two mechanisms in the mechanism space decreases when they areprojected to the permutation-equivariant mechanism (sub-)space.
This smaller distance implies asmaller covering number of the permutation-equivariant mechanism space, which further leads to asmall generalization bound [12].
We further provide an explanation for the learning process of non-permutation-equivariant neuralnetworks (NPE-NNs).
In learning the optimal auction mechanism by an NPE-NN, we show thatan extra positive term exists in the quadratic penalty of the ex-post regret based on the result (1) .
This term serves as a regularizer to penalize the “non-permutation-equivariance”.
Moreover, thisregularizer also interferes the revenue maximization, and thus affects the learning performance of
NPE-NNs.
This further explains the advantages of permutation-equivariance in auction design.
Experiments in extensive auction settings are conducted to verify our theory.
We design permutationequivariant versions of RegretNet (RegretNet-test and RegretNet-PE) by projecting the RegretNet [12]to the permutation-equivariant mechanism space in the training and test stage.
The empirical resultsshow that permutation-equivariance helps: (1) significantly improve the revenue while maintain thesame ex-post regret; (2) record the same revenue with a significantly lower ex-post regret; and (3)narrow the generalization gaps between the training revenue/ex-post regret and their test counterparts.
These results fully support our theory.
Related works.
To our best knowledge, this is the first work on theoretically explaining the benefits ofpermutation equivariance in auction design via deep learning.
Myerson completely solves the optimalauction design problem in one-item auctions [20].
However, solutions are not clear when the numberof bidders/items exceeds one [12].
Initial attempts have been presented on the characterization ofoptimal auction mechanisms [18, 23, 9] and algorithmic solutions [1, 2, 24].
Remarkable advanceshave been made on the required sample complexity for learning the optimal auction mechanism invarious settings, including single-item auctions [5, 19, 15], multi-item single-bidder auctions [11],combinatorial auctions [3, 29], and allocation mechanisms [21].
Machine learning-based auctiondesign (automated auction design) have obtained considerable progresses [6, 7, 27].
The optimalauction design is modelled as a linear programming problem [6, 7].
However, early works suffer fromthe scalablility issues that the number of the constraints grows exponentically when the bidder numberand the item numbers increase.
To address this issue, recent works propose to learn the optimalauction mechanism by deep learning.
RegretNet is designed for the multi-bidder and multi-itemsettings [12].
Then, RegretNet is developed to meet more restrictive constraints, such as the budgetcondition [14] and the cerifying strategyproof condition [8].
Equivariant models EquivariantNet [25]and TransNet [10] are presented with better performance and better out-of-setting generalization inlearning the near-optimal auction mechanism.
Notations and Preliminaries
Auction.
Suppose n bidders are bidding m items in an auction.
Every bidder i has her biddercontext (feature) x
i ∈ X , while every item j is associated with its item-context (feature) y j ∈ Y .
The bidder i has a private valuation v ij ∈ V ⊂ R ≥ 0 for the item j , which is sampled froma conditioned distribution P ( ·| x i , y j ) .
The value profile v i =
( v i 1 , . . .
, v im ) is unknown to theauctioneer.
For the simplicity, we define x = ( x T 1 , . . .
, x Tn ) T , y =
( y 1 , . . .
y m ) , v = ( v T 1 , . . .
, v Tn ) T ,v − i =
( v T 1 , . . .
, v Ti − 1 , v Ti +1 , . . .
, v Tn ) T , and ( v ′ i , v − i )
= ( v T 1 , . . .
, ( v ′ i ) T . .
. , v Tn ) T .
Every bidder submits a bid profile b i to the auctioneer according to her valuation profile.
Then, theauctioneer determines a feasible item allocation g ( b, x, y ) and corresponding payments p ( b, x, y ) asper an auction mechanism ( g, p ) .
Consequently, every bidder receives her utility u i ( v i , b, x, y ) =
94 (cid:80) mj
=1 g ij ( b, x, y ) · v ij − p i ( b, x, y ) .
The auction mechanism ( g, p ) consists of an allocation ruleg : R n × m × X n × Y m → R n × m and a payment rule p : R n × m × X n × Y m → R n × m , whereg ij is the probability of allocating item j to the bidder i , and p i =
(cid:80) mj
=1 p ij is the price thatthe bidder i should pay.
To avoid allocating an item over once, the allocation rule is constrainedsuch that (cid:80) ni
=
g ij ( b, x, y ) ≤ 1 for all j ∈ [ m ] .
Every v i in our notations can be replaced byb i .
Thus, we can define the similar notations: b − i =
( b T 1 , . . .
, b Ti − 1 , b Ti +1 , . . .
, b Tn ) T , ( b i , v − i ) =( v T 1 , . . .
, ( b i ) T . .
. , v Tn ) T and ( v i , b − i ) =
( b T 1 , . . .
, ( v i ) T . . .
, b Tn ) T .
Optimal auction mechanism.
An auction mechanism ( g, p ) is defined to be dominant strategyincentive compatible (DSIC), if truthful bidding is always a dominant strategy of every bidder;
i.e. ,u i ( v i , ( v i , b − i ) , x, y ) ≥ u i ( v i , b, x, y ) for all i ∈ [ n ] , v, b ∈ V n × m , x ∈ X n and y ∈ Y m .
Inaddition, an auction mechanism ( g, p ) is called individually rational (IR), if for any bidder-contextsx ∈ X n , any item-contexts
y ∈
Y m , and any bidder i ∈ [ n ] x ∈ X n , valuation profile and bid profilev, b ∈ V n × m , truthful bidding always leads to a non-negative utility; i.e. , u i ( v i , ( v i , b − i ) , x, y ) ≥ 0 .
If an auction mechanism is DSIC and IR, a rational bidder with an obvious dominant strategy willplay it (bidding truthfully).
Moreover, an optimal auction mechanism is required to maximize theauctioneer’s expected revenue rev = E ( v,x,y ) (cid:2) (cid:80)
ni
=1 p i ( v, x, y ) (cid:3) .
Auction design.
The ex-post regret reg i ( v, x, y ) for the bidder i is defined as
[Equation]
− u i ( v i , v, x, y ) .
An auction mechanism ( g, p ) is DSIC,
if andonly if (cid:80) ni
=1 reg i ( v, x, y ) =
for any value profile v ∈ V n × m , bidder-context x ∈ X n , and itemcontext
y ∈
Y m .
Suppose the payment rule p satisfy p
i ( b, x, y ) ≤ (cid:80) mj
=1 g ij ( b, x, y ) b ij .
Then, theauction design can be modeled as a linear programming problem that maximizes the expected revenue 115 E ( v,x,y )
[ (cid:80) ni =1 p i ( v, x, y )] subject to the expected ex-post regret E ( v,x,y )
[ (cid:80) ni
=1 reg i ( v, x, y )]
= 0 .
Without loss of generality, the ex-post regret may refer to the sum of all bidders’ ex-post regrets.
Definition 2.1.
Suppose the network’s parameter is ω , and
the bidder
i ’s empirical revenue and expost regret are defined as 1 L (cid:80)
L l =1 p ωi ( v ( l ) , x ( l ) , y ( l ) ) , and (cid:100) reg i ( ω )
=
1 L (cid:80)
L l =1 reg ωi ( v ( l ) , x ( l ) , y ( l ) ) ,where the sample set { ( v ( l ) , x ( l ) , y ( l ) ) }
Ll =1 is i.i.d.
sampled from the following prior distribution,
P ( v, x, y ) =
(cid:81) n,mi,j =1 P ( v ij | x i , y j )
P X
i ( x i ) P Y j ( y j ) .
Equivariant mapping.
We define a mapping f as G - equivariant if ψ g ◦ f
= f ◦ ρ g for two chosengroup linear representations ρ and ψ and any g in group G .
Definition 2.2.
(Permutaion-Equivariant Mapping)
A permutation-equivariant mapping is defined tobe f : R n × m → R n × m that for any instance x ∈ R n × m , and permutation matrices σ n ∈ R n × n andσ m ∈ R m × m , we have f ( σ n xσ m )
= σ n f ( x ) σ m .
In this paper, we consider the bidder-permutation σ n ∈ R n × n and item-permutation σ m ∈
R m × m .
Specifically, we define a mapping f is bidder-symmetric or item-symmetric, if f ( σ n x ) = σ n f ( x )or f ( xσ m ) =
f ( x ) σ m , respectively.
Moreover, we define an auction mechanism ( g, p ) as biddersymmetric or item-symmetric, if the allocation rule g and the payment rule p are both biddersymmetric or item-symmetric.
Orbit averaging.
For any feature mapping f : F → G , the orbit averaging Q on f is defined as
Q f = 1 | G | (cid:80) g ∈ G ψ − 1 g ◦ f ◦ ρ g , where ρ and ψ are two chosen group representations acting on thefeature spaces F and G , respectively.
Orbit averaging can project any mapping to be equivariant:
Proposition 2.3.
Orbit averaging Q is a projection to the equivariant mapping space { f : ψ ◦ f =f ◦ ρ } , i.e., ψ ◦ Q f
=
Q
f ◦ ρ and Q 2 = Q .
In particular, if f is already equivariant, then Q f = f .
Moreover, Q u and Q reg refer to the utility and the ex-post regret induced by Q g and Q p .
Forthe simplicity, we denote the orbit averagings that modify the auction mechanism to be biddersymmetric, item-symmetric, and bidder/item-symmetric by bidder averaging Q 1 , item averaging Q 2 ,and bidder-item aggregated averaging Q 3 .
The proof of the feasibility can be found in Appendix A.1.
Definition 2.4.
Let X be a feature space and F a space of some functions from X to R n .
Thel ∞ , 1 -distance on the space F is defined as l ∞ , 1 ( f, g ) = max x ∈X ( (cid:80) ni =1 | f i ( x ) −
g i ( x ) | ) .
Definition 2.5.
Covering number N ∞ , 1 ( F , r ) is the minimum number of balls with radius r that cancover F under l ∞ , 1 -distance.
Theoretical Results
This section presents the theoretical results.
For simplicity, we view p
=
( p 1 , . . .
, p n ) T as a n × 1matrix to present the prices the bidders should pay.
We first prove that the permutation-equivarianceinduces a same expected revenue and a smaller expected ex-post regret in Section 3.1.
Next in Section3.2, we prove that the permutation-equivariant mechanism space has a smaller covering number,which promises a smaller required sample complexity and a better generalization.
Detailed proofs areomitted from the main text and given in supplementary materials due to space limitation.
3.1 Benefits for Revenue and Ex-Post Regret
In this section, we discuss the benefits for the revenue and the ex-post regret in the conditions ofbidder-symmetry and item-symmetry separately, and then discuss the benefits when both of themhold.
Based on these results, we also study the learning process of non-permutation-equivariantneural networks for auction design.
3.1.1 Benefits in the Bidder/Item-Symmetry Condition
When the bidders come from the same distribution, the joint valuation distribution f is invariantunder bidder-permutation, i.e. f ( σ n v, σ n x, y ) =
f ( v, x, y ) for any σ n ∈ S n .
Meanwhile, whenthe items are indistinguishable, the joint distribution f is invariant under item-permutation, i.e. ,f ( vσ m , x, yσ m )
=
f ( v, x, y ) for any σ m ∈ S m .
Both conditions do not always hold simultaneously.
In this section, we study them separately.
To measure the “non-permutation-equivariance” of the mechanism, we introduce the conception ofregret gap between the projected mechanism and the original mechanism for as below,
[Equation]
where v is the valuation profiles, v i is valuation profile of bidder i , x is the bidder-context, y is theitem-context, and the orbit averaging Q · can be the bidder averaging Q 1 or the item averaging Q 2 .
The bidder averaging Q 1 and the item averaging Q 2 acting on the allocation rule g and the paymentrule p , respectively, are as below,
[Equation]
Q [Equation] [Equation] , x, y
[Equation] [Equation] x [Equation]
yσ
[Equation]
Q [Equation] p ( v, x, y [Equation]
[Equation]
(cid:88)
[Equation]
m
[Equation]
S
[Equation]
p [Equation] vσ m
[Equation]
x [Equation] yσ m
[Equation]
We thus can prove the following theorem that characterizes the benefits of permutation-equivariancefor revenue and ex-post regret in the condition of bidder/item-symmetry.
Theorem 3.1 (Benefits for revenue and ex-post regret in the condition of bidder/item-symmetry) .
When the valuation distribution is invariant under permutations of bidders/items, the projectedmechanism has the same expected revenue and a smaller expected ex-post regret, that is,
[Equation]
v,x,y
[Equation]
, and (1)
[Equation]
where p is the payment rule, reg is the ex-post regret, and Q · is the bidder/item averaging.
Proof Sketch.
The proof of Theorem 3.1 have the following three steps.
The first step is to provethe equation 1, and the second and third steps are to prove the inequality 2.
We consider the twoconditions at the same time and use Q to replace Q 1 and Q 2 .
Step 1: We prove that the permutation-equivariance maintains the expected revenue, i.e. ,
[Equation]
x, y ) (cid:21) .
By the definition of Q , the left-hand side equals ( n !)
− 1 (cid:80) ni =1 (cid:80) σ n ∈ S n E
[ p i ( σ n v, σ n x, y )] , or( m !)
− 1 (cid:80) ni
=1 (cid:80) σ m ∈ S
m E
[ p i ( vσ m , x, yσ m )] .
Notice that the distribution of ( v, x, y ) is invariantunder bidder/item-permutation, and therefore for each σ n and σ m , we have
[Equation]
=
E [ p
[Equation]
x,
[Equation]
[Equation]
The proof is completed.
Step 2: We prove that permutation-equivariance maintains the sum of all bidders’ utilities, i.e. ,
[Equation]
The proof can be similar with that in Step 1 if we define ˜ u as ˜
u
i ( v, x, y ) =
u i ( v i , v, x, y ) .
One cancheck that [ Q u ] i ( v i , v, x, y ) =
(cid:80) mj
=
[ Q g ] ij ( v, x, y ) v ij − [ Q p ] i ( v, x, y ) =
[ Q ˜ u ] i ( v, x, y ) immediately.
Then using ˜ u
i to replace p i , we can get the result.
Step 3: We prove that permutation-equivariance decreases the utility of the besting bidding:
[Equation]
We use another definition u as u i ( v, x, y ) =
max v ′
i u
i ( v i , ( v ′ i , v − i ) , x, y ) for the proof easier tounderstand.
Then the right-hand side becomes E [ (cid:80) ni
=1 u i ( v, x, y )] .
For bidder averaging Q 1 , the left-hand side becomes
[Equation]
The latter equals ( n !)
− 1 (cid:80) σ n ∈ S n E [ (cid:80) ni
=1 u i ( σ n v, σ n x, y )] .
According to the same process in Step 1,we know for each σ n , E [ (cid:80) ni =1 u
i ( σ n v, σ n x, y )]
=
E [ (cid:80) ni =1 u i ( v, x, y )] .
The proof process is similarfor Q 2 .
Combining the results in Step 2 and in Step 3, we obtain the inequality 2.
A smaller expected ex-post regret implies this mechanism is closer to the dominant strategy incentivecompatible condition.
Conversely, and equivalently, when the expected ex-post regrets are fixed,the projected auction mechanism has a larger expected revenue.
For any auction mechanism, in thebidder/item-symmetry condition, we can project it through the bidder/item averaging.
Remark 3.2.
The mechanism space can be decompose into the direct sum of the permutationequivariant mechanism space {M : QM =
M} and the complementary space {N : QN = 0 }[13].
Thus, an mechanism M has an unique decomposition: M = QM + N .
The pure permutationequivariant part QM contains all and only the “permutation-equivariance” of the mechanism M .
The pure non-permutation-equivariant part N is independent from the permutation-equivairance.
Inthis way, we may study the influence of permutation-equivariance by comparing the mechanism Mand its permutation equivariant part QM .
3.1.2 Interplay between Bidder-Symmetry and Item-Symmetry.
If the valuation distribution is invariant under both bidder-permutation and item-permutation, wecan project the mechanism to be permutation-equivariant with respect to both bidder and item intwo steps (by mapping Q 1 ◦ Q 2 or mapping Q 2 ◦ Q 1 ).
Consequently, the projected mechanism hasthe same expected revenue and a smaller expected ex-post regret.
Equivalently, we can also projectan auction mechanism to be bidder-symmetric and item-symmetric immediately by the bidder-itemaggregated averaging Q 3 as below,
[Equation]
and
Q
[Equation]
We can prove that the bidder-item aggregated averaging Q 3 is the composition of the orbit averagingoperators Q 1 and Q 2 , as shown in the following lemma.
This lemma shows that the order of Q 1 and
Q 2 would not influence their composition.
Lemma 3.3.
The bidder-item aggregated averaging is the composition of bidder averaging and itemaveraging: Q 3 = Q 1 ◦ Q 2 =
Q 2 ◦ Q 1 .
Based on this lemma, we can prove the following theorem on the benefits of permutation-equivariancefor revenue and ex-post regret in the condition of both bidder-symmetry and item-symmetry.
Theorem 3.4 (Benefits for revenue and ex-post regret in the condition of both bidder-symmetryand item-symmetry) .
When the valuation distribution is invariant under both item-permutation anditem-permutation, then the projected mechanism has a same expected revenue and a smaller expectedex-post regret, that is,
[Equation]
i = p
i ( v, x, y ) (cid:21) and
[Equation]
i
[Equation]
where p is the payment rule, reg is the ex-post regret, and Q 3 is the bidder-item aggregated averaging.
The difference between bidder-symmetry and item-symmetry is significant in practice.
For example,for a symmetric valuation distribution, when the mechanism is already bidder-symmetric but notitem-symmetric, we can project it to be item-symmetric to gain an extra benefit from item-symmetry.
That means, the two regret gaps induced by Q 1 and Q 2 are “additive” as below,
[Equation]
In general, E [∆ 2 ( g, p ; v, x, y )] ̸ = E [∆ 2 ( Q 1 g, Q 1 p ; v, x, y )] and thus E [∆ 3 ( g, p ; v, x, y )]
̸ = 229 E [∆ 1 ( g, p ; v, x, y )]
+ E [∆ 2 ( g, p ; v, x, y )] .
That means, the benefits from bidder-symmetry and itemsymmetry are “additive” but not “independent”.
3.1.3 Insights on Training Non-Permutation-Equivariant Mechanism
Because the expected revenue is always same for the original mechanism and the projectedpermutation-equivariant mechanism, we only consider the gredient caused by the expected expost regret.
We can decompose the original expected ex-post regret into the sum of the expectedex-post regret of the projected mechanism and the expectation of the regret gap as below,
[Equation]
x
[Equation]
y
[Equation]
The regret gap ∆ 3 ( g, p ; · ) follows from the “non-permutation-equivariance” of the mechanism M .
When the distance l ( M , QM ) tends to 0 , the regret gap converges to 0 .
When the auction mechanismhas a negligible ex-post regret, the expectation of the regret gap is also close to 0 .
That means, themechanism is closing to being permutation-equivariant.
However, even using symmetric dataset oradopting data augmentation in training, the learned mechanism will not be permutation-equivariantin general [17].
As a result, to achieve negligible ex-post regret, the non-permutation-equivariantmodels need to learn more samples to approach permutation-equivariance.
That is because thenon-permutation-equivariant part (expected regret gap) would mislead the gradient of the expectedregret but have no benefit to the expected revenue and the ex-post expected regret.
On the other hand, the regret gap can be viewed as a regularizer in the ex-post regret to penalizethe “non-permutation-equivariance” of the mechanism.
When the optimizer tries to minimize theex-post regret, the auction mechanism approaches to be permutation-equivariant.
Therefore, if themechanism achieves a negligible ex-post regret, it is almost to be permutationn-equivariant.
Thisresult can explain why RegretNet struggles to find permutation-equivariant auction mechanisms[25].
However, in complex settings, it will be harder for non-permutation-equivariant models toapproach the negligible ex-post regret.
It can explain why the permutation-equivariant models show asignificant improvement in complex setting, compared with that they have similar performances insimple settings [10], which shows the great importance of adopting permutation-equivariant modelsin complex settings.
3.2 Benefits for Generalization
In this section, we studies permutation-equivariance from the aspect of generalizability [28], whichcharacterizes the performance gap of a learned mechanism on collected training data and unseen data.
We first study the covering number of the permutation-equivariant mechanism space.
Let U ={ u ω : ω ∈ Ω } and P = { p ω : ω ∈ Ω } be the sets of all possible utilities and payment rules, and
Q · U = {Q · u : u ∈ U} and Q · P = {Q · p : p ∈ P} the spaces of all projected utilities and paymentrules.
In addition, let N ∞ , 1 ( U , r ) and N ∞ , 1 ( P , r ) be the minimum numbers of balls with radiusr that can cover U and P under l ∞ , 1 -distance, respectively.
We obtain the following result, whichindicates the projected permutation-equivariant mechanism space has smaller covering numbers.
Theorem 3.5 (Covering number of the permutation-equivariant mechanism space) .
The space of allprojected bidder-symmetric mechanisms has smaller covering numbers, that is,
[Equation]
r
[Equation]
The space of all projected item-symmetric mechanisms has smaller covering numbers, that is,
N ∞ , 1 ( Q 2 U , r ) ≤ N ∞ , 1 ( U , r ) and N ∞ , 1 ( Q 2 P , r ) ≤ N ∞ , 1 ( P , r ) .
Intuitively, the orbit averaging Q narrows the distance between two mechanisms: l ( QM , QM ′ ) ≤l ( M , M ′ ) , for any two mechanisms.
Then, any r -cover r > 0 A for space U or space P induces anr -cover QA for space QU or space QP .
Combining with Lemma 3.3, we have the following results,
[Equation]
We then prove that two generalization bounds of permutation-equivariant mechanisms which characterize the gap between the expected revenue/ex-post regret and their empirical counterparts.
Theorem 3.6 (Generalization bounds of permutation-equivariant mechanisms) .
If for any bidder, hervaluation satisfies that v i ( S ) ≤ 1 for any S ⊂ [ m ] , then with probability at least 1 − δ , we have thefollowing inequalities with ϵ ≥ (cid:113) 9 n 2 2 L (log 4 δ + max { log N ∞ , 1 ( P , ϵ 3 ) , log N ∞ , 1 ( U , ϵ 6 ) } ) ,
[Equation]
where L is the number of samples, U and P are the spaces of all possible utilities and payment rules.
Equivalently, we can rewrite this result in the form of the sample complexity,
Corollary 3.7.
For any ϵ > 0 , δ ∈ (0 , 1) , and mechanism parameter ω , when the sample complexity
L
[Equation]
1 ( U , ϵ 6 ) (cid:111)(cid:17) , with probability at least 1 − δ , thegeneralization bounds, eqs.
(3) and (4), hold.
Remark 3.8.
Combining Theorem 3.5, we have proved that the permutation-equivariance canimprove the generalizability.
Experiments
This section presents our experimental results.
More details and results are presented in the supplementary materails.
Code will be released publicly.
Model architecture.
We project RegretNet [12] to the permutation-equivariant mechanism spacevia orbit averaging: (1) using bidder averaging for the bidder-symmetry condition, (2) applying itemaveraging for the item-symmetry condition, and (3) employing bidder-item aggregated averaging forthe bidder-symmetry and item-symmetry condition.
The projected model is called RegretNet-PE.
Specifically, RegretNet is an auction mechanism defined as ( g ω , p ω ) , in which both of the allocationrule g ω and the payment rule p ω are neural networks that consist of three fully-connected layers, andω is the overall model parameter of the auction mechanism.
The detailed architecture is given in thesupplementary materials.
Auction settings.
We adopt two families of auction settings in our experiments: single-item auctionsand multi-item auctions.
The optimal auction mechanism for any single-item auction is known[20].
We thus compare our method with the optimal auction mechanisms in the single-item settings.
Besides, we employ a multivariate uniform distribution U [0 , 1] m to model the bidder valuationprofiles.
We sample 5,000 data points for training and 100 data batches of size 100 (totally 10,000data points) for test in every setting.
Model training.
We optimize the auction mechanism model via solving the following optimizationproblem, following the standard settings [12, 25, 10],
[Equation]
where λ ∈ R n is the Lagrange multiplier and ρ > 0 is the factor of the quadratic regularization term.
During the training process, the objective function L ρ ( ω, λ ) is minimized via Adam with learningrate 0.001 with respect to the model parameter ω and the Largrange multiplier λ is updated via SGDonce in every 100 iterations, until the ex-post regret is smaller than 0 .
The regularization factor ρis is set to 1000 and 5 initially for single-item settings and multiple and gradually increased along thetraining process.
In calculating the best bid profile v ′ i of every bidder
i , we first randomly initialize
[Table]
the bid profiles for 1,000 times, optimize each of them individually via Adam with the same settings,and take the best one as the approximated best bidding.
Evaluation.
We leverage three metrics to evaluate the performance of the auction mechanism, whichare: (1) the empirical revenue (cid:100) rev , (2) the empirical ex-post regret across all bidders, i.e. , (cid:80) ni =1 (cid:100) reg i ,and (3) the generalization error defined on top of regrets, i.e. , GE = | (cid:100) reg test − (cid:100) reg train | .
Computing resource.
The experiments are conducted on 1 GPU (NVIDIA ® Tesla ® V100 16GB)and 10 CPU cores (Intel ® Xeon ® Processor E5-2650 v4 @
2.20GHz).
Experimental results.
We train a RegretNet and a RegretNet-PE on the training data.
The welltrained RegretNet is then projected to be permutation-equivariant, denoted as “RegretNet-Test”.
Theresults are collected in Table 1.
From the table, we observe that (1) RegretNet-PE has a significanthigher revenue and a lower ex-post regret in the multi-item settings; (2) RegretNet-Test receivesthe same revenue with a significantly lower ex-post regret; and (3) RegretNet-PE narrows thegeneralization gap between the training revenue/ex-post regret and their test counterparts.
Theseresults fully supports our theoretical findings.
Conclusion and future works
In this paper, under additive valuation and symmetric valuation setting, we study the benefits ofpermutation-equivariance in auction mechanisms in two aspects: a better performance and a bettergeneralization.
First, we prove a smaller expected ex-post regret and a same expected revenue whenprojecting an mechanism to be permutation-equivariant.
Next, we propose the permutation-equivariantmechanism space has a smaller covering number, which promises the permutation-equivariant modelsa better generalization.
Extensive experiments are conducted to verify our theoretical results.
Ourresults help understand the optimal auction mechanisms’ characterization and the learning processes’difference between non-equivariant models and equivaraint models.
Beyond the additive valuation setting, an interesting direction is to extend our results to otherconditions, including the combinatorial and the unit-demand auctions.
Meanwhile, the understandingof the difference in the aspect of the training process between non-equivariant models and equivariantmodels is still elusive.
Social impact.
Our results can help understand and design optimal auction mechanisms for symmetricvaluation distribution.
As a result, our work could inspire more near-optimal auction mechanismsand promote economic growth.
No potential negative social impact is identified.
Checklist
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’scontributions and scope?
[Yes] Our contributions are stated in the 4th-7th sentences ofthe abstract, and the 4th-7th paragraphs of the introduction.
The scope is described inthe related work section.(b)
Did you describe the limitations of your work?
[Yes]
We describes the problem setting
we considered in the 3rd paragraph of the introduction.
Our contributions are for this
problem setting.
(c) Did you discuss any potential negative societal impacts of your work?
[Yes] Our work
is committed to learn the optimal auction mechanism.
No negative societal impact is
identified.
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
[Yes] We have read the ethics review guidelines.
We ensure that our paper
conforms all of them.
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes]
We stateall of our assumptions in the third paragraph of introduction, the first paragraph of preliminary, Theorem 3.1, and Theorem 3.2.
Wherever we use an assumption, we
always state it again.
(b) Did you include complete proofs of all theoretical results?
[Yes] We give a proof
sketch in Section 3 due to space limitation.
Full proofs are given in the supplemental material.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or as a URL)?
[No] We promise to
release our code publicly.
We are associated in a industrial organization.
The code is
currently under review of our organization.
(b) Did you specify all the training details (e
g., data splits, hyperparameters,
how they
were chosen)?
[Yes] We mostly follow the setting of the RegretNet.
Other details aregiven in Section 4.
(c) Did you report error bars (e
g., with respect to the random seed after running experi- ments multiple times)?
[No]
(d) Did you include the total amount of compute and the type of resources used (e.g., typeof GPUs, internal cluster, or cloud provider)?
[Yes] They are given in Section 4.
If you are using existing assets (e
g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[Yes] They are given in
Section 4.
(b) Did you mention the license of the assets?
[Yes] They are given in Section 4.
(c) Did you include any new assets either in the supplemental material or as a URL?
[No]
We promise to release our code publicly.
We are associated in a industrial organization.
The code is currently under review of our organization.
(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating?
[Yes] The data is synthetic in our data.
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[Yes] The data is synthetic in our data.
No personally
identifiable information or offensive content is contained.
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, if
applicable?
[N/A](b)
Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A](c)
Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation?
[N/A]