A recent line of work studies language generation in the limit, a formal model of language learning where an algorithm observes an adversarially generated enumeration of strings from an unknown target language $K$ and must eventually generate new, unseen strings from $K$. In this model, Kleinberg and Mullainathan (2024) proved that generation is achievable in surprisingly general settings; whenever $K$ belongs to a known countable collection of languages. However, their generator, while quite general, suffers from “mode collapse:” it generates from an ever-smaller subset of the target. To address this, Kleinberg and Wei (2025a) introduced a stronger notion of dense generation, requiring the output to asymptotically cover a positive fraction of the target, and showed it remains achievable for all countable collections. Both of these works rely on the crucial assumption of \textit{perfect} data: the adversary can neither insert strings from outside the target language (i.e., noise) nor omit strings from it (i.e., omissions). In practice, training data for language models is notoriously noisy, raising the fundamental question: \begin{center} \emph{How much contamination (either omissions or insertions) can language generation tolerate?} \end{center} Recent works have made partial progress on this question by studying (non-dense) generation with either finite amounts of noise (but no omissions) (Raman and Raman, 2025) or omissions (but no noise) (Bai et al., 2026). We characterize the contamination tolerance of both types of generation by proving the following results: \begin{itemize} \item \textbf{Generation under Contamination:} Language generation in the limit is achievable for all countable collections if and only if the fraction of contaminated examples converges to zero. When this condition fails, we characterize the collections which remain generable. \item \textbf{Dense Generation under Contamination:} Dense generation is achievable for all countable collections if and only if the amount of contamination is finite. For an infinite amount of contamination, we provide several characterizations of when dense generation is possible, showing it is strictly less robust than standard generation. \end{itemize} As a byproduct, we also resolve an open question of (Raman and Raman, 2025) on generation with membership oracle access under finite contamination.

We initiate the study of language generation in the limit, a model recently introduced by Kleinberg and Mullainathan (2024), under the constraint of differential privacy. We consider the \emph{continual release} model, where a generator must eventually output a stream of valid strings while protecting the privacy of the entire input sequence. Our first main result is that for countable collections of languages, privacy comes at no qualitative cost: we provide an $\varepsilon$-differentially-private algorithm that generates in the limit from \emph{any} countable collection. This stands in contrast to many learning settings where privacy renders learnability impossible. However, privacy does impose a quantitative cost: there are finite collections of size $k$ for which uniform private generation requires $\Omega(k/\varepsilon)$ samples, whereas just one sample suffices non-privately. We then turn to the harder problem of language \emph{identification} in the limit. Here, we show that privacy creates fundamental barriers. We prove that no $\varepsilon$-DP algorithm can identify a collection containing two languages with an infinite intersection and a finite set difference, a condition far stronger than the classical non-private characterization of identification. Next, we turn to the \emph{stochastic} setting where the sample strings are sampled i.i.d. from a distribution (instead of being generated by an adversary). Here, we show that private identification is possible if and only if the collection is identifiable in the adversarial model. Together, our results establish new dimensions along which generation and identification differ and, for identification, a separation between adversarial and stochastic settings induced by privacy constraints.

Learning curves are a fundamental primitive in supervised learning, describing how an algorithm’s performance improves with more data and providing a quantitative measure of its generalization ability. Formally, a learning curve plots the decay of an algorithm’s error for a fixed underlying distribution as a function of the number of training samples. Prior work on revenue-maximizing learning algorithms, starting with the seminal work of Cole and Roughgarden, adopts a distribution-free perspective (which parallels the PAC learning framework in learning theory). This approach evaluates performance against the hardest possible sequence of valuation distributions, one for each sample size, effectively defining the upper envelope of learning curves over all possible distributions, thus leading to error bounds that do not capture the shape of the learning curves.

Multiclass learnability is known to exhibit a properness barrier: there are learnable classes which cannot be learned by any proper learner. Binary classification faces no such barrier for learnability, but a similar one for optimal learning, which can in general only be achieved by improper learners. Fortunately, recent advances in binary classification have demonstrated that this requirement can be satisfied using aggregations of proper learners, some of which are strikingly simple. This raises a natural question: to what extent can simple aggregations of proper learners overcome the properness barrier in multiclass classification? We give a positive answer to this question for classes which have finite Graph dimension, $d_G$. Namely, we demonstrate that the optimal binary learners of Hanneke, Larsen, and Aden-Ali et al. (appropriately generalized to the multiclass setting) achieve sample complexity $O\left( \frac{d_G + \ln(1 / \delta)}{\varepsilon}\right)$. This forms a strict improvement upon the sample complexity of ERM. We complement this with a lower bound demonstrating that for certain classes of Graph dimension $d_G$, majorities of ERM learners require $\Omega \left( \frac{d_G + \ln(1 / \delta)}{\varepsilon}\right)$ samples. Furthermore, we show that a single ERM requires $\Omega \left(\frac{d_G \ln(1 / \varepsilon) + \ln(1 / \delta)}{\varepsilon}\right)$ samples on such classes, exceeding the lower bound of Daniel et al. (2015) by a factor of $\ln(1 / \varepsilon)$. For multiclass learning in full generality — i.e., for classes of finite DS dimension but possibly infinite Graph dimension — we give a strong refutation to these learning strategies, by exhibiting a learnable class which cannot be learned to constant error by any aggregation of a finite number of proper learners.

We study the problem of procurement auctions, in which an auctioneer seeks to acquire services from a group of strategic sellers with private costs. The quality of the services is measured through some submodular function that is known to the auctioneer. Our goal is to design computationally efficient procurement auctions that (approximately) maximize the difference between the quality of the acquired services and the total cost of the sellers, in a way that is incentive compatible (IC) and individual rational (IR) for the sellers, and generates non-negative surplus (NAS) for the auctioneer. Our contribution is twofold: i) we provide an improved analysis of existing algorithms for non-positive submodular function maximization and ii) we design computationally efficient frameworks that transform submodular function optimization algorithms to mechanisms that are IC and IR for the sellers, NAS for the auctioneer, and approximation-preserving. Our frameworks are general and work both in the offline setting where the auctioneer can observe the bids and the services of all the sellers simultaneously, and in the online setting where the sellers arrive in an adversarial order and the auctioneer has to make an irrevocable decision whether to purchase their service or not. We further investigate whether it is possible to convert state-of-art submodular optimization algorithms into descending auctions. We focus on the adversarial setting, meaning that the schedule of the descending prices is determined by an adversary. We show that a submodular optimization algorithm satisfying bi-criteria $(1/2,1)$-approximation in welfare can be effectively converted to a descending auction in this setting. We further establish a connection between descending auctions and online submodular optimization. Finally, we demonstrate the practical applications of our frameworks by instantiating them with different state-of-the-art submodular optimization algorithms and comparing their welfare performance through empirical experiments on publicly available datasets that consist of thousands of sellers.

Binary classification in the classic PAC model exhibits a curious phenomenon: Empirical Risk Minimization (ERM) learners are suboptimal in the realizable case yet optimal in the agnostic case. Roughly speaking, this owes itself to the fact that non-realizable distributions $\\mathcal{D}$ are more difficult to learn than realizable distributions -- even when one discounts a learner's error by $\\mathrm{err}(h^\\ast_\\mathcal{D})$, i.e., the error of the best hypothesis in $\\mathcal{H}$. Thus, optimal agnostic learners are permitted to incur excess error on (easier-to-learn) distributions $\\mathcal{D}$ for which $\\tau = \\mathrm{err}(h^\\ast_\\mathcal{D})$ is small. Recent work of Hanneke, Larsen, and Zhivotovskiy (FOCS '24) addresses this shortcoming by including $\\tau$ itself as a parameter in the agnostic error term. In this more fine-grained model, they demonstrate tightness of the error lower bound $\\tau + \\Omega \\left(\\sqrt{\\frac{\\tau (d + \\log(1 / \\delta))}{m}} + \\frac{d + \\log(1 / \\delta)}{m} \\right)$ in a regime where $\\tau > d/m$, and leave open the question of whether there may be a higher lower bound when $\\tau \\approx d/m$, with $d$ denoting $\\mathrm{VC}(\\mathcal{H})$. In this work, we resolve this question by exhibiting a learner which achieves error $c \\cdot \\tau + O \\left(\\sqrt{\\frac{\\tau (d + \\log(1 / \\delta))}{m}} + \\frac{d + \\log(1 / \\delta)}{m} \\right)$ for a constant $c \\leq 2.1$, matching the lower bound and demonstrating optimality when $\\tau =O( d/m)$. Further, our learner is computationally efficient and is based upon careful aggregations of ERM classifiers, making progress on two other questions of Hanneke, Larsen, and Zhivotovskiy (FOCS '24). We leave open the interesting question of whether our approach can be refined to lower the constant from 2.1 to 1, which would completely settle the complexity of agnostic learning.

We investigate language generation in the limit – a model by Kleinberg and Mullainathan and extended by Li, Raman, and Tewari. While Kleinberg and Mullainathan proved generation is possible for all countable collections, Li, Raman, and Tewari defined a hierarchy of generation notions (uniform, non-uniform, and generatable) and explored their feasibility for uncountable collections. Our first set of results resolve two open questions of Li et al. by proving finite unions of generatable or non-uniformly generatable classes need not be generatable. These follow from a stronger result: there is non-uniformly generatable class and a uniformly generatable class whose union is non-generatable. This adds to the aspects along which language generation in the limit is different from traditional tasks in statistical learning theory like classification, which are closed under finite unions. In particular, it implies that given two generators for different collections, one cannot combine them to obtain a single "more powerful" generator, prohibiting this notion of boosting. Our construction also addresses a third of Li et al.'s open questions on whether there are uncountable classes that are non-uniformly generatable and do not satisfy the eventually unbounded closure (EUC) condition introduced by Li et al. Our approach utilizes carefully constructed classes along with a novel diagonalization argument that could be of independent interest in the growing area of language generation.

In this work we initiate the study of regression in the universal rates framework of Bousquet et al. Unlike the traditional uniform learning setting, we are interested in obtaining learning guarantees that hold for all fixed data-generating distributions, but do not hold uniformly across them. We focus on the realizable setting and we consider two different well-studied loss functions: the cut-off loss at scale $\gamma > 0$, which asks for predictions that are $\gamma$-close to the correct one, and the absolute loss, which measures how far away the prediction is from the correct one. Our results show that the landscape of the achievable rates in the two cases is completely different. First we give a trichotomic characterization of the optimal learning rates under the cut-off loss: each class is learnable either at an exponential rate, a (nearly) linear rate or requires arbitrarily slow rates. Moving to the absolute loss, we show that the achievable learning rates are significantly more involved by illustrating that an infinite number of different optimal learning rates is achievable. This is the first time that such a rich landscape of rates is obtained in the universal rates literature.

We provide an efficient replicable algorithm for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell (STOC, 2022). We design the first dimension-independent replicable algorithm for this task which runs in polynomial time, is proper, and has strictly improved sample complexity compared to the one achieved by Impagliazzo et al. (STOC, 2022) with respect to all the relevant parameters. Moreover, our algorithm has sample complexity that is optimal with respect to the accuracy parameter $\epsilon$. Departing from the requirement of polynomial time algorithms, using the DP-to-Replicability reduction of Bun et al. (STOC 2023), we show how to obtain a replicable algorithm for large-margin halfspaces with improved sample complexity with respect to the margin parameter $\tau$, but running time doubly exponential in $1/\tau^2$ and worse sample complexity dependence on $\epsilon$ than our previous algorithm. We then design an improved algorithm with better sample complexity than both of our previous algorithms and running time exponential in $1/\tau^{2}.$

We study a setting where agents use no-regret learning algorithms to participate in repeated auctions. Recently, Kolumbus and Nisan [2022a] showed, rather surprisingly, that when bidders participate in second-price auctions using no-regret bidding algorithms, no matter how large the number of interactions $T$ is, the runner-up bidder may not converge to bidding truthfully. Our first result shows that this holds forall deterministictruthful auctions. We also show that the ratio of the learning rates of different bidders can qualitatively affect the convergence of the bidders. Next, we consider the problem of revenue maximization in this environment. In the setting with fully rational bidders, the seminal result of Myerson [1981] showed that revenue can be maximized by using a second-price auction with reserves. We show that, in stark contrast, in our setting with learning bidders, randomized auctions can have strictly better revenue guarantees than second-price auctions with reserves, when $T$ is large enough. To do this, we provide a black-box transformation from any truthful auction $A$ to an auction $A'$ such that: i) all mean-based no-regret learners that participate in $A'$ converge to bidding truthfully, ii) the distance between the allocation rule and the payment rule between $A, A'$ is negligible. Finally, we study revenue maximization in the non-asymptotic regime. We define a notion of auctioneer regret that compares the revenue generated to the revenue of a second price auction with truthful bids. When the auctioneer has to use the same auction throughout the interaction, we show an (almost) tight regret bound of $\tilde{\Theta}(T^{3/4})$. Then, we consider the case where the auctioneer can use different auctions throughout the interaction, but in a way that is oblivious to the bids. For this setting, we show an (almost) tight bound of $\tilde{\Theta}(\sqrt{T})$.

In this work we study the problem of actively learning binary classifiers from a given concept class, i.e., learning by utilizing unlabeled data and submitting targeted queries about their labels to a domain expert. We evaluate the quality of our solutions by considering the learning curves they induce, i.e., the rate of decrease of the misclassification probability as the number of label queries increases. The majority of the literature on active learning has focused on obtaining uniform guarantees on the error rate which are only able to explain the upper envelope of the learning curves over families of different data-generating distributions. We diverge from this line of work and we focus on the distribution-dependent framework of universal learning whose goal is to obtain guarantees that hold for any fixed distribution, but do not apply uniformly over all the distributions. We provide a complete characterization of the optimal learning rates that are achievable by algorithms that have to specify the number of unlabeled examples they use ahead of their execution. Moreover, we identify combinatorial complexity measures that give rise to each case of our tetrachotomic characterization. This resolves an open question that was posed by Balcan et al. (2010). As a byproduct of our main result, we develop an active learning algorithm for partial concept classes that achieves exponential learning rates in the uniform setting.

As ML models become increasingly complex and integral to high-stakes domains such as finance and healthcare, they also become more susceptible to sophisticated adversarial attacks. We investigate the threat posed by undetectable backdoors, as defined in Goldwasser et al. [2022], in models developed by insidious external expert firms. When such backdoors exist, they allow the designer of the model to sell information on how to slightly perturb their input to change the outcome of the model. We develop a general strategy to plant backdoors to obfuscated neural networks, that satisfy the security properties of the celebrated notion of indistinguishability obfuscation. Applying obfuscation before releasing neural networks is a strategy that is well motivated to protect sensitive information of the external expert firm. Our method to plant backdoors ensures that even if the weights and architecture of the obfuscated model are accessible, the existence of the backdoor is still undetectable. Finally, we introduce the notion of undetectable backdoors to language models and extend our neural network backdoor attacks to such models based on the existence of steganographic functions.

We study computational aspects of algorithmic replicability, a notion of stability introduced by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. Motivated by a recent line of work that established strong statistical connections between replicability and other notions of learnability such as online learning, private learning, and SQ learning, we aim to understand better the computational connections between replicability and these learning paradigms. Our first result shows that there is a concept class that is efficiently replicably PAC learnable, but, under standard cryptographic assumptions, no efficient online learner exists for this class. Subsequently, we design an efficient replicable learner for PAC learning parities when the marginal distribution is far from uniform, making progress on a question posed by Impagliazzo et al. [STOC, 2022]. To obtain this result, we design a replicable lifting framework inspired by Blanc, Lange, Malik, and Tan [STOC, 2023], that transforms in a black-box manner efficient replicable PAC learners under the uniform marginal distribution over the Boolean hypercube to replicable PAC learners under any marginal distribution, with sample and time complexity that depends on a certain measure of the complexity of the distribution. Finally, we show that any pure DP learner can be transformed in a black-box manner to a replicable learner, with time complexity polynomial in the confidence and accuracy parameters, but exponential in the representation dimension of the underlying hypothesis class.

We propose a new Markov Decision Process (MDP) model for ad auctions to capture the user response to the quality of ads, with the objective of maximizing the long-term discounted revenue. By incorporating user response, our model takes into consideration all three parties involved in the auction (advertiser, auctioneer, and user). The state of the user is modeled as a user-specific click-through rate (CTR) with the CTR changing in the next round according to the set of ads shown to the user in the current round. We characterize the optimal mechanism for this MDP as a Myerson's auction with a notion of modified virtual value, which relies on the value distribution of the advertiser, the current user state, and the future impact of showing the ad to the user. Leveraging this characterization, we design a sample-efficient and computationally-efficient algorithm which outputs an approximately optimal policy that requires only sample access to the true MDP and the value distributions of the bidders. Finally, we propose a simple mechanism built upon second price auctions with personalized reserve prices and show it can achieve a constant-factor approximation to the optimal long term discounted revenue.

When two different parties use the same learning rule on their own data, how can we test whether the distributions of the two outcomes are similar? In this paper, we study the similarity of outcomes of learning rules through the lens of the Total Variation (TV) distance of distributions. We say that a learning rule is TV indistinguishable if the expected TV distance between the posterior distributions of its outputs, executed on two training data sets drawn independently from the same distribution, is small. We first investigate the learnability of hypothesis classes using TV indistinguishable learners. Our main results are information-theoretic equivalences between TV indistinguishability and existing algorithmic stability notions such as replicability and approximate differential privacy. Then, we provide statistical amplification and boosting algorithms for TV indistinguishable learners.

In this work, we aim to characterize the statistical complexity of realizable regression both in the PAC learning setting and the online learning setting. Previous work had established the sufficiency of finiteness of the fat shattering dimension for PAC learnability and the necessity of finiteness of the scaled Natarajan dimension, but little progress had been made towards a more complete characterization since the work of Simon 1997 (SICOMP '97). To this end, we first introduce a minimax instance optimal learner for realizable regression and propose a novel dimension that both qualitatively and quantitatively characterizes which classes of real-valued predictors are learnable. We then identify a combinatorial dimension related to the graph dimension that characterizes ERM learnability in the realizable setting. Finally, we establish a necessary condition for learnability based on a combinatorial dimension related to the DS dimension, and conjecture that it may also be sufficient in this context. Additionally, in the context of online learning we provide a dimension that characterizes the minimax instance optimal cumulative loss up to a constant factor and design an optimal online learner for realizable regression, thus resolving an open question raised by Daskalakis and Golowich in STOC '22.

We design replicable algorithms in the context of statistical clustering under the recently introduced notion of replicability from Impagliazzo et al. [2022]. According to this definition, a clustering algorithm is replicable if, with high probability, its output induces the exact same partition of the sample space after two executions on different inputs drawn from the same distribution, when its internal randomness is shared across the executions. We propose such algorithms for the statistical $k$-medians, statistical $k$-means, and statistical $k$-centers problems by utilizing approximation routines for their combinatorial counterparts in a black-box manner. In particular, we demonstrate a replicable $O(1)$-approximation algorithm for statistical Euclidean $k$-medians ($k$-means) with $\operatorname{poly}(d)$ sample complexity. We also describe an $O(1)$-approximation algorithm with an additional $O(1)$-additive error for statistical Euclidean $k$-centers, albeit with $\exp(d)$ sample complexity. In addition, we provide experiments on synthetic distributions in 2D using the $k$-means++ implementation from sklearn as a black-box that validate our theoretical results.

We initiate the mathematical study of replicability as an algorithmic property in the context of reinforcement learning (RL). We focus on the fundamental setting of discounted tabular MDPs with access to a generative model. Inspired by Impagliazzo et al. [2022], we say that an RL algorithm is replicable if, with high probability, it outputs the exact same policy after two executions on i.i.d. samples drawn from the generator when its internal randomness is the same. We first provide an efficient $\rho$-replicable algorithm for $(\varepsilon, \delta)$-optimal policy estimation with sample and time complexity $\widetilde O\left(\frac{N^3\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$, where $N$ is the number of state-action pairs. Next, for the subclass of deterministic algorithms, we provide a lower bound of order $\Omega\left(\frac{N^3}{(1-\gamma)^3\cdot\varepsilon^2\cdot\rho^2}\right)$. Then, we study a relaxed version of replicability proposed by Kalavasis et al. [2023] called TV indistinguishability. We design a computationally efficient TV indistinguishable algorithm for policy estimation whose sample complexity is $\widetilde O\left(\frac{N^2\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$. At the cost of $\exp(N)$ running time, we transform these TV indistinguishable algorithms to $\rho$-replicable ones without increasing their sample complexity. Finally, we introduce the notion of approximate-replicability where we only require that two outputted policies are close under an appropriate statistical divergence (e.g., Renyi) and show an improved sample complexity of $\widetilde O\left(\frac{N\cdot\log(1/\delta)}{(1-\gamma)^5\cdot\varepsilon^2\cdot\rho^2}\right)$.

Consider the task of learning an unknown concept from a given concept class; to what extent does interacting with a domain expert accelerate the learning process? It is common to measure the effectiveness of learning algorithms by plotting the "learning curve", that is, the decay of the error rate as a function of the algorithm's resources (examples, queries, etc). Thus, the overarching question in this work is whether (and which kind of) interaction accelerates the learning curve. Previous work in interactive learning focused on uniform bounds on the learning rates which only capture the upper envelope of the learning curves over families of data distributions. We thus formalize our overarching question within the distribution dependent framework of universal learning, which aims to understand the performance of learning algorithms on every data distribution, but without requiring a single upper bound which applies uniformly to all distributions. Our main result reveals a fundamental trichotomy of interactive learning rates, thus providing a complete characterization of universal interactive learning. As a corollary we deduce a strong affirmative answer to our overarching question, showing that interaction is beneficial. Remarkably, we show that in important cases such benefits are realized with label queries, that is, by active learning algorithms. On the other hand, our lower bounds apply to arbitrary binary queries and, hence, they hold in any interactive learning setting.

In this paper we study the problem of multiclass classification with a bounded number of different labels $k$, in the realizable setting. We extend the traditional PAC model to a) distribution-dependent learning rates, and b) learning rates under data-dependent assumptions. First, we consider the universal learning setting (Bousquet, Hanneke, Moran, van Handel and Yehudayoff, STOC'21), for which we provide a complete characterization of the achievable learning rates that holds for every fixed distribution. In particular, we show the following trichotomy: for any concept class, the optimal learning rate is either exponential, linear or arbitrarily slow. Additionally, we provide complexity measures of the underlying hypothesis class that characterize when these rates occur. Second, we consider the problem of multiclass classification with structured data (such as data lying on a low dimensional manifold or satisfying margin conditions), a setting which is captured by partial concept classes (Alon, Hanneke, Holzman and Moran, FOCS'21). Partial concepts are functions that can be undefined in certain parts of the input space. We extend the traditional PAC learnability of total concept classes to partial concept classes in the multiclass setting and investigate differences between partial and total concepts.

In this paper, we study the problem of regret minimization for episodic Reinforcement Learning (RL) both in the model-free and the model-based setting. We focus on learning with general function classes and general model classes, and we derive results that scale with the eluder dimension of these classes. In contrast to the existing body of work that mainly establishes instance-independent regret guarantees, we focus on the instance-dependent setting and show that the regret scales logarithmically with the horizon $T$, provided that there is a gap between the best and the second best action in every state. In addition, we show that such a logarithmic regret bound is realizable by algorithms with $O(\log T)$ switching cost (also known as adaptivity complexity). In other words, these algorithms rarely switch their policy during the course of their execution. Finally, we complement our results with lower bounds which show that even in the tabular setting, we cannot hope for regret guarantees lower than $O(\log T)$.

We study the problem of selling information to a data-buyer who faces a decision problem under uncertainty. We consider the classic Bayesian decision-theoretic model pioneered by Blackwell. Initially, the data buyer has only partial information about the payoff-relevant state of the world. A data seller offers additional information about the state of the world. The information is revealed through signaling schemes, also referred to as experiments. In the single-agent setting, any mechanism can be represented as a menu of experiments. A recent paper by Bergemann et al.[8] present a complete characterization of the revenue-optimal mechanism in a binary state and binary action environment. By contrast, no characterization is known for the case with more actions. In this paper, we consider more general environments and study arguably the simplest mechanism, which only sells the fully informative experiment. In the environment with binary state and m≥3 actions, we provide an $O(m)$-approximation to the optimal revenue by selling only the fully informative experiment and show that the approximation ratio is tight up to an absolute constant factor. An important corollary of our lower bound is that the size of the optimal menu must grow at least linearly in the number of available actions, so no universal upper bound exists for the size of the optimal menu in the general single-dimensional setting. We also provide a sufficient condition under which selling only the fully informative experiment achieves the optimal revenue.

We investigate the algorithmic problem of selling information to agents who face a decision-making problem under uncertainty. We adopt the model recently proposed by Bergemann et al. [BBS18], in which information is revealed through signaling schemes called experiments. In the single-agent setting, any mechanism can be represented as a menu of experiments. Our results show that the computational complexity of designing the revenue-optimal menu depends heavily on the way the model is specified. When all the parameters of the problem are given explicitly, we provide a polynomial time algorithm that computes the revenue-optimal menu. For cases where the model is specified with a succinct implicit description, we show that the tractability of the problem is tightly related to the efficient implementation of a Best Response Oracle: when it can be implemented efficiently, we provide an additive FPTAS whose running time is independent of the number of actions. On the other hand, we provide a family of problems, where it is computationally intractable to construct a best response oracle, and we show that it is NP-hard to get even a constant fraction of the optimal revenue. Moreover, we investigate a generalization of the original model by Bergemann et al. [BBS18] that allows multiple agents to compete for useful information. We leverage techniques developed in the study of auction design (see e.g. [CDW12a], [AFHHM12], [CDW12b], [CDW13a], [CDW13b]) to design a polynomial time algorithm that computes the revenue-optimal mechanism for selling information.

We consider the black-box reduction from multidimensional revenue maximization to virtual welfare maximization. Cai et al. [12, 13, 14, 15] show a polynomial-time approximation-preserving reduction, however, the mechanism produced by their reduction is only approximately Bayesian incentive compatible (∊-BIC). We provide two new polynomial time transformations that convert any ∊-BIC mechanism to an exactly BIC mechanism with only a negligible revenue loss. Our first transformation applies to any mechanism design setting with downward-closed outcome space and only requires sample access to the agents' type distributions. Our second transformation applies to the fully general outcome space, removing the downward-closed assumption, but requires full access to the agents' type distributions. Both transformations only require query access to the original ∊-BIC mechanism. Other ∊-BIC to BIC transformations for revenue exist in the literature [23, 36, 18] but all require exponential time to run in both of the settings we consider. As an application of our transformations, we improve the reduction by Cai et al. [12, 13, 14, 15] to generate an exactly BIC mechanism.