Title: Sample Complexity of Posted Pricing for a Single Item
Venue: NeurIPS 2024
Accepted: yes
Keywords: sample complexity

revenue

welfare

pricing

online

prophet inequality
Abstract: Selling a single item to $n$ self-interested bidders is a fundamental problem in economics, where the two  objectives  typically considered are welfare maximization and revenue maximization.   Since the optimal auctions are often impractical and do not work for sequential bidders, posted pricing auctions, where fixed prices are set for the item for different bidders, have emerged as a practical and effective alternative. This paper investigates how many samples are needed from bidders' value distributions to find near-optimal posted prices, considering both independent and correlated bidder distributions, and welfare versus revenue maximization.  We obtain matching upper and lower bounds (up to logarithmic terms) on the sample complexity for all these settings.

Sample Complexity of Posted Pricing for a Single Item

Abstract

Selling a single item to n self-interested buyers is a fundamental problem in economics, where the two objectives typically considered are welfare maximization and revenue maximization. Since the optimal mechanisms are often impractical and do not work for sequential buyers, posted pricing mechanisms, where fixed prices are set for the item for different buyers, have emerged as a practical and effective alternative. This paper investigates how many samples are needed from buyers' value distributions to find near-optimal posted prices, considering both independent and correlated buyer distributions, and welfare versus revenue maximization. We obtain matching upper and lower bounds (up to logarithmic factors) on the sample complexity for all these settings.

1 Introduction

A fundamental problem in Economics is how to sell a single indivisible item to a set of n selfinterested buyers. This problem is challenging because the value associated to the item is private knowledge of each buyer and thus can be strategically misreported. The mechanism designer usually has one of two objectives: (a) welfare maximization where we want to maximize the value of the winning buyer, and (b) revenue maximization where we want to maximize the price paid by the winning buyer. The welfare maximization problem was resolved by Vickrey in 1961 [Vic61] and the revenue maximization problem was resolved (for independent distributions) by Myerson in 1981 [Mye81], both of whom were awarded the Nobel prize in Economics for their contributions. In either case the optimal mechanism is a truthful auction, in which buyers report their values and have no incentive to misreport.

Although we know optimal mechanisms for selling a single item, these auctions are often difficult/impossible to implement in practice. For instance, the buyer does not know exactly how much they will pay if they win the item and these auctions do not work for online settings where the buyers arrive one-by-one [AM + 06, HR09]. Thus, the last two decades has seen a lot of progress on understanding the power of simple but near-optimal mechanisms. In this regard, posted pricing mechanisms have been identified to be very successful (see books and surveys [Rou16, Har13, CFH + 19, Luc17]). Here, the mechanism designer puts a (carefully chosen) price π i on the item and then the i -th buyer takes the item if the item is not already sold and if they value it above π i . Posted pricing mechanisms have several advantages: they are truthful since π i does not depend on the i -th buyer's value; the buyer knows exactly how much they pay on winning an item; and they work even for online settings so the buyers do not have to be simultaneously present. Additionally, in the setting of Myerson's 1981 paper of independent buyer valuations, they are known to give a 2-approximation to both the optimal welfare (which is usually called Prophet Inequality) and the optimal revenue

[KS77, KS78, CHK07, CFPV19], as long as the posted prices are set correctly. This raises the main question of the current paper:

How many samples are necessary from the value distributions of the buyers to find near-optimal posted prices for single item? Is there a difference between independent vs. correlated distributions or between welfare vs. revenue maximization?

Despite being a natural question, the tight sample complexity bounds for single item posted pricing are unknown, both for independent/correlated and welfare/revenue settings.

1.1 Model

Before stating our results, we formally describe our model.

Posted pricing. There is single copy of an item. Buyers i = 1 , . . . , n arrive in order, each with a private value (willingness to pay) V i for that item. Price π i is offered to buyer i , and the first buyer (if any) for whom V i ≥ π i end up buying the item, 'winning' the auction. The objective is to maximize either welfare , which is the value of the item to the winner, or revenue , which is the price paid by the winner.

Distributions and policies. We will assume that the vector of values V = ( V 1 , . . . , V n ) is drawn from an unknown but fixed distribution D over [0 , 1] n . If each V i is drawn independently from some marginal distribution D i , then D is called a product distribution , written as D = D 1 ×··· × D n . Meanwhile, posted pricing policies are defined by a vector of prices π = ( π 1 , . . . , π n ) that must be fixed before the first buyer arrives, and possibly restricted to some subclass Π . The objective of policy π under buyer values V is denoted by π ( V ) , which equals V argmin { i : π i ≤ V i } and π argmin { i : π i ≤ V i } for the welfare and revenue objectives respectively, understood to be 0 if the item is not sold. We (abusively) let π ( D ) := E V ∼ D [ π ( V )] denote the expected objective of π , and let Π( D ) := sup π ∈ Π π ( D ) denote the best expected objective of a policy in Π knowing D .

Note that if we are given the product distribution on buyer values, the optimal policy (for either objective) can be easily computed using dynamic programming [CFPV19]. Consequently, for product distributions we consider the full policy class Π = [0 , 1] n , and omit the dependence on Π .

Learning problem. A learning algorithm takes as input samples V that are drawn independently and identically distributed (IID) from the unknown distribution D and an error parameter ε ∈ (0 , 1) , and seeks to return a policy π ∈ Π that satisfies π ( D ) ≥ Π( D ) -ε , which is called an ε -approximation . Given a failure probability δ ∈ (0 , 1) , the sample complexity is the minimum number of samples required for there to exist a learning algorithm that, under any distribution D , returns an ε -additive approximation with probability at least 1 -δ , noting that the randomness is over both the samples and any random bits in the algorithm. The sample complexity and learning algorithm will depend on the problem variant, defined by the objective (welfare or revenue), any parameters of the policy class Π , and whether D is restricted to be a product distribution or not. The sample complexity will also depend on the parameters ε, δ > 0 .

1.2 Our Contributions to Posted Pricing for Product Distributions

The study of sample complexity for posted pricing goes back to at least the seminal work of Kleinberg and Leighton [KL03] who study revenue maximization for selling a single item to a single buyer (welfare maximization is trivial for single buyer since we just allocate the item by setting 0 price). For the general posted pricing problem with n buyers, the best known sample complexity bounds were due to Guo et al. [GHTZ21], who showed that ˜ O ( n/ε 2 ) samples suffice from each buyer's distribution for both welfare and revenue maximization objectives. Although there is a simple Ω(1 /ε 2 ) lower bound for both welfare and revenue maximization settings, prior to our work it was unclear if polynomial dependency on the number of buyers n is necessary.

Our first result for product distribution shows that for welfare maximization using posted pricing (a.k.a. prophet inequality), the sample complexity is independent of the number of buyers n .

Theorem 1 (proved in Subsection 2.1) . For product distributions, the sample complexity of welfare maximization is O (1 /ε 2 · log 2 (1 /δ )) .

Proof sketch. We start by constructing the product empirical distribution using the samples, where for each i we take the uniform distribution on the ˜ O (1 /ε 2 ) samples and then take the product distribution for different i . Our main result is that the optimal policy (dynamic programming solution) on this product empirical distribution is an ε -approximation with high probability. Although one could use standard concentration bounds to bound the gap between the learned and optimal thresholds, such arguments lose a poly ( n ) factor. This is because the difference between successive thresholds is bounded by 1 , so naïvely the total variance of n thresholds could be Ω( n ) . Our main idea is to instead study a martingale that adds up the errors made in the dynamic programming solution. This way, too high values for one buyer and too low values for another buyer balance each other. On this martingale, we use Freedman's inequality (which is a martingale variant of Bernstein's inequality), which allows us to bound the total variance in terms of conditional variances. A careful application of another concentration bound allows us to bound these conditional variances in terms of the change in the optimal value, whose sum is always at most 1 . This latter concentration bound is what causes the additional factor of log(1 /δ ) in the sample complexity.

Our second result for product distribution shows a separation between welfare and revenue maximization using posted pricing, by proving that the ˜ O ( n/ε 2 ) sample-complexity result of [GHTZ21] for revenue maximization is tight up to log factors.

Theorem 2 (proved in Subsection 2.2) . For revenue maximization on product distributions, any learning algorithm requires Ω( n ε 2 ) samples to return an ε -additive approximation with probability greater than 6/7.

Proof sketch. We construct for each buyer i two possible value distributions that add the same amount (roughly 1 n ) to the value-to-go of the optimal dynamic program. However, these distributions have different optimal prices, and making a mistake (choosing an incorrect price) in isolation loses roughly ε n value. Although these mistakes accumulate in a non-linear fashion, we show that making

M mistakes must lose in total Ω( εM 2 n 2 ) . Finally, these value distributions have probabilities on the scale of 1 ± ε n with the same supports (essentially, only 1 /n of samples provide information), which means that Ω( n ε 2 ) samples are needed to avoid making a constant fraction of mistakes.

1.3 Our Contributions to Posted Pricing for Correlated Distributions

Independence among buyer valuations can be a strong modeling assumption for many applications. Although for arbitrary correlated distributions the optimal policy is not learnable, one could hope to learn the best policy in the class of all posted pricing policies. A recent work of Balcan et al. [BDD + 21] can be applied to this setting to show that ˜ O ( n/ε 2 ) samples are sufficient to learn ε -optimal posted pricing. As discussed in Theorem 2, this linear in n dependency is necessary for revenue maximization, even for product distributions. Our first observation is that for correlated distributions, we need to lose this factor even for welfare maximization.

Theorem 3 (corollary of Theorem 5) . For welfare maximization with correlated buyers, any learning algorithm requires Ω( n ε 2 ) samples to return an ε -additive approximation with constant probability.

Given this lower bound, a natural next question is to consider the subclass of posted pricing policies where the algorithm is only allowed to change its threshold at a small number of given locations. Can we now remove the linear in n dependency from sample complexity? The motivation to study this class comes from posted pricing applications where it is not possible for the algorithm designer to update the prices at each time step, e.g., due to business constraints.

Formally, for S ⊆ { 2 , . . . , n } , we say that posted pricing policy π respects change-points S if π i can differ from π i -1 only when i ∈ S . We let Π S denote the class of all policies that respect changepoints S , noting that policies in Π( ∅ ) post a static price for all buyers, and Π( { 2 , . . . , n } ) = [0 , 1] n . Our following result shows that one can obtain sample complexity that is independent of n , depending only on the size of S .

Theorem 4 (proved in Subsection 3.1) . For correlated distributions, the sample complexity of welfare or revenue maximization is O ( (1+ |S| ) log(1+ |S| )+log(1 /δ ) ε 2 ) when the policy is restricted to Π S , for any S ⊆ { 2 , . . . , n } .

Proof sketch. By existing results in learning theory, it suffices to bound the pseudo-dimension of Π S . This will boil down to understanding the structure of 'good sets', which are sets of the form { π ∈ Π S : π ( v ) ≥ z } , for some input v and some target z . We will show that for a natural parameterization of Π S , any good set can be expressed as the union and intersection of O ( |S| +1) halfspaces, which implies the pseudo-dimension of Π S is O (( |S| + 1) log( |S| + 1)) by a result of [BDD + 21]. This bound on the pseudo-dimension translates to the above sample complexity bound.

The learning algorithm which achieves the sample complexity bound in Theorem 4 is simply sample average approximation (SAA), which returns the policy in Π S with the highest objective value averaged over the samples. SAA can be computed in time O ( Tn ( Tn ) 1+ |S| ) . This is because there are 1 + |S| prices to decide, each of which can take Tn possible values (one for each realized value in the T samples of length n ), and evaluating each combination of prices over each of the T samples takes runtime linear in n . We leave as an open question whether there is a more efficient algorithm.

Finally, we complement our sample complexity upper bounds for correlated distributions by giving matching lower bounds (up to polylogs).

Theorem 5 (proved in Appendix A.5) . For welfare or revenue maximization on correlated distributions, a learning algorithm requires Ω( 1+ |S| ε 2 ) samples to return an ε -additive approximation with constant probability, when restricted to the policy class Π S for any S ⊆ { 2 , . . . , n } .

Proof sketch. In the construction for Theorem 5, on each trajectory exactly one of the 1 + |S| decision points (randomly selected) will be relevant, essentially diluting the samples by a factor of 1 + |S| and leading to a lower bound of Ω( 1+ |S| ε 2 ) .

1.4 Further Related Work

In 2007, Hajiaghayi et al. [HKS07] discovered connections between auction design and posted pricing via prophet inequalities. Since then, there is a long line of work on understanding the power of posted pricing for selling multiple items to combinatorial buyers [CMS10, FGL15, DKL20, AKS21, CC23]. For single item revenue maximization with known distributions, Correa et al. [CFPV19] showed the equivalence of welfare and revenue maximization objectives for single item posted pricing.

For background on sample complexity, we suggest Wainwright's excellent textbook [Wai19]. Sample complexity of auction design has been greatly studied; e.g., see [KL03, CR14, MR16]. We refer the readers to Guo et al. [GHZ19], who recently resolved single item revenue maximization in the offline setting, for an overview of the literature. In [GHTZ21], Guo et al. generalized their techniques for revenue maximization over product distributions to all 'strongly monotone' problems, which includes posted pricing for welfare and revenue maximization.

Recently, there is also a lot interest in learning auctions in limited feedback models like bandit and pricing queries [GKSW24, SW24, LSTW23]. We should note that sample complexity of optimal stopping (equivalent to our welfare maximization problem) has been previously studied in [GM22], who analyze linear stopping rules in a contextual setting. Our application of techniques from [BDD + 21] to online algorithms over correlated sequences is also similar in spirit to some results from [XMX24], who study a different application of inventory optimization.

Another related but tangential line of work focuses on prophet inequalities with samples [AKW14, RWW, CDFS22, CDF + 22, DKL + 24]. The key distinction in these works is that their benchmark is the expected hindsight optimum, rather than the optimal online policy. Notably, any online algorithm incurs at least a factor of 2 loss compared to the hindsight optimum, even in the case of single-item prophet inequalities. As a result, this line of research aims to achieve O (1) -approximation guarantees, rather than the sublinear regret guarantees pursued in the current paper. Furthermore, their techniques differ significantly, as they often assume unbounded distributions. Finally, there is also work that explores the (non-)robustness of algorithms for the prophet inequality problem to inaccuracies in the distributions [DK19] and to dependencies in distributions [ISW20, LPS24].

2 Product Distributions

In this section we first prove our improved upper bound on the sample complexity of welfare for product distributions, and then prove a new lower bounds on the sample complexity of revenue for product distributions.

2.1 Positive Result for Welfare: Proof of Theorem 1

The reward of the optimal policy is given by the following backward induction: r ∗ n +1 = 0 and r ∗ i = E [ max { r ∗ i +1 , V i } ] = E [ ( V i -r ∗ i +1 ) + ] + r ∗ i +1 . It sets π i = r ∗ i +1 .

When we do not know the distributions but only have T samples from each of them, we can consider the optimal policy on the product empirical distribution, which corresponds to replacing the expectations in the above definitions by the empirical average. That is, we will analyze the policy that sets π i = ˆ r i +1 , where ˆ r i is defined recursively by ˆ r n +1 = 0 , ˆ r i = 1 T ∑ T t =1 ( V ( t ) i -ˆ r i +1 ) + + ˆ r i +1 .

Let r i be the expected reward of this policy when starting with the i -th arrival. In order to prove the theorem, it is sufficient to show that with probability at least 1 -δ , we have r 1 ≥ r ∗ 1 -ϵ if T ≥ (5 ln(2 e/δ ) /ϵ ) 2 for any choices of ϵ, δ ∈ (0 , 1) .

Let us define η i = ˆ r i -E [max { ˆ r i +1 , V i } ] or equivalently as

That is, η i denotes the error introduced by using the empirical distribution for buyer i instead of the actual one. Note that η i can both be positive and negative.

The two key steps of our proof are as follows. We first show that

This inequality holds point-wise, that is for any samples drawn. Then, we show that for T ≥ (5 ln(2 e/δ ) /ϵ ) 2 samples, we have max j ≥ 1 ∣ ∣ ∣ ∑ j i =1 η i ∣ ∣ ∣ ≤ ϵ with probability at least 1 -δ .

In order to show (1), we first lower bound r 1 in terms of ˆ r 1 .

Proof. Let j ∈ { 0 , 1 , . . . , n } be the smallest index for which r j +1 ≥ ˆ r j +1 , which exists because 0 = r n +1 ≥ ˆ r n +1 = 0 . Note that we have r i < ˆ r i for all 1 ≤ i ≤ j . We rewrite r 1 as

Repeating this argument,

Inductively, we can also establish that

Combining these two equalities,

This implies ˆ r 1 -r 1 ≤ max j ∈{ 0 ,...,n } ∑ j η i i =1 , completing the proof.

A similar proof allows us to prove the following lemma.

Proof. Let j ∈ { 0 , 1 , . . . , n } be the smallest index for which ˆ r j +1 ≥ r ∗ j +1 , which exists because 0 = ˆ r n +1 ≥ r ∗ n +1 = 0 . Note that we have ˆ r i < r ∗ i for all 1 ≤ i ≤ j , allowing us to derive

This implies r ∗ 1 -ˆ r 1 ≤ max j ∈{ 0 ,...,n } ( -∑ j i =1 η i ) , completing the proof.

The last two lemmas imply (1). Thus, we need to bound max j | ∑ j i =1 η i | to complete the proof. Lemma 3. For every ϵ, δ > 0 , with probability at least 1 -δ , we have max j | ∑ j i =1 η i | ≤ ϵ if T ≥ (5 ln(2 e/δ ) /ϵ ) 2 .

Proof. Observe that

so it suffices to show that max τ | ∑ n i = τ η i | ≥ ϵ/ 2 with probability at most δ . Reversing the quantity of interest to max τ | ∑ n i = τ η i | allows us to define a martingale, and use Freedman's inequality, which is a martingale version of Bernstein's inequality.

Lemma 4 (Freedman, Theorem 1.6 in [Fre75]) . Consider a real-valued sequence { X t } t ≥ 0 such that X 0 = 0 and E [ X t +1 | X t , X t -1 , . . . , X 0 ] = 0 for all t . Assume that the sequence is uniformly bounded, i.e., | X t | ≤ M almost surely for all t . Now define the predictable quadratic variation process of the martingale to be W t = ∑ t j =0 E [ X 2 j ∣ ∣ X j -1 , . . . , X 0 ] for all t ≥ 1 . Then for all ℓ ≥ 0 and σ 2 ≥ 0 , and any stopping time τ , we have

.

A corollary of Freedman's inequality is that Pr [∣ ∣ ∣ ∑ τ j =0 X j ∣ ∣ ∣ ≥ ℓ ] ≤ 2 exp ( -ℓ 2 / 2 σ 2 + Mℓ/ 3 ) + Pr [ W τ ≤ σ 2 ] . In order to use Freedman's inequality, we consider nT random variables X 1 , . . . , X nT , where X iT + t = 1 T ( ( V ( t ) n -i -ˆ r n -i -1 ) + -E [ ( V ( t ) n -i -ˆ r n -i -1 ) + ]) for i ∈ { 0 , . . . , n -1 } and t ∈ { 1 , . . . , T } . By this definition,

because V ( t ) n -i and V n -i are identically distributed and independent of ˆ r n -i -1 . Moreover, for all j ,

Let E be the event that ∑ nT j =1 E [ X 2 j ∣ ∣ X 1 , . . . , X j -1 ] > σ 2 := e e +1 ln(2 e/δ ) T . By Freedman's inequality, using T ≥ (5 ln(2 e/δ ) /ϵ ) 2 , we have

It remains to show that Pr [ E ] ≤ δ 2 . To this end, we observe that for any i ∈ { 0 , . . . , n -1 } and t ∈ { 1 , . . . , T } ,

where the inequality uses E [ Y ] ≥ E [ Y 2 ] ≥ E [ Y 2 ] -( E [ Y ]) 2 = E [ ( Y -E [ Y ]) 2 ] for any random variable Y with 0 ≤ Y ≤ 1 almost surely. In total, we have

As ∑ n i =1 (ˆ r i -ˆ r i +1 ) = ˆ r 1 ≤ 1 almost surely, we have by Lemma 5 (see below) that

with probability at most δ 2 . Therefore, we have

Lemma 5. Let Y 1 , Y 2 , . . . , Y n be a sequence of (not necessarily independent) random variables in [0 , 1] such that ∑ n i =1 Y i ≤ 1 almost surely. Then, for any δ > 0 , with probability at most δ , we have

We defer the proof of Lemma 5 to the appendix.

2.2 Negative Result for Revenue: Proof of Theorem 2

Each buyer i = 1 , . . . , n has a marginal value distribution that could be D H i ('High') or D L i ('Low'):

which are valid distributions as long as ε ≤ 1 / 32 and n ≥ 2 . All 2 n configurations of whether each buyer has the High or Low distribution are possible.

Optimal policy Fix any configuration of whether each buyer has the High or Low distribution. Let r ∗ i denote the expected revenue to be earned under the optimal dynamic program, if buyer i is about to arrive and the item is not yet sold. We show inductively that r ∗ i = n +1 -i 4 n . By definition r ∗ n +1 = 0 , establishing r ∗ i = n +1 -i 4 n for i = n +1 . Now consider i = n, . . . , 1 , and assume r ∗ i +1 = n -i 4 n . Note that it is better to offer one of the prices 1 2 + n -i 4 n or 1 4 + n -i 4 n than to reject the buyer by offering a price of 1, because both of these prices are greater than r ∗ i (by the induction hypothesis). Thus, if buyer i has distribution D H i , then

Similarly, if buyer i instead has D L i , then

In either case, we have r ∗ i = n +1 -i 4 n , completing the induction. Note that r ∗ 1 = 1 / 4 .

Bounding a policy's objective by its number of mistakes Now, consider an arbitrary policy π = ( π 1 , . . . , π n ) decided by the learning algorithm. Let r i denote its expected revenue earned if buyer i is about to arrive and the item is not yet sold (under the true configuration of buyer distributions). We can without loss assume π to lie in { 1 2 + n -i 4 n , 1 4 + n -i 4 n } n , because both prices are higher than r i +1 , the expected revenue from rejecting (both prices are in fact higher than r ∗ i +1 , an upper bound on r i +1 ). We say that the policy makes a mistake for buyer i if either π i = 1 4 + n -i 4 n when i has the High distribution, or π i = 1 2 + n -i 4 n when i has the Low distribution. If the policy makes a mistake for i , then we have

on the other hand, if the policy does not make a mistake for i , then we have

Hence, if the policy makes M mistakes for some M ∈ { 1 , . . . , n } , then

Now, observe that n -i 4 n -r i +1 = r ∗ i +1 -r i +1 ≤ 4 ε n min { n -i, M } , because the loss from making a mistake is at most 4 ε n , and starting from buyer i + 1 , the number of mistakes can be at most n -( i +1) + 1 = n -i and also at most M . Therefore,

Recalling that r ∗ 1 = 1 / 4 , this shows the additive error is Ω( ε ) as long as the fraction of mistakes M/n is a constant.

Computing the Hellinger distance We first analyze the Hellinger distance H ( D H i , D L i ) between (a single observation of) D H i vs. D L i , which we note does not depend on the buyer i . The squared Hellinger distance can be bounded using 1 -H 2 ( D H i , D L i )

where the inequality applies Lemma 6 below to each square root. This shows that H 2 ( D H i , D L i ) = O ( ε 2 n ) , and the squared Hellinger distance is additive across independent samples. Using the fact that the Total Variation distance is upper-bounded by √ 2 times the Hellinger distance [GS02], we see that the Total Variation distance between T independent samples of D H i vs. T independent samples of D L i is O ( √ T n ε ) . The proof of Lemma 6 is deferred because it is elementary.

Completing the proof of Theorem 2 Suppose the distribution of each buyer is equally likely to be High or Low, independently across buyers. Fix any learning algorithm and the constant probability 1 / 2 . By the computation of Hellinger distance above, if the number of samples T is less than C n ε 2 for some constant C , then for any buyer i , the Total Variation distance between the samples observed under D H i vs. D L i is at most 1/2. ( C depends on the choice of constant 1/2.) This means that w.p. at least 1 -1 / 2 , the price π i decided by the learning algorithm cannot depend on whether buyer i had distribution D H i vs. D L i , which means that there exists an adversarial choice of D H i or D L i for each buyer i under which the probability of making a mistake for buyer i is at least (1 -1 / 2) / 2 = 1 / 4 .

Let M denote the (random) number of mistakes, under this adversarial configuration of whether each buyer i has distribution D H i or D L i . We have that E [ M n ] ≥ 1 / 4 . Although whether the algorithm makes a mistake could be arbitrarily correlated across buyers i , applying M n ≤ 1 , we can employ Markov's inequality on the random variable 1 -M n to see that

The LHS equals P [ M n ≤ 1 8 ] = 1 -P [ M n > 1 8 ] , and hence 1 7 ≤ P [ M n > 1 8 ] . We have shown that unless T = Ω( n ε 2 ) , there is probability at least 1/7 of making a constant fraction of mistakes, in which case we showed above that the additive error would be Ω( ε ) . This completes the proof of Theorem 2.

3 Correlated Distributions

In this section we prove our upper bound on the sample complexity of welfare/revenue maximization for correlated buyer distributions. We show nearly matching lower bounds in Appendix A.5.

3.1 Positive Result for Welfare and Revenue: Proof of Theorem 4

To bound the sample complexity of posted pricing for correlated distributions, it suffices to bound the pseudo-dimension of the policy class Π S . We use the same approach for both the welfare and revenue objectives. By standard learning theory results [BBL03], bounds on the pseudo-dimension translate to bounds on the sample complexity as follows.

Theorem 6. Let PDim(Π S ) denote the pseudo-dimension of Π S . For any ϵ > 0 , any δ ∈ (0 , 1) and any distribution D over [0 , 1] n , T = O ( 1 ϵ 2 (PDim(Π S ) + log 1 δ )) samples are sufficient to ensure that with probability at least 1 -δ over the draw of samples ( v 1 , . . . , v T ) ∼ D T , for all π ∈ Π S ,

We will show that PDim(Π S ) = O ( k log k ) , where k = |S| + 1 . This together with the above theorem immediately implies that Theorem 4 holds for the sample average approximation algorithm.

Let S = { i 1 , i 2 , i 3 , . . . , i k -1 } , where 1 < i 1 < i 2 < .. . < i k -1 . For each j = 1 , 2 , . . . , k , let I j = { i j -1 , . . . , i j -1 } , with the convention that i 0 = 1 and i k = n + 1 . In other words, I 1 , I 2 , . . . , I k are consecutive intervals that partition [ n ] , and Π S is the class of policies that offers every customer in I j the same price. Note that every policy π ∈ Π S can be parameterized by k prices ρ = ( ρ 1 , ρ 2 , . . . , ρ k ) , where ρ j is the price offered to customers in I j . In the rest of this proof, we will use π ρ to denote the policy in Π S parameterized by ρ .

By the definition of pseudo-dimension,

We will use a result from [BDD + 21] to bound the VC dimension of ˜ Π S . We state this result below in Definition 1 and Theorem 7. This result essentially says that the pseudo-dimension of the primal class is bounded if the dual class is well-structured. Here, 'well-structured' essentially means that the domain can be partitioned into pieces defined by a small number of boundary functions, and the function is simple on each piece.

Definition 1 (Definition 3.2 in [BDD + 21]) . A function class H ⊆ R Y that maps a domain Y to R is ( F , G , l ) -piecewise decomposable for a class G ⊆ { 0 , 1 } Y of boundary functions and a class F ⊆ R Y of piece functions if the following holds: for every h ∈ H , there are l boundary functions g (1) , . . . , g ( l ) ∈ G and a piece function f b ∈ F for each bit vector b ∈ { 0 , 1 } l such that for all y ∈ Y , h ( y ) = f b y ( y ) where b y = ( g (1) ( y ) , . . . , g ( l ) ( y )) ∈ { 0 , 1 } l .

The main theorem in [BDD + 21] states that if the dual class is ( F , G , l ) -piecewise decomposable, then the pseudo-dimension of the primal class is bounded.

Theorem 7 (Theorem 3.3 in [BDD + 21]) . Suppose that the dual function class U ∗ is ( F , G , l ) -piecewise decomposable with boundary functions G ⊆ { 0 , 1 } U and piece functions F ⊆ R U . The pseudo-dimension of U is bounded as follows:

We now apply Theorem 7 to bound the VC dimension of ˜ Π S . 5 To do so we must show that the dual class ˜ Π ∗ S is ( F , G , l ) -piecewise decomposable for 'nice' classes F and G . We will show that for both the welfare and revenue objectives, we can take F to be the family of constant functions and G to be the family of axis-aligned halfspaces. This will follow from the below lemma, whose proof is deferred to the Appendix for space reasons.

Lemma 7. Let v ∈ [0 , 1] n and let z ∈ R . Let G ( v , z ) = { ρ ∈ R k : π ρ ( v ) ≥ z } . For both the welfare and revenue objectives, there are l 1 , u 1 , · · · , l k , u k ∈ R ∪ {±∞} such that

Corollary 1. ˜ Π ∗ S is ( F , G , l ) -piecewise decomposable, where F is the set of constant functions, G is the set of axis-aligned halfspaces, and l = 2( |S| +1) .

Proof. Consider a function h ∈ ˜ Π ∗ S . By definition of ˜ Π ∗ S , there exist v ∈ [0 , 1] n and z ∈ R such that h ( ρ ) = 1 { π ρ ( v ) ≥ z } . By Lemma 7, there are l 1 , u 1 , . . . , l k , u k such that

For each j ∈ [ k ] , let L j = { ρ ∈ R k : ρ j > l j } and U j = { ρ ∈ R k : ρ j ≤ u j } . Note that L j and U j are axis-aligned halfspaces, and

Therefore, in the definition of ( F , G , l ) -piecewise decomposable, we may take the boundary functions to be the 2 k functions corresponding to the halfspaces L 1 , U 1 , . . . , L k , U k . On any given piece defined by these boundary functions, h is a constant function (equal to either 0 or 1).

For F the set of constant functions and G the set of axis-aligned halfspaces in R k , it is easy to check that PDim( F ∗ ) = 0 and VCdim( G ∗ ) = k . Combining Corollary 1 with Theorem 7, we get that

This completes the proof of Theorem 4.

Remark. If we directly apply Theorem 7 from [BDD + 21] to bound PDim(Π S ) , then we would get

A O ( k ln k ) pseudo-dimension bound for the revenue objective;

A O ( k ln( kn )) pseudo-dimension bound for welfare objective.

In particular, the bound for welfare would grow with n . This dependence on n is in fact unavoidable if one works with Π S directly, because for instances like v = ( 1 n , 2 n , . . . , 1) , the dual function corresponding to v is piecewise constant with Θ( n ) pieces, even if k = 1 . This is why our Corollary 1 focuses on the indicator functions ˜ Π ∗ S instead. Regardless, both the bounds for welfare and revenue require our Lemma 7 that analyzes the problem-specific structure of the 'good sets'.

Acknowledgement This work was done in part while the authors were visiting the Simons Institute for the Theory of Computing for the program on Data-Driven Decision Processes. The authors thank Zhuoxin Chen for identifying typos in an early version.

References

[AKS21] Sepehr Assadi, Thomas Kesselheim, and Sahil Singla. Improved truthful mechanisms for subadditive combinatorial auctions: Breaking the logarithmic barrier. In Proceedings of SODA , pages 653-661, 2021.

[AKW14] Pablo Daniel Azar, Robert Kleinberg, and S. Matthew Weinberg. Prophet inequalities with limited information. In Chandra Chekuri, editor, ACM-SIAM Symposium on Discrete Algorithms, SODA , pages 1358-1377, 2014.

[AM + 06] Lawrence M Ausubel, Paul Milgrom, et al. The lovely but lonely vickrey auction. Combinatorial auctions , 17(3):22-26, 2006.

[BBL03] Olivier Bousquet, Stéphane Boucheron, and Gábor Lugosi. Introduction to statistical learning theory. In Summer School on Machine Learning , pages 169-207. Springer, 2003.

[BDD + 21] Maria-Florina Balcan, Dan DeBlasio, Travis Dick, Carl Kingsford, Tuomas Sandholm, and Ellen Vitercik. How much data is sufficient to learn high-performing algorithms? generalization guarantees for data-driven algorithm design. In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing, STOC , page 919-932, 2021.

[CC23] José R. Correa and Andrés Cristi. A constant factor prophet inequality for online combinatorial auctions. In Proceedings of STOC , pages 686-697, 2023.

[CDF + 22] Constantine Caramanis, Paul Dütting, Matthew Faw, Federico Fusco, Philip Lazos, Stefano Leonardi, Orestis Papadigenopoulos, Emmanouil Pountourakis, and Rebecca Reiffenhäuser. Single-sample prophet inequalities via greedy-ordered selection. In ACM-SIAM Symposium on Discrete Algorithms, SODA , pages 1298-1325, 2022.

[CDFS22] José Correa, Paul Dütting, Felix A. Fischer, and Kevin Schewior. Prophet inequalities for independent and identically distributed random variables from an unknown distribution. Math. Oper. Res. , 47(2):1287-1309, 2022.

[CFH + 19] Jose Correa, Patricio Foncea, Ruben Hoeksma, Tim Oosterwijk, and Tjark Vredeveld. Recent developments in prophet inequalities. ACM SIGecom Exchanges , 17(1):61-70, 2019.

[CFPV19] José R. Correa, Patricio Foncea, Dana Pizarro, and Victor Verdugo. From pricing to prophets, and back! Oper. Res. Lett. , 47(1):25-29, 2019.

[CHK07] Shuchi Chawla, Jason D. Hartline, and Robert Kleinberg. Algorithmic pricing via virtual valuations. In Proceedings of the 8th ACM Conference on Electronic Commerce , EC '07, page 243-251, 2007.

[CMS10] Shuchi Chawla, David L. Malec, and Balasubramanian Sivan. The power of randomness in bayesian optimal mechanism design. In Proceedings of the 11th ACM Conference on Electronic Commerce , EC '10, page 149-158, 2010.

[CR14] Richard Cole and Tim Roughgarden. The sample complexity of revenue maximization. In Symposium on Theory of Computing, STOC , pages 243-252. ACM, 2014.

[DK19] Paul Dütting and Thomas Kesselheim. Posted pricing and prophet inequalities with inaccurate priors. In Conference on Economics and Computation, EC , pages 111-129. ACM, 2019.

[DKL20] Paul Dütting, Thomas Kesselheim, and Brendan Lucier. An o(log log m) prophet inequality for subadditive combinatorial auctions. In Proceedings of FOCS , pages 306-317, 2020.

[DKL + 24] Paul Duetting, Thomas Kesselheim, Brendan Lucier, Rebecca Reiffenhauser, and Sahil Singla. Online combinatorial allocations and auctions with few samples. In 65th IEEE Annual Symposium on Foundations of Computer Science, FOCS , 2024.

[FGL15] Michal Feldman, Nick Gravin, and Brendan Lucier. Combinatorial auctions via posted prices. In Proceedings of SODA , 2015.

[Fre75] David A Freedman. On tail probabilities for martingales. the Annals of Probability , pages 100-118, 1975.

[GHTZ21] Chenghao Guo, Zhiyi Huang, Zhihao Gavin Tang, and Xinzhi Zhang. Generalizing complex hypotheses on product distributions: Auctions, prophet inequalities, and pandora's problem. In Conference on Learning Theory, COLT , pages 2248-2288. PMLR, 2021.

[GHZ19] Chenghao Guo, Zhiyi Huang, and Xinzhi Zhang. Settling the sample complexity of single-parameter revenue maximization. In Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing, STOC , pages 662-673. ACM, 2019.

[GKSW24] Khashayar Gatmiry, Thomas Kesselheim, Sahil Singla, and Yifan Wang. Bandit algorithms for prophet inequality and pandora's box. In Proceedings of the thirty-sixth annual ACM-SIAM symposium on Discrete Algorithms, SODA , 2024.

[GM22] Xinyi Guan and Velibor V Miši´ c. Randomized policy optimization for optimal stopping. arXiv preprint arXiv:2203.13446 , 2022.

[GS02] Alison L Gibbs and Francis Edward Su. On choosing and bounding probability metrics. International statistical review , 70(3):419-435, 2002.

[Har13] Jason D Hartline. Mechanism design and approximation. Book draft. October , 122(1), 2013.

[HKS07] Mohammad Taghi Hajiaghayi, Robert D. Kleinberg, and Tuomas Sandholm. Automated online mechanism design and prophet inequalities. In Proceedings of the Twenty-Second AAAI Conference on Artificial Intelligence , pages 58-65. AAAI Press, 2007.

[HR09] Jason D. Hartline and Tim Roughgarden. Simple versus optimal mechanisms. In Proceedings 10th ACM Conference on Electronic Commerce (EC-2009) , pages 225-234. ACM, 2009.

[ISW20] Nicole Immorlica, Sahil Singla, and Bo Waggoner. Prophet inequalities with linear correlations and augmentations. In Conference on Economics and Computation, EC , pages 159-185, 2020.

[KL03] Robert D. Kleinberg and Frank Thomson Leighton. The value of knowing a demand curve: Bounds on regret for online posted-price auctions. In 44th Symposium on Foundations of Computer Science, FOCS , pages 594-605. IEEE Computer Society, 2003.

[KS77] U. Krengel and L. Sucheston. Semiamarts and finite values. Bulletin of the American Mathematical Society , 83:745-747, 1977.

[KS78] U. Krengel and L. Sucheston. On semiamarts, amarts, and processes with finite value. Advances in Probability and Related Topics , 4:197-266, 1978.

[LPS24] Vasilis Livanos, Kalen Patton, and Sahil Singla. Improved mechanisms and prophet inequalities for graphical dependencies. In Conference on Economics and Computation, EC , 2024.

[LSTW23] Renato Paes Leme, Balasubramanian Sivan, Yifeng Teng, and Pratik Worah. Pricing query complexity of revenue maximization. In Proceedings of the thirty-fourth annual ACM-SIAM symposium on Discrete Algorithms , 2023.

[Luc17] Brendan Lucier. An economic view of prophet inequalities. ACM SIGecom Exchanges , 16(1):24-47, 2017.

[MR16] Jamie Morgenstern and Tim Roughgarden. Learning simple auctions. In Proceedings of the 29th Conference on Learning Theory, COLT 2016 , volume 49 of JMLR Workshop and Conference Proceedings , pages 1298-1318. JMLR.org, 2016.

[Mye81] Roger B. Myerson. Optimal auction design. Math. Oper. Res. , 6(1):58-73, 1981.

[Rou16] Tim Roughgarden. Twenty lectures on algorithmic game theory . Cambridge University Press, 2016.

[RWW] Aviad Rubinstein, Jack Z. Wang, and S. Matthew Weinberg. Optimal single-choice prophet inequalities from samples. In 11th Innovations in Theoretical Computer Science Conference, ITCS , volume 151, pages 60:1-60:10.

[SW24] Sahil Singla and Yifan Wang. Bandit sequential posted pricing via half-concavity. In Conference on Economics and Computation, EC . ACM, 2024.

[Vic61] William Vickrey. Counterspeculation, auctions, and competitive sealed tenders. The Journal of finance , 16(1):8-37, 1961.

[Wai19] Martin J Wainwright. High-dimensional statistics: A non-asymptotic viewpoint , volume 48. Cambridge university press, 2019.

[XMX24] Yaqi Xie, Will Ma, and Linwei Xin. Vc theory for inventory policies. arXiv preprint arXiv:2404.11509 , 2024.

A Appendix

A.1 Proof of Lemma 2

Proof. Now, consider the smallest j so that ˆ r j ≥ r ∗ j . Again, we know such a j exists because 0 = ˆ r n +1 ≥ r ∗ n +1 = 0 . We claim that that ˆ r 1 ≥ r ∗ 1 +min { ∑ j -1 i =1 η i , 0 } .

Observe that there is nothing to be shown if j = 1 because then ˆ r 1 ≥ r ∗ 1 . So, let's consider the case that j > 1 . We can now write ˆ r 1 as a telescoping sum

Since ˆ r j ≥ r ∗ j , we have

Observe that for i +1 < j , we have ˆ r i +1 < r ∗ i +1 . Therefore

So, in combination with (4),

A.2 Proof of Lemma 5

Ignoring constants, a slick proof of this lemma is via another application of Freedman's inequality, where we define a martingale X i = Y i -E [ Y i | Y 1 , . . . , Y i -1 ] , and cut this martingale (stop) whenever the sum of conditional variances exceeds log(1 /δ ) . This bounds the sum of conditional variances by log(1 /δ ) , i.e. W τ ≤ σ 2 := log(1 /δ ) w.p. 1. Since the lemma is upper-bounding the probability of the event that ∑ i X i > log(1 /δ ) , we can show that this probability is O ( δ ) by substituting ℓ = log(1 /δ ) into Freedman's inequality.

Below we give another, elementary proof of this lemma.

Proof. Let Z = 1 if ∑ n i =1 E [ Y i | Y 1 , . . . , Y i -1 ] ≥ e e -1 ln ( e δ ) . By Markov's inequality, we have

Now, we have

For every single conditional expectation, we obtain by convexity

Now, consider a fixed realization of Y 1 , . . . , Y n . This realization fully determines Z . If Z = 0 , then

Otherwise, with Z = 1 , we have

So, we have an upper bound of δ e regardless of the realization. Taking an expectation, we get

which implies Pr [ ∑ n i =1 Y i ≤ 1 and Z = 1] ≤ E [ Ze -∑ n i =1 Y i ] · e ≤ δ e e = δ , completing the proof.

A.3 Proof of Lemma 6.

Proof. Let f ( x ) = √ C ( C + x ) = ( C 2 + Cx ) 1 / 2 , which is a continuous function over x ∈ [ -3 4 C, 3 4 C ] . We have f ′ ( x ) = 1 2 ( C 2 + Cx ) -1 / 2 C and f ′′ ( x ) = -1 4 ( C 2 + Cx ) -3 / 2 C 2 , both of which exist over x ∈ [ -3 4 C, 3 4 C ] . Applying Taylor's theorem around 0, we get

where y lies between 0 and x and the inequality holds because y ≥ -3 4 C , completing the proof.

A.4 Proof of Lemma 7

If z ≤ 0 then G ( v , z ) = R k , so we may take any values of l 1 , u 1 , . . . , l k , u k such that l 1 = l 2 = · · · = l k -∞ and u k = ∞ .

For the rest of the proof, assume z > 0 . First, consider the welfare objective. We show the lemma holds for l j and u j as follows:

Define l j to be -∞ if v ( I j ) 1 ≥ z , and otherwise to be max { v ( I j ) 1 , . . . , v ( I j ) m -1 } , where m is the smallest index such that v ( I j ) m ≥ z . (If v ( I j ) m < z for all m , set l j = max( v ( I j )) .)

Define u j to be max( v ( I j )) .

Case 1 ( G ⊆ G ′ ). If G = ∅ we are done, so assume otherwise. Let ρ ∈ G . Let j be the interval such that the algorithm (using thresholds ρ ) accepts a value in v ( I j ) . Since a value in interval j was accepted, ρ j ≤ max( v ( I j )) = u j . Also, since no value in the previous intervals were accepted, ρ i > u i for all i < j . Finally, we must have ρ j > l j , since otherwise the accepted value will be less than z . Thus ρ ∈ G ′ .

Case 2 ( G ′ ⊆ G ). Let ρ ∈ G ′ . Then ρ ∈ ( u 1 , ∞ ) ×··· × ( u j -1 , ∞ ) × ( l j , u j ] × R k -j for some j . Since ρ i > u i for all i < j , the algorithm using thresholds ρ does not accept any value in the intervals i < j . Since ρ j ∈ ( l j , u j ] , the definitions of l j and u j imply that a value in v ( I j ) is accepted, and this value is greater than or equal to z . Thus ρ ∈ G .

For revenue, the only difference is that we define l j to be min( z, max( v ( I j ))) . ( u j is still defined to be max( v ( I j )) .) With these definitions of l j and u j , a very similar analysis shows that G = G ′ in the revenue case as well.

A.5 Proof of Theorem 5

We let S ′ ⊆ { 1 } ∪ S be a subset of decision points such that price π i can be freely chosen for all i ∈ S ′ . For welfare maximization we require i + 1 ∈ { 1 , . . . , n } \ S ′ for all i ∈ S ′ , which can achieved while ensuring |S ′ | ≥ ⌊ 1+ |S| 2 ⌋ . Each decision point i ∈ S ′ has marginal value distribution(s) that could be 'High' or 'Low', separately for each decision point (i.e., all 2 |S ′ | combinations are possible). For welfare maximization, they are defined as

High: V i = 1 / 2 with probability (w.p.) 1, V i +1 = 1 w.p. 1 / 2 + ε , V i +1 = 0 w.p. 1 / 2 -ε ;

Low: V i = 1 / 2 w.p. 1, V i +1 = 1 w.p. 1 / 2 -ε , V i +1 = 0 w.p. 1 / 2 + ε .

For revenue maximization, they are defined as

High: V = 1 w.p. 1 / 2 + ε , V = 1 / 2 w.p. 1 / 2 -ε ;

Low: V = 1 w.p. 1 / 2 -ε , V = 1 / 2 w.p. 1 / 2 + ε .

The overall distribution over trajectories V is then correlated as follows. First, a decision point ˜ ı ∈ S ′ is drawn uniformly at random. Then, V ˜ ı (as well as V ˜ ı +1 in the case of welfare maximization) is drawn according to the distributions above, depending on whether decision point ˜ ı is High or Low. All other buyer values are 0 on this trajectory.

Given this, only the prices for decision point ˜ ı are relevant, which can without loss be restricted to { 1 / 2 , 1 } . The policy should optimize π i for each i ∈ S ′ as if ˜ ı = i . For either welfare or revenue maximization, the constructions above can be checked to satisfy the following properties:

Setting π i (and π i +1 in the case of welfare maximization) to 1 is optimal for High and earns objective 1 / 2 + ε in expectation; setting to 1/2 is optimal for Low and earns objective 1 / 2 ;

Setting the wrong price earns expected objective 1/2 for High and 1 / 2 -ε for Low, incurring a loss of ε compared to optimal in both cases.

The High and Low distributions have the same support and probabilities that differ by 2 ε .

We note that for welfare maximization, it does not matter whether π i +1 ∈ S , because the decision is on whether to accept buyer i when V i = 1 / 2 .

Finally, a policy has additive error Ω( ε ) as long as it has constant probability of setting the wrong price for a decision point. Due to property 3. above, the policy needs Ω( 1 ε 2 ) relevant observations for a given decision point to avoid setting the wrong price for it with constant probability. However, each sample contains a relevant observation for a given decision point only with probability 1 / |S ′ | , so Ω( |S ′ | ε 2 ) samples are necessary for the number of relevant observations to be Ω( 1 ε 2 ) with high probability. Because |S ′ | ≥ ⌊ 1+ |S| 2 ⌋ , this completes the proof of Theorem 5.

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Peer reviews:
comments: SUMMARY:
This paper studies the sample complexity of posted pricing problems. In such problems, a sequence of $n$ buyers arrive with valuations drawn from fixed distributions, and the goal is to post prices that maximize either the revenue or the welfare. In particular, the item is sold to the first buyer who accepts its proposed price. Notably, in the welfare maximizing case, this is the prophet inequality problem. 

Given access to the distributions of the buyers’ valuations, it is possible to find the optimal online strategy via dynamic programming. This paper studies this problem when the access to the distribution is given via samples. The authors provide matching sample complexity bounds for the revenue and welfare maximization tasks, and for both the independent and correlated setting.

STRENGTHS:
- Posted pricing is an important problem, vastly studied in both the EC and NeurIPS/ICML communities, with countless applications
- Studying its sample complexity is well-motivated
- the paper significantly improves on the state-of-the-art [Guo et al. COLT’21], removing the dependence on $n$
- The paper is well-written

WEAKNESSES:
The paper is a strong NeurIPS submission

QUESTIONS:
None

LIMITATIONS:
No potential negative societal impact of this work
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-10

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the problem of learning approximately revenue (welfare) optimal posted-prices single-item auctions from samples. The authors consider the setting where the valuation distributions of the agents are independent and the setting where the distributions are correlated. For both settings and both objectives they derive (almost) matching lower and upper bounds on the sample complexity of the learning task. All the results restrict the valuation distributions to be bounded in $[0,1]$ and consider additive approximations of the optimal posted-prices auctions.\n\nSTRENGTHS:\n-The problem of learning posted-prices auctions is of interest to the community.\n\n-The results derived are almost tight. \n\n-The technical results require some work (even though, conceptually, the approach relies heavily on prior work).\n\nWEAKNESSES:\n-The conceptual ideas of the work feel a bit incremental, given the long line of work in the setting of learning optimal auctions. \n\n-The results are limited to a very simple setting of single item auctions.\n\n-The running time of the algorithm for correlated distributions is exponential. \n\n-The presentation could be improved a bit. For example, I found the use of the word \"policy\" a bit confusing. Usually, this would imply some dynamic decision process, whereas here there is only a single-shot setting. It would be good to use different symbols for the revenue/welfare objectives. Line 61: instead of $V$ use something that denotes a set of samples. Line 104: what is the value-to-go? The proof sketch of Theorem 5 is a bit confusing. Lines 157-158 \"single item\" is repeated. The notation $r_i, \\hat{r}_i, r^*_i$ in Section 2.1 is a bit confusing. In Section 2.2 you can add a quick comment that choosing a price randomly will not help. In Line 23 \"know\" appears twice. In Line 173 drop \"a\". It might be useful to include the definition of the pseudo-dimension at least in the appendix.\n\nQUESTIONS:\n-What settings other than single-item auctions can the results be extended to?\n\n-For other types of distributions such as regular/MHR, can you derive optimal bounds?\n\n-For the bound on the pseudo-dimension, can't you use a direct argument like Morgenstern and Roughgarden (2016) for $t$-level auctions? I am not sure I understand the need to go through the dual class.\n\n-Can you comment a bit on the differences of your results/approach and [GHTZ'21]?\n\nLIMITATIONS:\nAddressed.", "TITLE": "Official Review", "RECOMMENDATION": 6, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 4, "DATE": "2024-07-15"}

comments: SUMMARY:
The paper studies a setup of a series posted-price auctions that tries to sell a single item up to the first price acceptance. The authors seeking for the number of samples from buyer value distributions (sample complexity) to be able to setup near-optimal posted-price in the auctions. They consider two objectives (social welfare and revenue maximization) and different dependencies of buyer distributions (independent and correlated). The paper contributes proven both upper and lower on the sample complexity for the all four settings: the bounds are close up to logarithmic factors.

STRENGTHS:
-	Clear math contribution

-	Well-structured proofs

WEAKNESSES:
-	No conclusions: the paper does not have a concluding discussion

-	No experimental evidence: asymptotic bounds are clear measures for growing (shrinking) parameters, but, for practice, constant factors are more important. It would be nice to understand how they work in practical cases.

QUESTIONS:
-	What are the factors (multiplicative constants) in the upper and lower bounds in Theorems 1-4?  What is behind O(...) and \Omega(...)?

LIMITATIONS:
the authors adequately addressed the limitations
TITLE: Official Review
SOUNDNESS_CORRECTNESS: 4
DATE: 2024-07-15

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper considers the problem of statistically estimating the optimal posted price mechanism for selling one item to multiple buyers. Here, it is assumed that the buyers appear sequentially, a price is presented to each of them and the sale is completed if the price posted to them is lower than their valuation. The mechanism aims to optimize one of two targets -- the welfare, defined here as the value of the bidder who wins the auction and revenue of the auctioneer which is the price paid by the winning bidder. It is assumed that the valuations of the bidders are drawn from a distribution and the paper aims to analyze the sample complexity of determining near-optimal posted prices for each of these settings where it is assumed that the learner receives iid samples from the bid distributions. Depending on the degree of dependence between the valuations of the bidders, the paper considers two settings -- firstly, the independent setting where the values are determined independently for each bidder and the dependent setting where the valuations are correlated across the bidders. \n\nIn the independent setting, it is shown that there exists a separation between the sample complexities of welfare and revenue maximization. They show that essentially $1 / \\epsilon^2$ samples suffice for welfare maximization whereas revenue maximization necessarily \\emph{requires} $\\Omega (n / \\epsilon^2)$ samples where $n$ is the number of bidders and $\\epsilon$ an additive error parameter. In this case, they obtain near-optimal characterization of the sample complexity while the upper and lower bounds from prior work are $n / \\epsilon^2$ and $1 / \\epsilon^2$ respectively. In contrast, when the bids are correlated, the paper shows that $\\Omega (n / \\epsilon^2)$ is \\emph{required} for both revenue and welfare maximization. To partially address this pessimistic bound, the paper also considers the setting where the class of possible posted prices is restricted. The particular restriction considered in the paper is by restricting the change points of the price schedule; that is, the price is only allowed to change at $k$ points in the sequence of prices. Here, it is shown that sample complexities independent of $n$ are obtainable and instead only depend on $k$. \n\nThe proof of the upper bound independent of $n$ for welfare maximization in the independent valuation setting is quite interesting. Essentially, the paper analyzes the dynamic programming algorithm for welfare maximization. Observing that the (welfare of the) posted prices have a closed-form solution in terms of the welfare of the next step, the paper proves that the sub-optimality of the posted prices may be bounded by a sum of error terms which satisfy a backwards martingale structure. The use of standard Martingale concentration bounds then yield the required bound. This proof is elegant and appears to be novel. Unfortunately, such martingale structure does not hold in the presence of dependencies between the valuations. Instead, they adopt a classical uniform convergence based approach and show that the statistical complexity may be controlled by the pseudo-dimension of the policy class.\n\nOverall, this is a nice paper that obtains nearly optimal statistical characterizations in several fundamental settings, making several interesting contributions. The proofs of the results, the welfare maximization for independent valuations in particular, are elegant and well-presented. My main concerns are with the assumptions underlying the learning problem. The paper assumes that one obtains independent samples from the \\emph{valuations} of the bidders. It is not clear how such samples are obtained -- it is not clear how one may obtain such samples in practice. More exposition on this point would be helpful. Furthermore, it would be helpful if the authors could comment on alternative settings circumventing the $\\Omega (n / \\epsilon^2)$ sample complexities incurred by the paper -- perhaps, by bounding the degree of dependence between the bidders?\n\nSTRENGTHS:\nSee main review\n\nWEAKNESSES:\nSee main review\n\nQUESTIONS:\nSee main review\n\nLIMITATIONS:\nSee main review", "TITLE": "Official Review", "RECOMMENDATION": 7, "REVIEWER_CONFIDENCE": 4, "SOUNDNESS_CORRECTNESS": 3, "DATE": "2024-07-15"}