SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Our objective is to design a communication-efficient, active learning query protocol for the seller that interacts with each bidder  $i$ , and infers their type  $\mathbf{t}_i \in \mathbb{R}^d$  by accessing only a small subset of elements of the type vector (as  $d$  is very large). We use  $\mathcal{Q}$  to denote the query protocol, whose output is a vector in the low-dimensional latent space  $\mathbb{R}^k$ . A bidder interacts with the query protocol truthfully if it is in her best interest to evaluate functions requested by the protocol on her true private type  $\mathbf{t}_i$ . We use  $\mathcal{Q}(\mathbf{t}_i) \in \mathbb{R}^k$  to denote the output of  $\mathcal{Q}$  when interacting with a truthful bidder with type  $\mathbf{t}_i$ . We now define the notion of an  $(\varepsilon_{\mathrm{mdl},p}, \zeta_p, p)$ -query protocol and the notion of query noise.
Definition 3  $((\varepsilon_{\mathsf{mdl},p},\zeta_p,p)$ -query protocol).  $\mathcal{Q}$  is called an  $(\varepsilon_{\mathsf{mdl},p},\zeta_p,p)$ -query protocol, if, for all  $\mathbf{t} \in \mathbb{R}^d$  and  $\mathbf{z} \in \mathbb{R}^k$  satisfying  $\| \mathbf{t} - \mathbf{A}\mathbf{z}\|_p \leq \varepsilon_{\mathsf{mdl},p}$ , we have  $\| \mathbf{z} - \mathcal{Q}(\mathbf{t})\|_p \leq \zeta_p$ .
Definition 4 (Query noise). Let  $\mathbf{t}_i$  be the true type of a bidder. Our query protocol can access entries of  $\mathbf{t}_i + \epsilon_{nq,p}$ , where  $\epsilon_{nq,p}$  is an (unknown) vector. The query noise  $\varepsilon_{nq,p}$  satisfies  $\| \epsilon_{nq,p} \|_p \leq \varepsilon_{nq,p}$ .
The query noise depends on the specifics of the interactions between the seller and the bidder. For example, if the seller is only allowed to ask queries of the form "what is your value for the subset  $S$ ?", the query noise  $\varepsilon_{\mathrm{sq},p}$  is equal to zero. Our bounds will also depend on the model error.
Definition 5 (Model error). Given a valuation profile  $\mathbf{t} \in \mathbb{R}^{nd}$ , the model error is  $\varepsilon_{mdl,p}$  if, for all  $i \in [n]$ , there exists a  $\mathbf{z}_i \in \mathbb{R}^k$  such that  $\| \mathbf{t}_i - \mathbf{A}\mathbf{z}_i \|_p \leq \varepsilon_{mdl,p}$ .
Note that, we don't have bounds of the form “ $\|\mathbf{t}_i - \mathbf{A}\mathbf{z}_i\| \leq \varepsilon$ ” for individual types, but for the distributions  $\mathcal{D}_i$  and  $\mathbf{A} \circ \hat{\mathcal{D}}_{z,i}$ . The characterization of Prokhorov distance (Lemma 1) allows us to relate the two quantities in the proofs that follow.
5. Notice that we consider the more general case of having  $d$  number of rows (instead of  $m$ ).
We now rephrase the above discussion in order to cast it in the framework of Randomized Linear Algebra (RLA) and active learning. Dropping the index  $i$  for notational simplicity, we assume that  $\mathbf{A}\mathbf{z} \approx \mathbf{t}$  and we seek to recover an approximate solution vector  $\mathcal{Q}(\mathbf{t})$  such that the  $\ell_p$  norm error between the approximate and the optimal solution is bounded. Importantly, the query protocol  $\mathcal{Q}$  is not allowed full access to the vector  $\mathbf{t}$  in order to construct the approximate solution vector. This is a well-studied problem in the context of RLA: the learner is given a large collection of  $k$ -dimensional data points (the  $d \gg k$  rows of the design matrix  $\mathbf{A} \in \mathbb{R}^{d \times k}$ ), but can only query a small subset of the real-valued labels associated with each data point (elements of the vector  $\mathbf{t} \in \mathbb{R}^d$ ). Prior work in RLA and active learning has studied this problem in order to identify the optimal number of queries that allow efficient, typically relative error, approximations of the loss function. In our parlance, prior work has explored the existence of query protocols that construct a vector  $\mathcal{Q}(\mathbf{t})$  such that
$$
\left\| \mathbf {t} - \mathbf {A} \mathcal {Q} (\mathbf {t}) \right\| _ {p} \leq \gamma_ {p} \| \mathbf {t} - \mathbf {A} \mathbf {z} \| _ {p}, \tag {1}
$$
where  $\gamma_p > 1$  is an error parameter that controls the approximation accuracy. Of particular interest in the RLA literature are relative error approximations, with  $\gamma_p = 1 + \epsilon$ , for some small  $\epsilon > 0$ ; see [MMWY21, MMWY22] for a detailed discussion. However, relative error approximations are less important in our setting, since our protocols in Section 3.2 necessitate  $\zeta_p \geq \varepsilon_{\mathrm{mdl},p}$ . For  $p = 2$ , the underlying problem is active learning for least-squares regression: [DMMS11] analyzed its complexity (namely, the number of queries) of query protocols in this setting, eventually providing matching upper and lower bounds. Similarly, for  $p = 1$ , the underlying problem is active learning for least absolute deviation regression, a robust version of least-squares regression: [MD21] analyzed the complexity of query protocols in this setting. The query protocols of [DMMS11, MD21] are straightforward: they sample a small set of labels (i.e., bidder types) and elicit the bidder's preferences for this set. Then, the respective  $\ell_p$  norm regression problem is solved on the smaller set and the resulting solution is returned as an approximation to the original solution. The types to be sampled (see Appendix D.1 for details) are selected using distributions that can be constructed by accessing only the design matrix  $\mathbf{A}$ . Specifically, for the  $p = 2$  case, one needs to compute or approximate the leverage scores of the rows of the matrix  $\mathbf{A}$ . For the  $p = 1$  case, one needs to compute or approximate the Lewis weights of the design matrix  $\mathbf{A}$ . (The Lewis weights are an extension of the leverage scores to  $\ell_p$  norms for  $p \neq 2$ .) The work of [DW17, DWH18, DM21, DW18, CP19] for the  $p = 2$  case involves more elaborate query protocols, using primitives such as volume sampling and the Batson-Spielman-Srivastava sparsifier to improve the query complexity. Finally, the  $p > 2$  case for active learning for regression problems was recently resolved in [MMWY22, MMWY21]; we discuss their approach in our context in Appendix D.2.
To the best of our knowledge, our work is the first one to formulate connections between mechanism design and Randomized Linear Algebra for active learning. Two technical points of departure that are needed in order to adapt the RLA work for active learning to the mechanism design framework are: (i) we need to derive bounds of the form of eqn. (1) for the  $\ell_p$  norm distance between the exact and approximate solutions, whereas prior work typically bounds the error of the loss function when an approximate solution is used; and (ii) the entries of the bidder's type vector  $\mathbf{t}$  might not be known exactly, but only up to a small error. The latter assumption corresponds to the use of noisy queries in the model of [CD22] and is known to be equivalent, up to logarithmic factors, to threshold queries via binary search. Our work addresses both technicalities and seamlessly combines the RLA work for active learning with mechanism design.
Prior to stating our main result, we need to define a fundamental property of the design matrix  $\mathbf{A} \in \mathbb{R}^{d \times k}$  that will affect the approximation error. Let
$$
\sigma_ {\min , p} (\mathbf {A}) = \min  _ {\mathbf {x} \in \mathbb {R} ^ {k}, \| \mathbf {x} \| _ {p} = 1} \| \mathbf {A x} \| _ {p}. \tag {2}
$$
For  $p = 2$ , this is simply the smallest singular value of the matrix  $\mathbf{A}$ . For other values of  $p$ , the above definition is the standard generalization of the smallest singular value of  $\mathbf{A}$  for the induced matrix  $\ell_p$  norm. Notice that  $\sigma_{\min,p}(\mathbf{A})$  is a property of the matrix  $\mathbf{A}$  and can be computed a priori via, say, the QR factorization or the Singular Value Decomposition (SVD) for  $p = 2$  and via linear programming for  $p = 1$ . As we will see in Theorem 1 below, smaller values of  $\sigma_{\min,p}(\mathbf{A})$  result in increased sample complexity for our query protocols.
To be precise, multiple smaller problems have to be solved and a "good enough" solution has to be chosen in order to boost the success probability. See Appendix D for details.
Theorem 1. Let  $\mathbf{A} \in \mathbb{R}^{d \times k}$  be the design matrix, and recall the definitions of the model error  $\varepsilon_{\text{mdl},p}$  (Definition 5) and the query noise (Definition 4). For all integers  $1 \leq p < \infty$ , there exist query protocols  $\mathcal{Q}$  using  $s_p$  queries for each bidder  $i \in [n]$ , such that, with probability at least  $1 - \delta$ ,
$$
\left\| \mathbf {z} _ {i} - \mathcal {Q} (\mathbf {t} _ {i}) \right\| _ {p} \leq \frac {c _ {p} \left(\varepsilon_ {m d l , p} + \varepsilon_ {n q , p}\right)}{\sigma_ {\operatorname* {m i n} , p} (\mathbf {A})} = \zeta_ {p}
$$
holds for all  $n$  bidders  $i \in [n]$ . Here  $c_p$  is a small constant that depends on  $p$ . The respective query complexities for  $p = 1$  and  $p = 2$  are (asymptotically) identical:
$$
s _ {1} = s _ {2} = O \left(k \cdot \ln k \cdot \ln^ {n} / \delta\right). \tag {3}
$$
For  $p \geq 3$ , the query complexity is
$$
s _ {p} = O \left(k ^ {p / 2} \cdot \ln^ {3} k \cdot \ln^ {n} / \delta\right). \tag {4}
$$
Several comments are in order. (i) The error  $\zeta_{p}$  is a small constant times the modelling error plus the error due to noisy queries. In the limit case where the modelling error is equal to zero and the queries are noiseless, the bidders' types can be recovered exactly in our framework. However, as the modelling error and the query noise increase, approximating user types becomes harder and less accurate. (ii) Importantly, the approximation accuracy of Theorem 1 grows linearly with the inverse of the smallest  $\ell_{p}$  norm singular value of the design matrix  $\mathbf{A}$ . Our results indicate that the approximation accuracy of the query model  $\mathcal{Q}$  depends on this simple property of the design matrix  $\mathbf{A}$ . For example, focusing on the  $p = 2$  case, our theorem shows that as the archetypes (columns of the matrix  $\mathbf{A}$ ) become more linearly dependent and the smallest singular value approaches zero, the error of our approximation worsens. This is quite reasonable: if archetypes are linearly dependent, then it is increasingly difficult to approximate the respective entries of the vector  $\mathbf{z}$ . (iii) The query complexities  $s_1$  and  $s_2$  are asymptotically identical, growing linearly with  $k\ln k$ , where  $k$  is the number of archetypes. They both depend on the log of the number of bidders (due to a union bound) and on the log of  $1/\delta$ , where  $\delta$  is the failure probability. The query complexity for  $p \geq 3$  is larger and is dominated by the  $k^{p/2}$  term. Importantly, the query complexity remains independent of  $d$ , the number of bidder types, which, in worst case, could be exponential to the number of underlying items. (iv) Improving the sampling complexities  $s_1$  and  $s_2$  has been a topic of intense interest in the RLA community and we defer the reader to [MMWY21, MMWY22], which has essentially provided matching upper and lower bounds for various values of  $p$ . We just note that for the well-studied  $p = 2$  case, volume sampling approaches [DW17, DWH18, DM21, DW18] achieve essentially matching bounds, while the work of [CP19] removes (at least in expectation) the  $\ln k$  factor from  $s_2$ , at the expense of significant additional protocol complexity. From a technical perspective, we note that  $\zeta_p \geq \varepsilon_{\mathrm{md1}, p}$ , as necessitated in Theorem 2 and that our query protocols are all one-round protocols.
Finally, notice that our theorem works for all  $p \geq 1$ , but not for  $p = \infty$ , which was the setting of [CD22]. In Appendix D.4, we present a (modest) improvement of the result of [CD22] and explain why it seems improbable that the  $p = \infty$  case can be generalized to a much broader class of design matrices. This is a strong motivating factor to explore properties of mechanism design for the recommender system setting for other  $\ell_p$  norms, as we do in this work.
3.2 The Mechanism Design component
The goal of the mechanism design component is to transform a mechanism  $\hat{\mathcal{M}}$  for  $\hat{\mathcal{D}}$  into a mechanism  $\mathcal{M}$  for  $\mathcal{D}_z$ . We first define exactly the type of access to  $\hat{\mathcal{D}}_z$  and  $\hat{\mathcal{M}}$  our construction requires.
Definition 6 (Access to  $\hat{\mathcal{M}}$ ). By "query access to  $\hat{\mathcal{M}}$  " we mean access to an oracle which, given a valuation profile  $\mathbf{t}$ , outputs the allocation and payments of  $\hat{\mathcal{M}}$  on input  $\mathbf{t}$ .
Definition 7 (Access to  $\hat{\mathcal{D}}_z$ ). By "oracle access to  $\hat{\mathcal{D}}$  " we mean access to (1) a sampling algorithm  $S_i$  for each  $i \in [n]$ , where  $S_i(\mathbf{x}, \delta)$  draws a sample from the conditional distribution of  $\hat{\mathcal{D}}_{z,i}$  on the  $k$ -dimensional cube  $\times_{j \in [k]}[x_j, x_j + \delta_j)$ , and (2) an oracle which, given as input a type  $\mathbf{t}_i$  for bidder  $i$ , outputs the type in the support of  $\hat{\mathcal{D}}_{z,i}$  that is closest to  $\mathbf{t}_i$  in  $\ell_p$  distance, i.e., outputs  $\text{argmin}_{\mathbf{t}_i' \in \text{supp}(\hat{\mathcal{D}}_{z,i})} \| \mathbf{t}_i - \mathbf{t}_i' \|_p$ .
<sup>7</sup>We make no attempt to optimize constants and focus on simplicity of presentation. In our proofs,  $c_{1} = 2.5$ ;  $c_{2} = 7.5$ ; and for  $p \geq 3$ ,  $c_{p} = 18 \cdot (200)^{1/p} + 3$ . Notice that the last constant converges to 21 as  $p$  increases.
If the allocation is randomized, our approach works even if the query to the oracle returns a (correct) deterministic instantiation of the randomized allocation.<sup>8</sup>
In Definition 7, the first part of our oracle access (sampling from the conditional) is also necessary in [CD22]. The second part is new to our work, and replaces a strong requirement in [CD22]. In more detail, given a type  $\mathbf{t}_i$ , Cai and Daskalakis [CD22] (as well as Brustle et al. [BCD20]) need to know if  $\mathbf{t}_i \in \text{supp}(\widetilde{\mathcal{D}}_{z,i})$ , and, if not, need access to  $\text{argmax}_{\mathbf{t}_i' \in \text{supp}(\widetilde{\mathcal{D}}_{z,i})} \mathbb{E}_{\mathbf{b}_{-i} \sim \widetilde{\mathcal{D}}_{z,-i}}[u_i(v_i, \hat{\mathcal{M}}(\mathbf{t}_i', \mathbf{b}_{-i}))]$  where  $\widetilde{\mathcal{D}}_{z,i}$  is a rounded-down version of  $\hat{\mathcal{D}}_{z,i}$ . However, for arbitrary distributions and mechanisms, this task might be computationally inefficient, or simply infeasible. In our work, we need access to  $\text{argmin}_{\mathbf{t}_i' \in \text{supp}(\widehat{\mathcal{D}}_{z,i})} \| \mathbf{t}_i - \mathbf{t}_i' \|_p$  in the "no" case.
Given these definitions, our main theorem for this component is stated as follows.
Theorem 2. Let  $\mathcal{D} = \times_{i=1}^{n}\mathcal{D}_{i}$  be the bidders' type distribution and  $v_{i}: \mathbb{R}^{d} \times 2^{[m]} \to \mathbb{R}_{+}$  be a  $\mathcal{L}$ -Lipschitz valuation function for each bidder  $i \in [n]$ . Also, let  $\mathbf{A} \in \mathbb{R}^{d \times k}$  be a design matrix and  $\hat{\mathcal{D}}_{z} = \times_{i=1}^{n}\hat{\mathcal{D}}_{z,i}$ , where  $\hat{\mathcal{D}}_{z,i}$  is a distribution over  $\mathbb{R}^{k}$  for each  $i \in [n]$ .
Suppose that we are given (1) query access to a mechanism  $\hat{\mathcal{M}}$  that is IR and BIC w.r.t.  $\hat{\mathcal{D}}_z$  and valuations  $\{v_i^{\mathbf{A}}\}_{i\in [n]}$ , (2) oracle access to  $\hat{\mathcal{D}}_z$ , and (3) any  $(\varepsilon_{mdl,p},\zeta_p,p)$ -query protocol  $\mathcal{Q}$  with  $\zeta_p\geq \varepsilon_{mdl,p}$ . Then, we can construct a mechanism  $\mathcal{M}$  that is oblivious to  $\mathcal{D}$  and  $v(\cdot ,\cdot)$ , such that for all  $\mathcal{D}$  that satisfy  $\pi_p(\mathcal{D}_i,\mathbf{A}\circ \hat{\mathcal{D}}_{z,i})\leq \varepsilon_{mdl,p}$  for all  $i\in [n]$ , the following hold: (1)  $\mathcal{M}$  only interacts with every bidder using  $\mathcal{Q}$  once, (2)  $\mathcal{M}$  is IR and  $(\eta ,\mu)$ -BIC w.r.t.  $\mathcal{D}$ , where  $\mu = O(\sqrt{\zeta_p})$  and  $\eta = O(n\| \mathbf{A}\|_{\infty}\mathcal{L}\sqrt{\zeta_p})$ , and (3) the expected revenue of  $\mathcal{M}$  is at least  $Rev(\hat{\mathcal{M}},\hat{\mathcal{D}}_z) - O(n\eta)$ .
Note that  $\mathcal{M}$  is an indirect mechanism, so it is slightly imprecise to call it  $(\eta ,\mu)$ -BIC. Formally, interacting with  $\mathcal{Q}$  truthfully is an approximate Bayes-Nash equilibrium.
In order to prove Theorem 2, we use a key lemma, Lemma 2, which establishes the robustness guarantees of Theorem 2, but in the space of latent types. Intuitively, let  $\mathbf{t}_i$  be the type of bidder  $i$ , and  $\mathbf{z}_i$  be a random variable distributed according to  $\hat{\mathcal{D}}_{z,i}$ . We know that  $\pi_p(\mathcal{D}_i,\mathbf{A} \circ \hat{\mathcal{D}}_i) \leq \varepsilon_{\mathrm{md1},p} \leq \zeta_p$ . Due to Lemma 1, there exists a coupling such that with probability greater than  $1 - \zeta_p$ ,  $\| \mathbf{t}_i - \mathbf{A}\mathbf{z}_i \|_p \leq \zeta_p$ . Since the seller uses a  $(\varepsilon_{\mathrm{md1},p}, \zeta_p, p)$ -query protocol, with probability at least  $1 - \zeta_p$ ,  $\| \mathcal{Q}(\mathbf{t}_i) - \mathbf{z}_i \|_p \leq \zeta_p$ . Note that, this implies that  $\mathbf{z}_i$  and  $\mathcal{Q}(t_i)$  are distributed such that their Prokhorov distance is at most  $\zeta_p$ . At this step, Lemma 2 provides us a mechanism  $\widetilde{\mathcal{M}}$ , constructed from  $\hat{\mathcal{M}}$ , that we can execute on types  $\mathcal{Q}(\mathbf{t}_1), \ldots, \mathcal{Q}(\mathbf{t}_n)$ , obtained by interacting with the bidders via the query protocol. With probability at least  $1 - \zeta_p$ , we have that  $\| \mathbf{t}_i - \mathbf{A}\mathbf{z}_i \|_\infty \leq \| \mathbf{t}_i - \mathbf{A}\mathbf{z}_i \|_p \leq \varepsilon_{\mathrm{md1},p}$  and thus, using the fact that the query protocol ensures  $\| \mathcal{Q}(\mathbf{t}_i) - \mathbf{z}_i \|_p \leq \zeta_p$  as well, we have  $\| \mathbf{t}_i - \mathbf{A}\mathcal{Q}(\mathbf{t}_i) \|_\infty \leq \varepsilon_{\mathrm{md1},p} + k \| \mathbf{A} \|_\infty \zeta_p$ . The guarantees of  $\widetilde{\mathcal{M}}$  for the distribution over  $\mathcal{Q}(\mathbf{t}_i)$ s are therefore translated into guarantees (with a small error) of the overall mechanism for the  $\mathcal{D}$ .
The proof of Lemma 2 is quite involved and is the main focus of our analysis. Here, we sketch the key ideas behind the proof, and defer all formal arguments to Appendix C. The proof uses the following notion of a rounded distribution.
Definition 8 (Rounded Distribution). Let  $\mathcal{F}$  be a distribution supported on  $\mathbb{R}_{\geq 0}^k$ . For any  $\delta > 0$  and  $\ell \in [0, \delta]^k$ , we define function  $r^{(\ell, \delta)}: \mathbb{R}_{\geq 0}^k \mapsto \mathbb{R}_{\geq 0}^k$  such that  $r_i^{(\ell, \delta)}(\mathbf{x}) = \max \left\{\frac{x_i - \ell_i}{\delta} \cdot \delta + \ell_i, 0\right\}$  for all  $i \in [k]$ . Let  $\mathbf{x}$  be a random vector sampled from  $\mathcal{F}$ . We define  $\lfloor \mathcal{F} \rfloor_{\ell, \delta}$  as the distribution for the random variable  $r^{(\ell, \delta)}(\mathbf{x})$ , and we call  $\lfloor \mathcal{F} \rfloor_{\ell, \delta}$  the rounded distribution of  $\mathcal{F}$ .
We follow an approach similar to Brustle et al. [BCD20]. The main idea is that arguing directly about mechanisms for distributions that are close in Prokhorov distance is difficult. On the flip side, arguing about mechanisms for distributions that are close in total variation distance is much easier, since the total variation is a more stringent (and hence more tamable) notion of distance. The key observation is that, if two distributions are close in Prokhorov distance then, in expectation over the random parameter  $\ell$ , their rounded-down versions are close in total variation distance.
That is, if, for example,  $\hat{\mathcal{M}}$  allocates item  $j$  to bidder  $i$  with probability  $1/2$  and with the remaining probability item  $j$  is not allocated, our construction does not need to know this distribution/fractional allocation and works even if nature samples and returns an integral allocation for item  $j$ .
Our overall construction is via three reductions. First, in Lemma 3, given a mechanism for  $\hat{\mathcal{F}}_z$  we design a mechanism for the rounded-down version. Second, in Lemma 4, given a mechanism for the rounded-down  $\hat{\mathcal{F}}_z$  we design a mechanism for  $\lfloor \mathcal{F}_z\rfloor_{\ell ,\delta}$ , which maintains its guarantees if  $\pi_p(\mathcal{F}_z,\hat{\mathcal{F}}_z)$  is small. Third, in Lemma 5, given a mechanism for  $\lfloor \mathcal{F}_z\rfloor_{\ell ,\delta}$  we design a mechanism for  $\mathcal{F}_z$ . Fig. 1 presents a detailed overview of the overall design architecture, and how the RLA and different mechanism design components interact.
Figure 1: Agents interact with the query protocol  $\mathcal{Q}$ , which learns their latent types  $\mathcal{Q}(\mathbf{t}_i)\mathrm{s}$ . The mechanism design component (which is oblivious to the distributions of the true agents' types  $\mathcal{D}$ ) then uses these to produce the final allocation and payments, utilizing only query access to  $\hat{\mathcal{M}}$  and sampling access to  $\hat{\mathcal{D}}_z$  such that the overall framework is approximately  $(\eta, \mu)$ -BIC wrt  $\mathcal{D}$ .
Overall Design of the RLA for Active Learning-based Mechanism Design Framework
Our proofs for Lemmas 3 and 5 are adaptations of the corresponding lemmas of [BCD20], where our main task is to flesh out the exact dependence on the dimensionality of the latent space and the  $\ell_p$ -norm (versus the  $\ell_1$ -norm in [BCD20]). The novelty of our approach comes in the construction and analysis of Lemma 4. The difficulty of transforming mechanisms for  $\lfloor \hat{\mathcal{F}}_z \rfloor_{\ell, \delta}$  into mechanisms for  $\lfloor \mathcal{F}_z \rfloor_{\ell, \delta}$  is that the two distributions might not share the same support. Thus, we need a way to map bids that are not in the support of  $\lfloor \hat{\mathcal{F}}_z \rfloor_{\ell, \delta}$  to bids that are. Brustle et al. [BCD20] do this by "optimally misreporting" on behalf of the bidder, by calculating  $\text{argmax}_{\mathbf{z} \in \text{supp}(\lfloor \hat{\mathcal{F}}_z, i \rfloor_{\ell, \delta})} \mathbb{E}_{b_{-i}} \sim \lfloor (\hat{\mathcal{F}}_z)_{-i} \rfloor_{\ell, \delta} [u_i(v_i, \hat{\mathcal{M}}(\mathbf{z}, b_{-i}))]$ , and then picking matching payments that make the overall mechanism IR. Our approach leverages the fact that  $\hat{\mathcal{F}}_{z,i}$  and  $\mathcal{F}_{z,i}$  are close in Prokhorov distance, and thus any point on the support of one distribution is close to a point on the support of the other, with high probability. An ideal construction would map a report  $\mathbf{w}_i$  to the "valid" report (i.e., a report in the support of  $\lfloor \hat{\mathcal{F}}_z, i \rfloor_{\ell, \delta}$ ) that minimizes the  $\ell_p$  distance to  $\mathbf{w}_i$ . This operation is linear on the support of  $\lfloor \hat{\mathcal{F}}_z, i \rfloor_{\ell, \delta}$ , and does not need any information on the valuation functions, nor on the actual probabilities that elements of the distribution are sampled with. Unfortunately, our assumption on what "oracle access" means does not allow us to do this operation (finding the closest point w.r.t.  $\ell_p$ ) on  $\lfloor \hat{\mathcal{F}}_z, i \rfloor_{\ell, \delta}$ , but only on  $\hat{\mathcal{F}}_{z,i}$ ; we prove that, by occurring a small loss, our assumption suffices.
3.3 Putting everything together
Combining Theorems 1 and 2, we can give mechanisms for concrete settings. Formally, we have the following theorem.