SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Theorem 3. Under the same setting as in Theorem 2, for bidders with arbitrary valuation functions, we can construct mechanism  $\mathcal{M}$  using only query access to the mechanism  $\hat{\mathcal{M}}$  (Definition 6) and oracle access to distribution  $\hat{\mathcal{D}}$  (Definition 7), and oblivious to the true type distribution  $\mathcal{D}$ . We consider queries (to each bidder  $i$ ) of the form "What is your value for the subset of items  $S$ ?"
Mechanism  $\mathcal{M}$  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 the expected revenue of  $\mathcal{M}$  is at least  $Rev(\hat{\mathcal{M}},\hat{\mathcal{D}}_z) - O(n\eta)$ . Additionally, with probability at least  $1 - \delta$ ,
$$
\zeta_ {p} = c _ {p} \cdot \left(\sigma_ {\min , p} (\mathbf {A})\right) ^ {- 1} \cdot \varepsilon_ {m d l, p}
$$
for a small constant  $c_p$  that depends on the parameter  $p$  (see footnote 7). The number of queries is  $O\left(k \cdot \ln k \cdot \ln^{(n / \delta)}\right)$  and  $O\left(k^{\frac{p}{2}} \cdot \ln^{3} k \cdot \ln^{(n / \delta)}\right)$  for  $p = 1, 2$  and  $p \geq 3$ , respectively.
The proof of Theorem 3 follows from Theorem 2 and Theorem 1, and is deferred to Appendix B.
As we've already discussed, the main mechanism of Cai and Daskalakis [CD22] requires bidders to have constrained-additive valuations<sup>9</sup>, as well  $\mathbf{A}$  to satisfy a number of restrictions. Here, we completely remove both conditions. On the flip-side, [CD22] ask bidders weaker queries, of the form "are you willing to pay price  $\tau$  for item  $j$ ?" Using such queries, one can binary search over  $\tau$ , and drive down the query noise (see Definition 4). For  $\ell_{\infty}$ , the extra cost of such an operation would be  $\ln (\| \mathbf{A} \|_{\infty} / \varepsilon)$ , where  $\varepsilon$  is the desired accuracy. However, for other  $p$ -norms, for the same target accuracy, this operation requires an extra factor of  $\Theta (\log (d^{1/p}))$  queries, giving a dependence on  $d$ .
4 Conclusions and Future Work
In this paper, we study mechanism design for prior distributions close to a topic model, inspired from the recommender systems literature. We formulate connections between mechanism design and Randomized Linear Algebra for active learning for regression problems, importing state-of-the-art results from Randomized Linear Algebra to mechanism design, and alleviate or relax restrictive assumptions of prior work. Developing a deeper understanding of such connections is an important direction for future research. For example, one could study this and other topic models in the context of mechanism design for correlated bidders, two-sided markets, information structure design, etc. Additionally, another interesting open problem would be to develop a framework for proving lower bounds for mechanism design (e.g., lower bounds on the query complexity for single-round or multi-round protocols used to communicate with the bidders) using known limitations of algorithms in active learning, and vice-versa.
Christos Boutsikas, Petros Drineas and Marios Mertzanidis are supported in part by a DOE award SC0022085, and NSF awards CCF-1814041, CCF-2209509, and DMS-2152687. Alexandros Psomas and Paritosh Verma are supported in part by an NSF CAREER award CCF-2144208, a Google Research Scholar Award, and a Google AI for Social Good award.
${}^{9}$  A valuation function is constrained-additive if  $v\left( {t,S}\right)  = \max  _{{T \in  {\mathcal{I}} \cap  {2}^{S}}}\mathop{\sum }\limits_{{{j \in  T}}}{t}_{j}$  ,where  $\mathcal{I}$  is a downward-closed set system.