Benefits of Permutation-Equivariance in Auction Mechanisms
Authors: Tian Qin
Fengxiang He
Dingfeng Shi
Wenbing Huang
Dacheng Tao
Venue: NeurIPS 2022 Accept (2022)
Keywords: Permutation-Equivariance
Auction Mechanisms

Abstract: Designing an incentive-compatible auction mechanism that maximizes the auctioneer's revenue while minimizes the bidders’ ex-post regret is an important yet intricate problem in economics. Remarkable progress has been achieved through 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 the set, and the valuation distribution is invariant when the bidders and/or the items are permutated. We prove that permutation-equivariant neural networks have significant advantages: the permutation-equivariance decreases the expected ex-post regret, improves the model generalizability, while maintains the expected revenue invariant. This implies that the permutation-equivariance helps approach the theoretically optimal dominant strategy incentive compatible condition, and reduces the required sample complexity for desired generalization. Extensive experiments fully support our theory. To our best knowledge, this is the first work towards understanding the benefits of permutation-equivariance in auction mechanisms. 

Benefits of Permutation-Eqqiivarinne in Auction Mechanisms
Tian Qin Fengxiang H Dingfeng Shi²
Wenbing Huang Dacheng Tao?
University of Science and Technology of China 2ID Explore Academy, JD.com Inc.
GGoling School of Artificial Intelligence, Renmin University of China *Beijing Key Laboratory of Big Data Management and Analysis Methods tqin@mail... ustc. edu. cn, fengxiang. f .he@gmail. com, shidingfengobuaa. edu, en, hwenbing0126. com, dacheng. taoßgmail.com
Abstract
Designing an incentive-compatiiy auction mechanism that maximizes the tioneer's revenue while minimizes the bidders' ex-post regret an important yet intricate problem in economics. Remarkable progress has been achieved through 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 set of items is defined as the sum of all items' valuations in the set, and the valuation distribution is invariant when the bidders and/or the items are permutated. We prove that permutation-eqqivaiinn neural networks have sig- nificant advantages: the permutation-qquivariaane regret, improves the model generalizability. while maintains the expected revenue invariant. This implies that the eermutatio-equiiariinn helps approach the theo- retically optimal dominant strategy incentive compatible condition, and reduces the required sample complexity for desired generalization. Extensive experiments fully support our theory. To our best knowledge, this is the first work towards understanding the benefits of permuuation-qquiiariacce in auction mechanisms.
equivariance decreases the expected ex-por ex-post
1 Introduction
Optimal auction design 30 has wide applications in economics, including computational advertising [18], resource allocation (16). and supply chain [4. In an auction, every bidder has a private valuation profile over all items, and accordingly, submits her bid profile. An auctioneer collects the bids from all bidders, and determines feasible item allocation the bidders well as the prices the bidders need to pay. Consequently, every bidder receives her utility. From the auctioneer's perspective, the optimal auction mechanism is required to maximize her revenue, defined the sum of all bidders' payments. From the aspect of the bidders, the optimal auction mechanism needs to incentivize every bidder to bid their truthful valuation profiles (truthful bidding). This is summarized the dominant strategy incentive compatible (DSIC) condition; i.e., truthful bidding always the dominant strategy for every bidder [25].
The optimal auction mechanism can be approximated via neural networks 12 10]. The "approx- imation error", or the "distance" to the DSIC condition, is usually measured by the ex-post regret, defined as the gap between the bidder's 's utility of truthful bidding and the utility when her bid profile is only the best to herself (selfish bidding). while the bid profiles of all other bidders fixed in both cases 12] When a bidder's 's ex ex-post regret is O, truthful bidding is her dominant strategy. Therefore, the optimal auction design be modeled maximize the expected revenue subject to the expected ex-post regret being 0 for all bidders [1]
linear programming problem, where the object is
* Both authors contributed equally.
36th Conference on Neural Information Processing Systems (NcurIPS 2022).

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Another major consideration in learning the optimal mechanism is the generalizability to unseen data, usually measured by the generalization bound, i.e., the upper bound of the gap between the expected revenue/ex-pone 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 29 [0]. The additive valuation condition defines the valuation for set of items as the sum of the valuations for all items in this set. The symmetric valuation condition assumes the joint distribution of all bidders' valuation profiles to be invariant when bidders and/or items are permutated. This setting covers many applications in practice. For example, when items are independent, the additive valuation condition holds. Moreover, if the auction is anonymous or the order of the items is not prior- known, the symmetric valuation condition holds.
We demonstrate that permutation-equivariaa models have significant advantages in learning the optimal auction mechanism as follows. (1) We prove that the permutation-equivarianee in auction mechanisms decreases the expected ex-post regret while maintaining the expected revenue invariant. Conversely, and equivalently, the permutation-cqiivaraane promises larger expected revenue, when the expected ex-post regret is fixed. (2) We show that the permutation-cquiiariaane of auction mechanisms reduces the required sample complexity for desirable generalizability. We prove that the /o0. -distance between any two mechanisms in the mechanism space decreases when they are projected to the permutation -equivarian mechanism suba)sspace. This smaller distance implies a smaller covering number of the permutation-equivariann mechanism space, which further leads to a small generalization bound [12.
We further provide an explanation for the learning process of networks (NPE-NNs). In learning the optimal auction mechanism by NPE-NN, we show that an extra positive term exists in the quadratic penalty of the ex- ex-post regret based on the result (1), This term serves as regularizer penalize the permuuation--qquivrraancc regularizer also interferes the revenue maximization, and thus affects the learning performance of NPE-NNs. This further explains the advantages of permutation-equivariacce in auction design.
peermutation-equivariie neural
Moreover, this
Experiments in extensive auction settings are conducted to verify our theory. We design permutation- equivariant versions of RegretNet (RegretNet-PE and RegretNet-test) by projecting the RegretNet mechanism space in the training and test stage respectively. The
112 to the permutation-equivariin empirical results show that permuration-eqquiariie while maintain the same ex- ex-post regret: (2) record the same revenue with a significantly lower ex-post regret; and (3) narrow the generalization gaps between the training ex ex-post regret and its test counterpart. These results fully support our theory.
helps: (1) significantly improve the revenue
Related works. Myerson completely solves the optimal auction design problem in one-item auctions 23 However, solutions are not clear when the number of bidders/items exceeds one [12. Initial
attempts have been presented on the characterization of optimal auction mechanisms [20 26 9
and algorithmic solutions [ 2 27]. Remarkable advances have been made in the required sample complexity for learning the optimal auction mechanism in various settings, including single-item
auctions 5 21 16), multi-item single-bidder auctions 11] combinatorial auctions B 32), and
allocation mechanisms 24). Machine learning-bact based auction design (automated auction design) have obtained considerable progress [6 31]. The optimal auction design is modeled as a linear programming problem 6 77 However, early works suffer from the scalability issues that the number of the constraints grows exponentially when the bidder number and the item numbers increase. To address this issue, recent works propose to learn the optimal auction mechanism by deep learning. RegretNet is designed for multi-bidder and multi-item settings [12.. Then, RegretNet is developed to meet more restrictive constraints, such as the budget condition [4] and the certifying strategyproof condition [8]. Rahme et al. [29] propose the first equivariant neural network- based auction mechanism design method with significant empirical advantages. Ivanov et al. [17 propose a RegretFormer which (1) introduces attention layers to RegretNet to learn permutation auction mechanisms, and (2) adopts a new interpretable loss function to control the revenue-regret trade-off. Duan et al. [10] extend the applicable domain to contextual settings. All these works make remarkable contributions in designing new algorithms from the empirical aspect only. However, the theoretical foundations are still elusive. To our best knowledge, our paper is the first work on theoretically studying the benefits of permutation equivariance in auction design via deep learning.
2

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2 Notations and Preliminaries
Auction. Suppose n bidders are bidding m items in an auction. Every bidder has her bidder- context (feature) E X. while every item j is associated with its item- (feature) yj E V. The bidder i has a private valuation Big
C Ryo for the item j. which sampled from a conditioned distribution P(-[21, Uj). The value profile
(Un. Vim) is unknown to the (y1 Fn 1 v= (u)
auctioneer. For the simplicity, we define = (xi....,, xT) v-; (of ....,... - 02 1'41...., "T,", and (0/, v-:) (of (.))
Every bidder submits a bid profile b; to the auctioneer according to her valuation profile. Then, the auctioneer determines feasible item allocation 9b, x. y) and corresponding payments p(b, xr, y) as per an auction mechanism (g. p). Consequently, every bidder receives her utility as
m
u. (, b, x.
giy (g,b,, vij - p: x. x)
The auction mechanism (g. p) consists of an allocation rule R x X" x 3m + R
and a payment rule p Ruxm x X x ym + I_= where gij is the probability of allocating item j to the bidder i, and p;
Dj=1 Pij is the price that the bidder should pay. To avoid allocating an item over once, the allocation rule constrained such that E (b, 2, S 1 for all j E [m]. Every v; in our notations can be replaced by b;. Thus, we can define the similar notations: b-, = (b7 .........
67 ...,....., bT)T, (b,, E-1) (oT,...,,b,)) (b.)...............
.0."....0 2 and
(v,, b_:) = (6π (w)) ..,,) 6T)
Optimal auction mechanism. An auction mechanism (g.p) is defined to be dominant strategy incentive compatible (DSIC), if truthful bidding is always dominant strategy of every bidder; i.e.,
(1,, (2t, b-s), x,,) 2 u,(u,, 6, x., v),
for all i E In], v. E vuxm x E X and E y" In addition, an auction mechanism (g. p) is called individually rational (IR), if for any bidder-contexts E X", any item-contexts y E"" any bidder i E [n] æ E X", valuation profile and bid profile v, b E ymkm, truthful bidding always leads to non-negative utility, i.e.,
u,(0;, (15, b-i),,,) I. 2 0.
If an auction mechanism DSIC and IR, a rational bidder with an obvious dominant strategy will play it (bidding truthfully). Moreover, an optimal auction mechanism is required maximize the auctioneer's s expected revenue rec = B( [>> Pc(U, x. v)]
Auction design. The ex ex-post regret reg, (v, x, y) for the bidder is defined
max uz(r;, (b/, u-s), x, v) us us(Ui, U. KE
An auction mechanism (g.p) is DSIC, if and only if E reg, (v, (1,,,y) 0 for any value pro- X", and item-context y E ym Suppose the payment file v E ym bidder-context m (b, x, y)b,,, which implies that each bidder has a non- rule p satisfies v)) I EØ=
negative utility. Then, the auction design can be modeled as lincar programming problem that maximizes the expected revenue ((v.,,v) |E/= Pa (v, (,,,,)) y)] subject to the expected ex-post regret (( En regi(v, x. y)] O. Without loss of generality, the ex ex-post regret may refer to the average of all bidders' ex-post regrets.
Definition 2.1. Suppose the network's parameter is w, and the bidder i's empirical payment and ex- post regret are defined as T, ΣL= pf (v(0, (0) ())), and (w) 1, Σ regi (o(),,(() (")))) where the sample set {v((), (1) (, ()) is f.ddd sampled from the following prior distribution, P(O, y) = =-j=1 P(Dcy 22, Px, Py, (Vz).
Equivariant mapping. We define a mapping f as G-equivariant if bhg o = fo Py for two chosen group linear representations p and d and any g in group G.
Definition 2.2 (Permutation--quuinl Equivariaan Mapping). A permutation-eqquivatii mapping is defined to be f Ruxmm -> nR that for any instance x E Raxm and permutation matrices on E R and Om E mm we have f( o (
3

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In this paper, we consider the bidder- permutation on E = and item- item- pemmmuetttaiion om E R Rmxm Specifically, we define a mapping f bidder--ymmmetri or itm--symeetric. if f( on (a) or / ( = [( respectively. Moreover, we define an auction mechanism (g. p) as bidder- symmetric or item-symmetrie, if the allocation rule g and the payment rule p are both bidder- symmetric or item-symmetric.
Orbit averaging. For any feature mapping f F + G. the orbit averaging Q on f is defined as Q Zgec v o Pg: where and v/ are two chosen group representations acting on the feature spaces F and G, respectively. Orbit averaging can project any mapping to be equivariant:
Proposition 2.3. Orbit averaging Q is projection to the equivariant mapping space {f 1 o f = fop), ie.. 1 Qf Q p and Q Q. In particular, if f already equivariant, then Q = f.
Moreover, Qu and Qreg refer to the utility and the ex-post regret induced by Qy and Op. For the simplicity. we denote the orbit averagings that modify the auction mechanism to be bidder- symmetric, item-symmetric, and bidder/ite--ymmeetre by bidder averaging Q1. item averaging 22. and bidder-item aggregated averaging Q3. Besides, a detailed proof of the feasibility of the projected mechanisms be found in Appendix A.1
Hypothesis complexity. The generalizatbility to unseen data is usually measured by the generalization bound, which depends on the hypothesis set's complexity. To characterize the complexity of the hypothesis set, we introduce the following definitions covering number No.. and its corresponding distance 1. Based on the covering number, we can obtain a generalization bound in Theorem 3.6
Definition 2.4 (1 -distance). Let X' be a feature space and F space of functions from X' to R". The &0,1 distance on the space defined as loc, 1 gf. max lfe(z) (z))).
Definition 2.5 (Covering number). Covering number Noc. (F,r) is the minimum number of balls with radius r that can cover F under loc, -iittancce
3 Theoretical Results
This section presents the theoretical results. For simplicity, we view = (pi....,PP)) as a X 1 matrix to present the prices the bidders should pay. We first prove that the crrmuuation-cquivariinee induces the same expected revenue and a smaller expected ex-post regret in Section 3.1 Next in Section 3.2 prove that the permutation-eqquiaaiii number, which promises a smaller required sample complexity and a better generalization. Detailed proofs are omitted from the main text and given in supplementary materials due to space limitation.
mechanism space has smaller covering
3.1 Benefits for Revenue and Ex-Post Regret
In this section, we discuss the benefits for the revenue and the ex- ex-post regret in the conditions of bidder-symmetry and item-symmetry separately, and then discuss the benefits when both of them hold. Based on these results, we also study the learning process of non-permutation-cquivaann neural networks for auction design.
3.1.1 Benefits in the Bidder/Item-Symnnn
Condition
When the bidders come from the same distribution, the joint valuation distribution f is invariant under iidder-permutationn i.e. [(o, v, &n y) = f æ; y) for any o, E Sn, Meanwhile, when the items are indistinguishable, the joint distribution f is invariant under neemppernttation, i.e., [(vam, x, ) = [w, x. y) for any am E Sm- Both conditions do not always hold simultaneously. In this section, we study them separately.
To measure the regret gap between the projected mechanism and the original mechanism as below,
ppermutation-equiirriaan of the mechanism, we introduce the conception of
A (g.
MAX O'EVOX
Lie u)) (1/, v.) x),, Σ uclvi, (V/, v-)), "'VEn
max
where is the valuation profiles, on the valuation profile of bidder ar is the bidder-context, y the item-context, and the orbit averaging Q. can be the bidder averaging O the item averaging O2.

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The bidder averaging 21 and the item averaging Q2 acting on the allocation rule and the payment rule p. respectively, are below,
Q10(e,2,1) = no! on a
Σ
" 21p(U,2,V)
Σ on
n!
on En
22g(v,z,8) = m' ESm
g(vo v, yom)om and 22p(v, x,
pm y0m).
m!!
m!!
ESm
We thus can prove the following theorem that characterizes the benefits of permutationn-equiariaaee for revenue and ex ex-post regret in the condition of bidder/item-st -ymmm/ttrm-mmmmmttrr....
Theorem 3.1 (Benefits for revenue and ex post regret in the condition of bidder/item bidder/item-symeette symmetry). When the valuation distribution is invariant under permutations of bidders/items, the projected mechanism has the same expected revenue and smaller expected ex-post regret, that is,
(v,,,,))
Σ ((
Σ pa (a(u,,,)) and
(1)
E (v,22.))
Σ (v, 1=1
- ((
[erreg) 1=1
= E(
( (s.
2
where p the payment rule, reg is the ex-post regret, and Q is the bidder/item averaging.
A smaller expected ex-post regret implies this mechanism is closer to the dominant strategy incentive computible condition. Conversely, and equivalently, when the expected ex-post regrets are fixed, the projected auction mechanism has larger expected revenue. For any auction mechanism, in the bidder/item -eremmtttmmmymmetyyy condition, we can project it through the bidder/item averaging.
Remark 3.2. The mechanism space can be decomposed into the direct of the permutation- equivariant mechanism space {M QM = M) and the complementary space {W QN = 0} [13. Thus, mechanism M has a unique decomposition: M QM + W. The pure permutation- cquivariant part QM contains all and only the The pure aon--emmtatiion-qgeeeee -equivariini part N is independent from the of the mechanism M. this way, we may study the influence of permutation-equivariiiee and its permutation equivariant part Q.M.
In
by comparing the mechanism M
3.1.2 Interplay between Bidder-Symmetry and Item-Symmetry.
If the valuation distribution is invariant under both bidder-permunee permutation and item item- permutation, we can project the mechanism to be permutation-equivariin with respect to both bidder and item in two steps (by mapping Q Q2 or mapping 22 Q1). Consequently, the projected mechanism has same expected revenue and a smaller expected ex-post regret. Equivalently, we can also project an auction mechanism to be bidder-symmetric and item-symmetric immediately by the bidder-item aggregated averaging 23 as below,
Qsg(v,I, = n!m!
Σ
Σ
on g(0 UO J. )om and
on E.. Em
Qsp(u, 2.0) = nlm!
Σ
Σ
1 p(a,, on yo
on O.ESA ESm
We can prove that the bidder-item aggregated averaging Ca is the composition of the orbit averaging operators Qi and Q2, shown in the following lemma. This lemma shows that the order of Q1 and 22 would not influence their composition.
Lemma 3.3. The bidder-item aggregated averaging the composition of bidder averaging and item averaging: 23 Q O2 Q2 o Q1-
Based on this lemma, we can prove the following theorem on the benefits of permutation-equivaainnn for revenue and ex ex-post regret in the condition of both bidder-symmetry and item-symmetry.
5

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Theorem 3,4 (Benefits for revenue and ex ss-pott regret in the condition of both bidder-symmett symmetry and -ymmmetttrry) When the valuation distribution invariant under both item nem-pernntttion and fimmpermuttttin, then the projected mechanism has a same expected revenue and smaller expected ex-post regret, that is,
Σ (1,,(1,,,,,) B((,a..))
Σ Em( and
Σ reg,(v, 2, - (oaa.))))
Σ ((Qrreg)( = E( As(o. w| o,
(v,2,))
where is the payment rule, reg is the ex-post regret, and Q3 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 not item-symmetric, we can project
be item-symmetric gain an extra benefit from item-symmetry. That means, the two regret gaps induced by 21 and Q2 are "additive" as below,
As(g. Pi U, x. v) (g. U. + (2 2ip: v,
# (919 Qipru, x, y)] and thus R[As(g, P; u, x, y) * E[A1 (g,15 u, v) + (9, U, x. v) Thus, the benefits from bidder-symmetry and item- symmetry are "additive" but not strictly "independent".
In general, E[A2(g.r...2..))
3.1.3 Insights on Training www.-ermutattion-Eqqivrvrrir Mechanism
Because the expected revenue is always the same for the original mechanism and the projected permutation--quivariiee mechanism, we only consider the gradient caused by the expected ex- ex-post regret. We can decompose the original expected ex ex-post regret into the sum of the expected ex post regret of the projected mechanism and the expectation of the regret gap as below.
Σ reg. (v, x., w) (v.a,))))
((erreg)((taa.))) + (v,a,))) As(o.
The regret gap As p:: -) follows from the pooo--prmuuattionnuivvvaiannnn When the distance ([M, 2M) tends to 0, the regret gap converges to O. When the auction mechanism has negligible ex-post regret, the expectation of the regret gap is also close to That means, the mechanism is close to being permutation-eqquiaaiiai
of the mechanism M.
However, even using a symmetric dataset or training, the learned mechanism will not be permutation -equavviiiiiiiiiiiiani in general [19]. As a result, to achieve negligible ex ex-post regret, the penn--emutaatooeqguaaaee eqqivvaainnn models need learn more samples approach part (expected regret gap) would mislead the gradient of the expected regret but have no benefit to the expected revenue and the expected ex ex-post regret.
adopting data augmentation
That is because the
non-permutaaiooneequvaaaaa
On the other hand, the regret gap can be viewed as a regularizer in the ex-post regret to penalize the
of the mechanism. When the optimizer tries to minimize the Therefore, if the This auction mechanisms [29]. However, in complex settings, will be harder for non-pernnn wen---emtattoo-eeeeeee eeqivvarranii models to approach the negligible ex ex-post regret. It can explain why the permutation-eqqivariaa models show a significant improvement in complex settings, compared with that they have similar performances in simple settings [10 17). which shows the great importance of adopting permutation-equivariin models in complex settings.
ex-post regret, the auction mechanism approaches to be permutation-equivaaias mechanism achieves negligible ex-post regret, it is almost to be permutation-equivariie result can explain why RegretNet struggles to find
3.2 Benefits for Generalization
In this section, we study
from the aspect of generalizability [5],,, which characterizes the performance gap of learned mechanism on collected training data and unseen data.
We first study the covering number of the permutation-equivarian mechanism space. Let l {u" E l2} and P {p" w 2) be the spaces of all possible utilities and payment rules, and

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Quu:QQu:EEl U} and - pE the spaces of all projected utilities and payment rules. In addition, let a (U,r) and x be the minimum numbers of balls with radius r that can cover U and P under 10.1 -distance, respectively. We obtain the following result, which mechanism space has smaller covering numbers.
indicates the projected cemmutation-quivvriinn
Theorem 3.5 (Covering number of the permutation-ccuivariaan equivariant mechanism space). The space of all projected bidder-symmetric mechanisms has smaller covering numbers, that is,
No,, (U,r) and No.,1 (
The space of all projected itmm-symmetric mechanisms has smaller covering numbers, that is, N r) (U, and N 1
Intuitively, the orbit averaging Q narrows the distance between two mechanisms: 1QM, QM') S M(M, for any two mechanisms. Then, any r-cover A for space L/ or space P induces an r-cover QA for space QU or space QP.
Combining with Lemma 3.3 we have the following results,
No.,, I (Q,U, No,, (2)
224, (9, (Q,,)) S No., (2aP, = (9.2.,))) NN... z No,, (P,r).
and
We then prove that two generalization bounds of permutation-eqqiiariinen terize the gap between the expected revenue/ex-pe -pot regret and their empirical counterparts. Similar generalization results existing in previous works [10 [2].
mechanisms, which charac-
Theorem 3.6 (Generalization bounds of permutation-equiiariann mechanisms). for any bidder, her valuation satisfie that (5) for any 5 (m], then with probability at least 6, we have the
following inequalities with 2
9n2 2L
(log + max Moe 3), so. {)
L
n
E
Σ
pf
(v, x.
p (10)
100, y00)
sc, and
(3)
12
n
E
rege (v, xr, w)
reg,(w)
Se
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 the form of the sample complexity.
Corollary 3.7. For any > 0, å E (0, 1), and mechanism parameter when the sample complexity (log $ + max { log Noc, (P, 5), lo (4,,))) {)}) with probability at least 1 - 6, the
L
9n²
2
generalization bounds, eqs. and + hold.
Remark 3.8. Combining Theorem 3.5
we have proved that the permutation-equiiarianee can
improve the generalizability.
Experiments
This section presents experimental results. More details and results presented in the supple- mentary materials.
We project RegretNet [12] to the cermutation-cquivarine mechanism space via employing bidder-item aggregated averaging for the bidder-symmetry and item-symmetry condition. The projected model is called RegretNet-PE. We also project the well-trained RegretNet, called RegretNet- Test. Specifically, RegretNet is an auction mechanism defined as (g",p")) ), in which both the allocation rule 9d and the payment rule pd neural networks that consist of three fully- connected layers, and w is the overall model parameter of the auction mechanism. The detailed architecture given in the supplementary materials.
Model architecture.
Comparison with EquivariantNet. RegretNet uses two feed-forward fully-connected networks to learn the allocation rule and payment rule, respectively. We denote the weight matrix in the
7

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Table 1: Experimental results. "n x m Uniform" refers that there are bidders and m items, and the valuations are i.i.d. drawn from the uniform distribution U|0 1]. To simplify, we multiply all results by a factor of 10 for the ex ex-post regret and generalization error (GE).
2 x 1 Uniform
1 Uniform
5 Uniform
Method
Revenue
Regret
GE
Revenue
Regret
GE
Revenue
Regret
GE
Optimal
0.417
0.531
0
0.672
RegretNet
0.415
17.4
6.00
0.535
18.3
11.4
0.658
15.9
6.40
RegretNet-Test
0.415
16.3
0.535
13.3
0.658
6.50
RegretNet-PE 0.420
14.6
3.90
0.541
16.4
10.2
0.677
13.2
5.10
Table 2: Experimental results. "n x m Uniform" refers that there are n bidders and m items, and the valuations are i.i.d. sampled from the uniform distribution (10 1).
1 x 2 Uniform
2 x 2 Uniform
Method
Revenue
Regret
Revenue
Regret
RegretNet
0.562
0.00061
0.870
0.00070
EquivariantNet
0.551
0.00013
0.873
0.00100
RegretNet-Test
0.562
0.00052
0.870
0.00054
RegretNet-PE 0.563
0.00037
0.913
0.00067
layer É as W( Both EquivariantNet and RegretNet-PE inherit the architecture of RegretNet (with some modifications), but utilize different approaches to realize the permutatoo--qquiiariancc
EquivariantNet applies parameter-sharing every layer during training, to constrain W(() to be equivariant. In contrast, RegretNet-PE employs orbit averaging to be permutation-eqquivriin Specifically, RegretNet-PE adopts a weight matrix 1x @ ('( ...'"."""" )" in the first layer, weight matrices T W(O) in the following layers, and multiples matrix (p ,... to the output layer, where K is the scale of the group G (91...... :.......K''' 9K) }. Pss represents the permutation operator
on bidders and items, J is an identity matrix, and @ is the Kronecker product. It is worth noting that RegretNet-PEI is only designed for verifying our theory.
Auction settings. We first adopt the two-bidder single-item, two-bidder single-item, bidder single-item, and five-bidder single-item settings in the experiments that compare the learned mech- anisms with theoretical optimal mechanisms. The optimal auction mechanism for any single-iten item auction is known (23]. We thus compare the mechanisms leaned by our method with the optimal auction mechanisms in the single-item settings. Also, we compare RegretNet-PP and EquivariantNet in the one-bidder, two-item setting, and the two-bidder, two-item setting. Besides, multivariate uniform distribution U (0, 1]m² to model the bidder valuation profiles. In all settings, we sample 640,000 data points for training and 5,000 points for test. Due to the space limitation, place the results of two-bidder five-item and five-bidder three-item settings in Appendix B.2
employ
Model training. We optimize the auction mechanism model via solving the following optimization problem, following the standard settings
12
L ΣΣ
Colw, A) = -
1
P. (v
()+ + Σ Avreg, (w) +
P Σ reg.(w)
where À E 12" is the Lagrange multiplier and p 0 is the factor of the quadratic regularization term. During the training process, the objective function [w, A) is minimized via Adam with a learning rate of 0.001 with respect to the model parameter w and the Lagrange multiplier À updated once in every 100 iterations, until the ex ex-post regret is smaller than 0.001. The regularization factor is set to 1.0 initially and gradually increased along the training process. In calculating the best bid profile of of every bidder i, we first randomly initialize the bid profiles once in training and 1,000 times in test, optimize each of them individually via Adam with the same settings, and take the best one as the approximated best bidding.
8

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Evaluation. We leverage three metrics to evaluate the performance of the auction mechanism, which are: (1) the empirical revenue reV, (2) the empirical ex-post regret averaging across all bidders, Le., reg = Li= - and (3) the generalization error defined on top of regrets, i.e. GF = PEerest - regerain where regrest and reftroin are the empirical ex-post regrets during test and training, respectively.
Computing resource. The experiments are conducted on I GPU (NVIDIA Tesla 10 V100 16GB) and 10 CPU cores (Intel Xeon Processor E5-2650 v4 @ 2.20GHz).
Experimental results. We train RegretNet and RegretNet-PE on the training data. The well- trained RegretNet is then projected to be commutation-quivaarre denoted as "RegretNet- The results collected in Tables and 2
From Tables D and 2 we observe that (1) compared to RegretNet, RegretNet-PE: has a significantly
higher revenue with lower ex-post regret, and narrows the generalization gap between the training ex-post regret and its test counterpart: (2) compared RegretNet, RegretNet- receives the same revenue with significantly lower ex-post regret; and (3) under comparable ex-post regrets, RegretNet-PE has considerably higher revenue than EquivariantNer, while all cermutation-equivarinn models (RegretNet-Tess and RegretNet-PE) can outperform RegretNet. These results show significant benefits of cranuutttion-quivariinnn on revenue, ex-post regret, and generalizability. which fully
supports our theoretical findings in Theorems 3.1 3.4 and 3.5
5 Conclusion and future works
In this paper, under additive valuation and symmetric valuation setting, we study the benefits of cemmutationequivvrrinne in auction mechanisms in two aspects: a better performance and better generalization. First, we prove smaller expected ex ex-post regret and the same expected revenue when projecting a mechanism to be permutation-equivaat mechanism space has a smaller covering number, which promises the permutation -qquivaaiinn models better generalization. Extensive experiments are conducted to verify our theoretical results. Our results help understand the optimal auction mechanisms' characterization and the learning processes difference between non-equivariant models and equivariant models.
Next, we propose the
equivariant
Beyond the additive valuation setting, an interesting direction to extend our results to other conditions, including the combinatorial and the unil-demand auctions. Meanwhile, the understanding of the difference in the aspect of the training process between non non-equivariant models and equivariant models is still elusive.
Social impact. Our results can help understand and design optimal auction mechanisms for symmetric valuation distribution. As result, our work could inspire more near-optimal auction mechanisms and promote economic growth. No potential negative social impact is identified.
Acknowledgments and Disclosure of Funding
This work was supported in part by the Major Science and Technology Innovation 2030 "New Generation Artificial Intelligence" Key Project (No. 2021ZD01 1 1700), the National Natural Science Foundation of China (No. 62006137), and the Beijing Outstanding Young Scientist Program (No. HWWWZYJHH0220110002000
We sincerely appreciate Hang Yu, Xiaowen Wei, Kaifan Yang, Shaopeng Fu, and Qingsong Zhang for the valuable comments and the anonymous NeurIPS reviewers for the helpful feedback.
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[2] Alaei, Fu, N. Haghpanah, and Hartline. The simple economics of approximately optimal auctions. The 16th Annual Symposium on Foundations of Computer Science,, 2013.
[3] M.-F. Balean, Sandholm, and Vitercik. Sample complexity of automated mechanism design. Neural Information Processing Systems (NeurIPS), 2016.

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[4] L. Chen, L. Wang, and Y. Lan. Auction models with resouree pooling in modern supply chain management. Modern Supply Chain Research and Applications, 2019.
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[8] M. Curry, P.-Y. Chiang, T. Goldstein, and J. Dickerson. Certifying strategyproof auction networks. Neural Information Processing Systems (NeurIPS), 2020.
[9] C. Daskalakis, A. Deckelbaum, and C. Tzamos. Strong duality for multiple-good monopolist. Econometrica, 2017.
[10] Z. Duan, J. Tang, Y. Yin, Z. Feng, X. Yan, M. Zaheer, and X. Deng. A context-integrated transformer-base based neural network for auction design. arXiv preprint arXiv:2201. 12489, 2022.
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[13] B. Elesedy and S. Zaidi. Provably strict generalisation benefit for equivariant models. In International Conference on Machine Learning (ICML), 2021.
[14] Z. Feng, H. Narasimhan, and D. C. Parkes. Deep learning for revenue-optimal auctions with budgets. In the 17th International Conference on Autonomous Agents and Multiagent Systems, 2018.
[15] He and D. Tao. Recent advances in deep learning theory. arXiv preprint arXiv:2012.10931, 2020.
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[19] C. Lyle, M. van der Wilk, M. Kwiatkowska, Y. Gal, and B. Bloem- Reddy. On the benefits of invariance in neural networks. arXiv preprint 200.00000 2020.
[20] A. M. Manelli and D. R. Vincent. Bundling as optimal selling mechanism for multiple good monopolist. Journal of Economic Theory, 2006.
[21] M. Mohri and A. M. Medina. Learning theory and algorithms for revenue optimization in second-price auctions with reserve. International Conference Machine Learning (ICML), 2013.
[22] M. Mohri, A. Rostamizadeh, and A. Talwalkar, Foundations of machine learning. MIT press, 2018.
[23] R. B. Myerson. Optimal auction design. Mathematics of operations research, 1981.
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[25] N. Nisan, Roughgarden, Tardos, and V. Vazirani. Algorithmic game theory. Cambridge University Press, 2007.
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[26] G. Pavlov. Optimal mechanism for selling two goods. The BE Journal of Theoretical Economics, 2011.
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[29] Rahme, S. Jelassi, Bruna, and M. Weinberg. permutation-equivant neural network architecture for auction design. In Proceedings of the AAAI Conference on Artificial Intelligence, 2021.
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Checklist
1. For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper's contributions and scope? [Yes] Our contributions are stated in the 4th-Sth sentences of the abstract, and the 4th-6th paragraphs of the introduction. The scope is described in the related work section.
(b) Did you describe the limitations of your work? [Yes] We describes the problem setting considered in the 3rd paragraph of the introduction. Our contributions for this problem setting.
(c) Did you discuss any potential negative societal impacts of your work? [Yes] Our work committed learn the optimal auction mechanism. No negative societal impact is identified.
(d) Have you read the ethies review guidelines and ensured that your paper conforms them? [Yes] We have read the ethics review guidelines. We ensure that our paper conforms all of them.
2. If you including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results? [Yes] We state all 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] Full proofs are given in the supplemental material.
3. If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main ex- perimental results (either in the supplemental material as URL)? [No] We are associated in 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 are
given Sections 4 and B
(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., type of GPUs, internal cluster, cloud provider)? [Yes] They given in Section
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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
(c) Did you include any new assets either in the supplemental material or as URL? [No] We are associated in 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 data.
(e) Did you discuss whether the data you are using/curating contains personally identifiable information offensive content? [Yes] The data is synthetic in our data. No personally identifiable information or offensive content is contained.
5. 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, applicable? [N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amount spent participant compensation? [N/A]
12


Reviews:
[{"review_id": "woo44flUFVy", "reviewer": "Reviewer_1aRG", "summary": "This paper considers the learning problem arising from seller revenue optimization in repeated auctions, with an approximate incentive compatibility constraint, under the assumption that the joint distribution of context and values is available to the learner (at least an empirical version). It studies whether averaging the learnt payment and allocation functions over permutations of items (resp. bidders) reduces bias error term and statistical error term when the distribution is invariant by permutation of the items (resp. bidders).", "questions": "How does the statistical error provided in Th. 3.6 compares to those of empirical Myerson auctions ?", "limitations": "Suggestions:\n\n1. I feel the proof of Th. 3.1 could be deferred to appendix or at least shortened as it mostly comes from the definitions + convexity of max function.\n\n2. About the case $Q_3$ (both item and user symmetry), isn't the optimal auction a second-price auction with one optimized reserve price? A sanity check could be to compare the error term of the auction learnt by the RegNet to the learning of this unique reserve price.\n\n3. As an additional direction, the authors could compare the aggregation method to other classes of equivariant models they cite in the introduction, either theoretically or empirically (or both).", "rating": 5, "confidence": 2, "soundness": 2, "presentation": 2, "contribution": 1, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. The structure of the paper and of the exposition is clear.\n\n2. The writing itself should be improved as a certain number of quantities are not formally defined or even not defined at all (or just in the appendix).\n    * l. 125 I would have expected a more formal definition of $f(x)\\sigma_m$\n    * Th 3.5 introduces covering numbers ${\\cal N}_{\\infty,1}(\\cdot,\\cdot)$. I guess $\\infty,1$ refers to some distance, but it is only provided in appendix\n    * $v^{(l)}, x^{(l)}, y^{(l)}$ are probably samples in Th. 3.6\n    * $\\omega$ is again only defined in appendix (Th. 3.6) \n    * $\\widehat{reg}_i(\\omega)$ (Th. 3.6) is not defined at all (even in appendix)\n    * I guess $rgt_i = reg_i$ ?\nMost of these are \"_guessable_\" or defined somewhere in the appendix, but remember the paper should be self-contained.\n\n3. I find the contributions are a bit light. Indeed, Th. 3.1 follows quickly from the definition, Th. 3.6 is mostly a corollary of results in [10]. So the strongest contribution is Th. 3.5 on the dominance of covering number that comes from the projection contracting the distances. Still, in the end, the message is clear. ", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "unW_8EhBQir", "reviewer": "Reviewer_SPL9", "summary": "This paper studied permutation-equivariant auction mechanisms, which, roughly, are auctions that do not depend on the labeling of the bidders/items. Permutation equivariant auctions are a promising solution concept for settings where the bidder/item distributions are symmetric. The authors prove that permutation equivariant auctions never decrease revenue, and always yield better ex-post regret (bringing the auction closer to being perfectly incentive compatible). The authors explore generalization guarantees, and provide experimental evidence for the benefits of permutation-equivariant auctions.", "questions": "How do the results of this paper compare to Rahme et al. [25]? My understanding is that the approach in this paper involves a conversion of any auction mechanism to a permutation-equivaraint one via orbit averaging. Is this a subset of the set of permutation equivariant mechanisms considered by Rahme et al? \n\nIn Theorem 3.5, wouldn’t covering number inequalities follow from the simple fact that the set of all permutation equivariant auctions is a subset of the set of all auctions? (I think this is what lines 261-263 is saying.) Is there a more quantitative gap that can be obtained? \n", "limitations": "N/A", "rating": 5, "confidence": 2, "soundness": 3, "presentation": 1, "contribution": 3, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This paper studies a relevant, important, and difficult question: the design of optimal auctions. However, I think there are many points that need to be addressed. First, I found the paper to generally be difficult to read. Permutation equivariance is never given a clear definition in the auction context – instead it is abstractly defined for arbitrary maps via group representations (I don’t know if the authors should assume that the NeurIPS audience is intimately familiar with group theory jargon). Then, my understanding was that permutation equivariance is obtained by projecting any map via an orbit-averaging operator (but I wasn’t sure why this is the case). *I think that lines 157-158 more or less make this clear: the allocation function and payment are being averaged over all permutations of the bidders (or items) to obtain a symmetric auction. \n\nThis brings me to the second main concern that there is almost no comparison to previous work on symmetric auctions/study of permutation equivariance. Permutation equivariance is the main contribution of Rahme et al. [25] in the context of auction design, but there is nearly zero discussion or comparison to their results.\n\nMy reading is that the main contribution of this paper is a novel way to view permutation equivariance: any auction can be converted into a permuation equivariant one via the orbit averaging operator. If this is indeed the main contribution, I think the intuition for what is going on ought to be much more clearly spelled out (because it seems like an elegant and novel contribution!). \n\nThe overall writing style (flow/sentence structure/grammar/spelling) could also use a lot of improvement which would make the paper easier to read and follow.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "pbkF6LVE4Cn", "reviewer": "Reviewer_FBeS", "summary": "The use of deep neural networks for automated mechanism design is increasingly important. Several recent papers have observed that when valuation distributions are symmetric, a symmetric auction is optimal. Thus, neural networks for mechanism design ought to respect this symmetry by using architectures (i.e. DeepSets, attention) that are permutation-equivariant. The current paper theoretically analyzes the effect of equivariance: they give a technique for projecting non-equivariant mechanisms onto the set of equivariant mechanisms, and show that it has several nice properties. Most importantly, it always preserves auction revenue while weakly improving expected violations of strategyproofness. From a learning theory perspective, learning over the class of the projected mechanisms also has better generalization guarantees. And, the projection technique is constructive so the authors can actually train these mechanisms and show they perform decently.", "questions": "Do you have any thoughts or speculation on how orbit averaging interacts with the gradient-based computation of best untruthful bids? Is it possible that it is somehow \"harder to optimize\" the misreports when orbit averaging, compared to normal RegretNet?\n\nYou consider both bidder-permutation and item-permutation. In addition to comparing against the Myerson auction in the 1 item setting and looking at bidder-permutation, it might make sense to compare to some of the known multi-item mechanisms in the 1 bidder setting (e.g. the Manelli-Vincent mixed bundling mechanism as in the original RegretNet paper), and look at item-permutation. Training a 2-item 1 agent orbit averaged mechanism could be interesting in this case (where the optimal mechanism is in fact item-equivariant).", "limitations": "In addition to focusing a little more on previous work about equivariant auctions in the background sections, it would enhance the experiment section to include results from other equivariant auction networks where the comparison makes sense. I don't think running lots of new experiments is necessary, but where the numbers can be compared, I think this should be done.\n\nI think the authors should further spell out how orbit averaging interacts with the allocation constraints (items sum to one). This was a question I had as I was reading the paper. I checked and saw that orbit averaging should preserve feasibility of allocations, but I think it makes sense to explicitly reassure the reader that this holds, rather than leaving the reader to check it.", "rating": 7, "confidence": 4, "soundness": 4, "presentation": 3, "contribution": 4, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "*Originality*\n\nI'm not convinced this is hugely original. It seems mainly to borrow other techniques related to equivariant ML. I think that's fine -- it's putting pieces together in an interesting and important way, but there is no really new technique here.\n\nThe paper claims in one sentence to be the first work explaining the benefits of permutation equivariance in auction design via deep learning. This is arguably not true depending on how one parses the word \"explaining\". Previous works looking at equivariant networks for auction design are at least pointed out and cited, but I think they should be given more discussion and the claim about being \"the first\" should be weakened or qualified. Doing this would not detract from the significance of the paper, in fact it would enhance the motivation for the results in this paper.\n\n*Quality*\n\nThe quality of the paper seems high, the theoretical results are reasonable, and the empirical results are convincing enough given they are not the main focus. \n\n*Clarity*\n\nThe paper is clear. There are good proof sketches in the main text and the full proofs are easy to follow. Some of the early definitions may be too compressed and hard to follow, but I think this is the fault of the conference page limits and not the fault of the authors.\n\n*Significance*\n\nThis is a thoughtful contribution to an important topic in auction design. There are already several papers looking at symmetry in automated mechanism design, but this is a new, clear, and useful perspective.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}, {"review_id": "5cnYWIrUJSn", "reviewer": "Reviewer_99t4", "summary": "This work studies the benefits of incorporating permutation-equivariance into a neural network (NN) trained to function as an auction mechanism in auctions with additive valuations and symmetric bidders and/or items. The authors prove that incorporating permutation-equivariance which reflects the bidder/item symmetries 1) decreased the expected ex-post regret (measures \"distance\" to the space of dominant strategy incencitve compatible, \"truthful\", auctions) of the bidders, 2) maintains the expected auctioneer revenue, and 3) improves mechanism generalizability (e.g, to other bidder valuation profiles), intuitively via a reduction of the auction mechanism hypothesis space. Permutation-equivariance is achieved by \"orbit\" averaging the NN outputs (item allocations and bidder payments) over all possible permutations of bidders/items. Experiments are conducted in both single-item (theoretically optimal mechanism is known) and multi-item auctions. Properties 1, 2, and 3 are supported by the experiments in all 2, 3, and 5 bidder auctions.", "questions": "- In Sec 3.1.2, should there be a $\\sigma_m^{-1}$ on the right side of the definition of $Q_3 p(v, x, y)$? Same with $Q_2 p(v, x, y)$ in Sec 3.1.1? If not, why not?\n- In Table 1, does the, for example, \"2 x 1 Uniform\" header mean \"number of bidders x number of items Distribution of valuations\"? I did not see this explained anywhere.\n- Is RegretNet-PE your model? I did not see your model named in the text anywhere (maybe I missed it) and had to infer this.\n- The conclusions you draw from the experimental results appear to be over simplifications. Your first claim (1) appears to be violated in the 5 x 1 setting (your revenue is 0.633 vs optimal is 0.667). Your second claim (2) is also violated in the 5 x 3 setting (ex-post regret increases vs RegretNet, yours is 31.2 vs RegretNet 28.4). Please report accurate conclusions. If I'm misunderstanding your table, please clarify.\n- How would you apply your approach to settings where subsets of the items/bidders are symmetric? Can you just average over the permutations within the subset? This might be a nice note to add to the exposition.\n- Can you please compare/contrast RegretNet-PE with EquivariantNet? More generally, can you list the contributions of your work that go beyond [25]?", "limitations": "Can you comment on the computational expense of your approach? Averaging over all possible permutations presumably increases the number of forward passes of your NN by a factorial amount. Therefore, this approach is only feasible when the number of symmetric bidders/items is small.", "rating": 6, "confidence": 4, "soundness": 3, "presentation": 3, "contribution": 2, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "This work proves several properties conjectured and partially explored empirically in previous work [25]. Most of the exposition is clear. Overall, given the existence of the previous work [25] at AAAI, I find the work a bit incremental and, therefore, lacks significance beyond [25]. The authors do not compare / contrast the permutation-equivariant NN in [25] with their own (unclear if RegretNet-PE is original), and only include one comparison with that model at the end of their appendix (also without any discussion of EquivariantNet). The theorems proven in this paper are interesting, but not very surprising (esp generalizability, reducing the hypothesis space typically improves generalizability). It's unclear how useful the sample complexity bounds are and experiments are not conducted to empirically confirm the complexity bound.", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"ethics_review_area": ""}, "extra_text": {}}]

Author rebuttal:

Decision: Accept