Average treatment effect estimation is the most central problem in causal inference with application to numerous disciplines. While many estimation strategies have been proposed in the literature, the statistical optimality of these methods has still remained an open area of investigation, especially in regimes where these methods do not achieve parametric rates. In this paper, we adopt the recently introduced structure-agnostic framework of statistical lower bounds, which poses no structural properties on the nuisance functions other than access to black-box estimators that achieve some statistical estimation rate. This framework is particularly appealing when one is only willing to consider estimation strategies that use non-parametric regression and classification oracles as black-box sub-processes. Within this framework, we prove the statistical optimality of the celebrated and widely used doubly robust estimators for both the Average Treatment Effect (ATE) and the Average Treatment Effect on the Treated (ATT), as well as weighted variants of the former, which arise in policy evaluation.

Estimates of heterogeneous treatment effects such as conditional average treatment effects (CATEs) and conditional quantile treatment effects (CQTEs) play an important role in real-world decision making. Given this importance, one should ensure these estimators are calibrated. While there is a rich literature on calibrating estimators of non-causal parameters, very few methods have been derived for calibrating estimators of causal parameters, or more generally estimators of quantities involving nuisance parameters. In this work, we develop general algorithms for reducing the task of causal calibration to that of calibrating a standard (non-causal) predictive model. Throughout, we study a notion of calibration defined with respect to an arbitrary, nuisance-dependent loss $\ell$, under which we say an estimator $\theta$ is calibrated if its predictions cannot be changed on any level set to decrease loss. For losses $\ell$ satisfying a condition called universal orthogonality, we present a simple algorithm that transforms partially-observed data into generalized pseudo-outcomes and applies any off-the-shelf calibration procedure. For losses $\ell$ satisfying a weaker assumption called conditional orthogonality, we provide a similar sample splitting algorithm the performs empirical risk minimization over an appropriately defined class of functions. Convergence of both algorithms follows from a generic, two term upper bound of the calibration error of any model that decouples the error in estimating unknown nuisance parameters from the calibration error in a hypothetical world where the learned nuisances are true. We demonstrate the practical applicability of our results in experiments on both observational and synthetic data. Our results are exceedingly general, showing that essentially any existing calibration algorithm can be used in causal settings, with additional loss only arising from errors in nuisance estimation.

We address the problem of estimating heterogeneous treatment effects in panel data, adopting the popular Difference-in-Differences (DiD) framework under the conditional parallel trends assumption. We propose a novel doubly robust meta-learner for the Conditional Average Treatment Effect on the Treated (CATT), reducing the estimation to a convex risk minimization problem involving a set of auxiliary models. Our framework allows for the flexible estimation of the CATT, when conditioning on any subset of variables of interest using generic machine learning. Leveraging Neyman orthogonality, our proposed approach is robust to estimation errors in the auxiliary models. As a generalization to our main result, we develop a meta-learning approach for the estimation of general conditional functionals under covariate shift. We also provide an extension to the instrumented DiD setting with non-compliance. Empirical results demonstrate the superiority of our approach over existing baselines.

Structure-agnostic causal inference studies how well one can estimate a treatment effect given black-box machine learning estimates of nuisance functions (like the impact of confounders on treatment and outcomes). Here, we find that the answer depends in a surprising way on the distribution of the treatment noise. Focusing on the partially linear model of Robinson (1988), we first show that the widely adopted double machine learning (DML) estimator is minimax rate-optimal for Gaussian treatment noise, resolving an open problem of Mackey et al. (2018). Meanwhile, for independent non-Gaussian treatment noise, we show that DML is always suboptimal by constructing new practical procedures with higher-order robustness to nuisance errors. These ACE procedures use structure-agnostic cumulant estimators to achieve r-th order insensitivity to nuisance errors whenever the (r+1)-st treatment cumulant is non-zero. We complement these core results with novel minimax guarantees for binary treatments in the partially linear model. Finally, using synthetic demand estimation experiments, we demonstrate the practical benefits of our higher-order robust estimators.

This paper investigates the integration of response time data into human preference learning frameworks for more effective reward model elicitation. While binary preference data has become fundamental in fine-tuning foundation models, generative AI systems, and other large-scale models, the valuable temporal information inherent in user decision-making remains largely unexploited. We propose novel methodologies to incorporate response time information alongside binary choice data, leveraging the Evidence Accumulation Drift Diffusion (EZ) model, under which response time is informative of the preference strength. We develop Neyman-orthogonal loss functions that achieve oracle convergence rates for reward model learning, matching the theoretical optimal rates that would be attained if the expected response times for each query were known a priori. Our theoretical analysis demonstrates that for linear reward functions, conventional preference learning suffers from error rates that scale exponentially with reward magnitude. In contrast, our response time-augmented approach reduces this to polynomial scaling, representing a significant improvement in sample efficiency. We extend these guarantees to non-parametric reward function spaces, establishing convergence properties for more complex, realistic reward models. Our extensive experiments validate our theoretical findings in the context of preference learning over images.

Causal effect estimation seeks to determine the impact of an intervention from observational data. However, the existing causal inference literature primarily addresses treatment effects on frequently occurring events. But what if we are interested in estimating the effects of a policy intervention whose benefits, while potentially important, can only be observed and measured in rare yet impactful events, such as extreme climate events? The standard causal inference methodology is not designed for this type of inference since the events of interest may be scarce in the observed data and some degree of extrapolation is necessary. Extreme Value Theory (EVT) provides methodologies for analyzing statistical phenomena in such extreme regimes. We introduce a novel framework for assessing treatment effects in extreme data to capture the causal effect at the occurrence of rare events of interest. In particular, we employ the theory of multivariate regular variation to model extremities. We develop a consistent estimator for extreme treatment effects and present a rigorous non-asymptotic analysis of its performance. We illustrate the performance of our estimator using both synthetic and semi-synthetic data.

Accurate estimation of conditional average treatment effects (CATE) is at the core of personalized decision making. While there is a plethora of models for CATE estimation, model selection is a non-trivial task, due to the fundamental problem of causal inference. Recent empirical work provides evidence in favor of proxy loss metrics with double robust properties and in favor of model ensembling. However, theoretical understanding is lacking. Direct application of prior theoretical works leads to suboptimal oracle model selection rates due to the non-convexity of the model selection problem. We provide regret rates for the major existing CATE ensembling approaches and propose a new CATE model ensembling approach based on $Q$-aggregation using the doubly robust loss. Our main result shows that causal $Q$-aggregation achieves statistically optimal oracle model selection regret rates of $\log(M)/n$ (with $M$ models and $n$ samples), with the addition of higher-order estimation error terms related to products of errors in the nuisance functions. Crucially, our regret rate does not require that any of the candidate CATE models be close to the truth. We validate our new method on many semi-synthetic datasets and also provide extensions of our work to CATE model selection with instrumental variables and unobserved confounding.

Indirect experiments provide a valuable framework for estimating treatment effects in situations where conducting randomized control trials (RCTs) is impractical or unethical. Unlike RCTs, indirect experiments estimate treatment effects by leveraging (conditional) instrumental variables, enabling estimation through encouragement and recommendation rather than strict treatment assignment. However, the sample efficiency of such estimators depends not only on the inherent variability in outcomes but also on the varying compliance levels of users with the instrumental variables and the choice of estimator being used, especially when dealing with numerous instrumental variables. While adaptive experiment design has a rich literature for \textit{direct} experiments, in this paper we take the initial steps towards enhancing sample efficiency for \textit{indirect} experiments by adaptively designing a data collection policy over instrumental variables. Our main contribution is a practical computational procedure that utilizes influence functions to search for an optimal data collection policy, minimizing the mean-squared error of the desired (non-linear) estimator. Through experiments conducted in various domains inspired by real-world applications, we showcase how our method can significantly improve the sample efficiency of indirect experiments.

We study the problem of model selection in causal inference, specifically for conditional average treatment effect (CATE) estimation. Unlike machine learning, there is no perfect analogue of cross-validation for model selection as we do not observe the counterfactual potential outcomes. Towards this, a variety of surrogate metrics have been proposed for CATE model selection that use only observed data. However, we do not have a good understanding regarding their effectiveness due to limited comparisons in prior studies. We conduct an extensive empirical analysis to benchmark the surrogate model selection metrics introduced in the literature, as well as the novel ones introduced in this work. We ensure a fair comparison by tuning the hyperparameters associated with these metrics via AutoML, and provide more detailed trends by incorporating realistic datasets via generative modeling. Our analysis suggests novel model selection strategies based on careful hyperparameter selection of CATE estimators and causal ensembling.

We study causal representation learning, the task of recovering high-level latent variables and their causal relationships in the form of a causal graph from low-level observed data (such as text and images), assuming access to observations generated from multiple environments. Prior results on the identifiability of causal representations typically assume access to single-node interventions which is rather unrealistic in practice, since the latent variables are unknown in the first place. In this work, we consider the task of learning causal representation learning with data collected from general environments. We show that even when the causal model and the mixing function are both linear, there exists a surrounded-node ambiguity (SNA) [Varici et al. 2023] which is basically unavoidable in our setting. On the other hand, in the same linear case, we show that identification up to SNA is possible under mild conditions, and propose an algorithm, LiNGCReL which provably achieves such identifiability guarantee. We conduct extensive experiments on synthetic data and demonstrate the effectiveness of LiNGCReL in the finite-sample regime.

We study the problem of online sequential decision-making given auxiliary demonstrations from _experts_ who made their decisions based on unobserved contextual information. These demonstrations can be viewed as solving related but slightly different tasks than what the learner faces. This setting arises in many application domains, such as self-driving cars, healthcare, and finance, where expert demonstrations are made using contextual information, which is not recorded in the data available to the learning agent. We model the problem as a zero-shot meta-reinforcement learning setting with an unknown task distribution and a Bayesian regret minimization objective, where the unobserved tasks are encoded as parameters with an unknown prior. We propose the Experts-as-Priors algorithm (ExPerior), an empirical Bayes approach that utilizes expert data to establish an informative prior distribution over the learner's decision-making problem. This prior enables the application of any Bayesian approach for online decision-making, such as posterior sampling. We demonstrate that our strategy surpasses existing behaviour cloning and online algorithms, as well as online-offline baselines for multi-armed bandits, Markov decision processes (MDPs), and partially observable MDPs, showcasing the broad reach and utility of ExPerior in using expert demonstrations across different decision-making setups.

Recent progress in Neural Causal Models (NCMs) showcased how identification and partial identification of causal effects can be automatically carried out via training of neural generative models that respect the constraints encoded in a given causal graph [Xia et al. 2022, Balazadeh et al. 2022]. However, formal consistency of these methods has only been proven for the case of discrete variables or only for linear causal models. In this work, we prove the consistency of partial identification via NCMs in a general setting with both continuous and categorical variables. Further, our results highlight the impact of the design of the underlying neural network architecture in terms of depth and connectivity as well as the importance of applying Lipschitz regularization in the training phase. In particular, we provide a counterexample showing that without Lipschitz regularization this method may not be asymptotically consistent. Our results are enabled by new results on the approximability of Structural Causal Models (SCMs) via neural generative models, together with an analysis of the sample complexity of the resulting architectures and how that translates into an error in the constrained optimization problem that defines the partial identification bounds.

In a variety of applications, including nonparametric instrumental variable (NPIV) analysis, proximal causal inference under unmeasured confounding, and missing-not-at-random data with shadow variables, we are interested in inference on a continuous linear functional (e.g., average causal effects) of nuisance function (e.g., NPIV regression) defined by conditional moment restrictions. These nuisance functions are generally weakly identified, in that the conditional moment restrictions can be severely ill-posed as well as admit multiple solutions. This is sometimes resolved by imposing strong conditions that imply the function can be estimated at rates that make inference on the functional possible. In this paper, we study a novel condition for the functional to be strongly identified even when the nuisance function is not; that is, the functional is amenable to asymptotically-normal estimation at root-n-rates. The condition implies the existence of debiasing nuisance functions, and we propose penalized minimax estimators for both the primary and debiasing nuisance functions. The proposed nuisance estimators can accommodate flexible function classes, and importantly they can converge to fixed limits determined by the penalization regardless of the identifiability of the nuisances. We use the penalized nuisance estimators to form a debiased estimator for the functional of interest and prove its asymptotic normality under generic high-level conditions, which provide for asymptotically valid confidence intervals. We also illustrate our method in a novel partially linear proximal causal inference problem and a partially linear instrumental variable regression problem.

In this paper, we study nonparametric estimation of instrumental variable (IV) regressions. Recently, many flexible machine learning methods have been developed for instrumental variable estimation. However, these methods have at least one of the following limitations: (1) restricting the IV regression to be uniquely identified; (2) only obtaining estimation error rates in terms weak metrics (e.g., projected norm) rather than strong metrics (e.g., L_2 norm); or (3) imposing the so-called closedness condition that requires a certain conditional expectation operator to be sufficiently smooth. In this paper, we present the first method and analysis that can avoid all three limitations, while still permitting general function approximation. Specifically, we propose a new penalized minimax estimator that can converge to a fixed IV solution even when there are multiple solutions, and we derive a strong L_2 error rate for our estimator under lax conditions. Notably, this guarantee only needs a widely-used source condition and realizability assumptions, but not the so-called closedness condition. We argue that the source condition and the closedness condition are inherently conflicting, so relaxing the latter significantly improves upon the existing literature that requires both conditions. Our estimator can achieve this improvement because it builds on a novel formulation of the IV estimation problem as a constrained optimization problem.

Many causal and policy effects of interest are defined by linear functionals of high-dimensional or non-parametric regression functions. $\sqrt{n}$-consistent and asymptotically normal estimation of the object of interest requires debiasing to reduce the effects of regularization and/or model selection on the object of interest. Debiasing is typically achieved by adding a correction term to the plug-in estimator of the functional, which leads to properties such as semi-parametric efficiency, double robustness, and Neyman orthogonality. We implement an automatic debiasing procedure based on automatically learning the Riesz representation of the linear functional using Neural Nets and Random Forests. Our method only relies on black-box evaluation oracle access to the linear functional and does not require knowledge of its analytic form. We propose a multitasking Neural Net debiasing method with stochastic gradient descent minimization of a combined Riesz representer and regression loss, while sharing representation layers for the two functions. We also propose a Random Forest method which learns a locally linear representation of the Riesz function. Even though our method applies to arbitrary functionals, we experimentally find that it performs well compared to the state of art neural net based algorithm of Shi et al. (2019) for the case of the average treatment effect functional. We also evaluate our method on the problem of estimating average marginal effects with continuous treatments, using semi-synthetic data of gasoline price changes on gasoline demand.

Estimation and inference on causal parameters is typically reduced to a generalized method of moments problem, which involves auxiliary functions that correspond to solutions to a regression or classification problem. Recent line of work on debiased machine learning shows how one can use generic machine learning estimators for these auxiliary problems, while maintaining asymptotic normality and root-$n$ consistency of the target parameter of interest, while only requiring mean-squared-error guarantees from the auxiliary estimation algorithms. The literature typically requires that these auxiliary problems are fitted on a separate sample or in a cross-fitting manner. We show that when these auxiliary estimation algorithms satisfy natural leave-one-out stability properties, then sample splitting is not required. This allows for sample re-use, which can be beneficial in moderately sized sample regimes. For instance, we show that the stability properties that we propose are satisfied for ensemble bagged estimators, built via sub-sampling without replacement, a popular technique in machine learning practice.

We consider the problem of partial identification, the estimation of bounds on the treatment effects from observational data. Although studied using discrete treatment variables or in specific causal graphs (e.g., instrumental variables), partial identification has been recently explored using tools from deep generative modeling. We propose a new method for partial identification of average treatment effects (ATEs) in general causal graphs using implicit generative models comprising continuous and discrete random variables. Since ATE with continuous treatment is generally non-regular, we leverage the partial derivatives of response functions to define a regular approximation of ATE, a quantity we call uniform average treatment derivative (UATD). We prove that our algorithm converges to tight bounds on ATE in linear structural causal models (SCMs). For nonlinear SCMs, we empirically show that using UATD leads to tighter and more stable bounds than methods that directly optimize the ATE.

For many inference problems in statistics and econometrics, the unknown parameter is identified by a set of moment conditions. A generic method of solving moment conditions is the Generalized Method of Moments (GMM). However, classical GMM estimation is potentially very sensitive to outliers. Robustified GMM estimators have been developed in the past, but suffer from several drawbacks: computational intractability, poor dimension-dependence, and no quantitative recovery guarantees in the presence of a constant fraction of outliers. In this work, we develop the first computationally efficient GMM estimator (under intuitive assumptions) that can tolerate a constant $\epsilon$ fraction of adversarially corrupted samples, and that has an $\ell_2$ recovery guarantee of $O(\sqrt{\epsilon})$. To achieve this, we draw upon and extend a recent line of work on algorithmic robust statistics for related but simpler problems such as mean estimation, linear regression and stochastic optimization. As a special case, we apply our algorithm to instrumental variables linear regression with heterogeneous treatment effects, and experimentally demonstrate that it can tolerate as much as $10$ -- $15\%$ corruption, significantly improving upon baseline methods.

Abstract. We present a general framework for designing approximately revenue-optimal mechanisms for multi-item additive auctions, which applies to both truthful and nontruthful auctions. Given a (not necessarily truthful) single-item auction format [Formula: see text] satisfying certain technical conditions, we run simultaneous item auctions augmented with a personalized entry fee for each bidder that must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types and in particular are independent of realized bids. We bound the revenue of the resulting two-part tariff mechanism using a novel geometric technique that enables revenue guarantees for many common nontruthful auctions that previously had none. Our approach adapts and extends the duality framework of Cai, Devanur, and Weinberg [ SIAM J. Comput., 50 (2021), pp. STOC16-160–STOC16-200] beyond truthful auctions. Our framework can be used with many common auction formats, such as simultaneous first-price, simultaneous second-price, and simultaneous all-pay auctions. Our results for first-price and all-pay are the first revenue guarantees of nontruthful mechanisms in multidimensional environments, addressing an open question in the literature [T. Roughgarden, V. Syrgkanis, and E. Tardos, J. Artificial Intelligence Res., 59 (2017), pp. 59–101]. If all-pay auctions are used, we prove that the resulting mechanism is also credible in the sense that the auctioneer cannot benefit by deviating from the stated mechanism after observing agent bids. This is the first static credible mechanism for multi-item additive auctions that achieves a constant factor of the optimal revenue. If second-price auctions are used, we obtain a truthful [Formula: see text]-approximate mechanism with fixed entry fees that are amenable to tuning via online learning techniques.

Randomized experiments can be susceptible to selection bias due to potential non-compliance by the participants. While much of the existing work has studied compliance as a static behavior, we propose a game-theoretic model to study compliance as dynamic behavior that may change over time. In rounds, a social planner interacts with a sequence of heterogeneous agents who arrive with their unobserved private type that determines both their prior preferences across the actions (e.g., control and treatment) and their baseline rewards without taking any treatment. The planner provides each agent with a randomized recommendation that may alter their beliefs and their action selection. We develop a novel recommendation mechanism that views the planner’s recommendation as a form of instrumental variable (IV) that only affects an agents’ action selection, but not the observed rewards. We construct such IVs by carefully mapping the history –the interactions between the planner and the previous agents– to a random recommendation. Even though the initial agents may be completely non-compliant, our mechanism can incentivize compliance over time, thereby enabling the estimation of the treatment effect of each treatment, and minimizing the cumulative regret of the planner whose goal is to identify the optimal treatment.

In recent years, both academic research and industry applications see an increased effort in using machine learning methods to measure granular causal effects and design optimal policies based on these causal estimates. Open source packages such as CausalML and EconML provide a unified interface for applied researchers and industry practitioners with a variety of machine learning methods for causal inference. The tutorial will cover the topics including conditional treatment effect estimators by meta-learners and tree-based algorithms, model validations and sensitivity analysis, optimization algorithms including policy leaner and cost optimization. In addition, the tutorial will demonstrate the production of these algorithms in industry use cases.

One of the most commonly used methods for forming confidence intervals is the empirical bootstrap, which is especially expedient when the limiting distribution of the estimator is unknown. However, despite its ubiquitous role in machine learning, its theoretical properties are still not well understood. Recent developments in probability have provided new tools to study the bootstrap method. However, they have been applied only to specific applications and contexts, and it is unclear whether these techniques are applicable to the understanding of the consistency of the bootstrap in machine learning pipelines. In this paper, we derive general stability conditions under which the empirical bootstrap estimator is consistent and quantify the speed of convergence. Moreover, we propose alternative ways to use the bootstrap method to build confidence intervals with coverage guarantees. Finally, we illustrate the generality and tightness of our results by examples of interest for machine learning including for two-sample kernel tests after kernel selection and the empirical risk of stacked estimators.

Policy makers often need to estimate the long-term effects of novel treatments, while only having historical data of older treatment options. We propose a surrogate-based approach using a long-term dataset where only past treatments were administered and a short-term dataset where novel treatments have been administered. Our approach generalizes previous surrogate-style methods, allowing for continuous treatments and serially-correlated treatment policies while maintaining consistency and root-n asymptotically normal estimates under a Markovian assumption on the data and the observational policy. Using a semi-synthetic dataset on customer incentives from a major corporation, we evaluate the performance of our method and discuss solutions to practical challenges when deploying our methodology.

We consider the estimation of treatment effects in settings when multiple treatments are assigned over time and treatments can have a causal effect on future outcomes. We propose an extension of the double/debiased machine learning framework to estimate the dynamic effects of treatments and apply it to a concrete linear Markovian high-dimensional state space model and to general structural nested mean models. Our method allows the use of arbitrary machine learning methods to control for the high dimensional state, subject to a mean square error guarantee, while still allowing parametric estimation and construction of confidence intervals for the dynamic treatment effect parameters of interest. Our method is based on a sequential regression peeling process, which we show can be equivalently interpreted as a Neyman orthogonal moment estimator. This allows us to show root-n asymptotic normality of the estimated causal effects.

An agent has access to multiple information sources, each modeled as a Brownian motion whose drift provides information about a different component of an unknown Gaussian state. Information is acquired continuously---where the agent chooses both which sources to sample from, and also how to allocate attention across them---until an endogenously chosen time, at which point a decision is taken. We demonstrate conditions on the agent's prior belief under which it is possible to exactly characterize the optimal information acquisition strategy. We then apply this characterization to derive new results regarding: (1) endogenous information acquisition for binary choice, (2) the dynamic consequences of attention manipulation, and (3) strategic information provision by biased news sources.

We consider the problem of bid prediction in repeated auctions and evaluate the performance of econometric methods for learning agents, using a dataset from a mainstream sponsored search auction marketplace. Sponsored search auctions are a billion dollar industry and the main source of revenue of several tech giants. A critical problem in optimizing such marketplaces is understanding how bidders will react to changes in the auction design. We propose the use of no-regret-based econometrics for bid prediction, modeling players as no-regret learners with respect to a utility function, unknown to the analyst. We propose econometric approaches to simultaneously learn the parameters of a player’s utility and her learning rule, and apply these methods to a real-world dataset from the BingAds sponsored search auction marketplace. We show that the no-regret econometric methods perform comparably to state-of-the-art time-series machine-learning methods when there is no co-variate shift, but significantly outperform machine-learning methods when there is a co-variate shift between the training and test periods. This portrays the importance of using structural econometric approaches for predicting how players will respond to changes in the market. Moreover, we show that among structural econometric methods, approaches based on no-regret learning outperform more traditional, equilibrium-based, econometric methods that assume that players continuously best respond to competition. Finally, we demonstrate how the prediction performance of the no-regret learning algorithms can be further improved by considering bidders who optimize a utility function with a visibility bias component.

Regression Trees [Breiman et al. 1984] and Random Forests [Breiman 2001], are one of the most widely used estimation methods by machine learning practitioners. Despite their widespread use, their theoretical underpinnings are far from being fully understood. Recent breakthrough advances have shown that such greedily built trees are asymptotically consistent [Biau et al. 2010, Denil et al. 2014, Scornet et al. 2015] in the low dimensional regime, where the number of features is a constant, independent of the sample size. Also, the works of [Mentch et al. 2016, Wager et al. 2018] provide asymptotic normality results for honest versions of Random Forests. In this work, we analyze the performance of regression trees and forests with binary features in the high-dimensional regime, where the number of features can grow exponentially with the number of samples. We show that trees and forests built greedily based on the celebrated CART criterion, provably adapt to sparsity: when only a subset $R$, of size $r$, of the features are relevant, then the mean squared error of appropriately shallow trees, or fully grown honest forests, scales exponentially only with the number of relevant features and depends only logarithmically on the overall number of features. More precisely, we identify three regimes, each providing different dependence on the number of relevant features. When the relevant variables are “weakly” relevant (in the sense that there is not strong separation between the relevant and irrelevant variables in terms of their ability to reduce variance), then shallow trees achieve “slow rates” on the mean squared error of the order of $2^r/\sqrt{n}$, when variables are independent, and $1/n^{1/(r+2)}$, when variables are dependent. When the relevant variables are “strongly” relevant, in that there is a separation in their ability to reduce variance as compared to the irrelevant ones, by a constant $\beta_{\min}$, then we show that greedily built shallow trees and fully grown honest forests can achieve fast parametric mean squared error rates of the order of $2^r/(\beta_{\min}\,{n})$. When variables are strongly relevant, we also show that the predictions of sub-sampled honest forests have an asymptotically normal distribution centered around their true values and whose variance scales at most as $O(2^r \log(n)/(\beta_{\min}\,{n}))$. Thus, sub-sampled honest forests are provably a data-adaptive method for non-parametric inference, that adapts to the latent sparsity dimension of the data generating distribution, as opposed to classical non-parametric regression approaches. Our results show that, at least for the case of binary features, forest based algorithms can offer immense improvement on the statistical power of non-parametric hypothesis tests in high-dimensional regimes.

We provide excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate a target parameter depends on an unknown parameter that must be estimated from data (a "nuisance parameter"). We analyze a two-stage sample splitting meta-algorithm that takes as input two arbitrary estimation algorithms: one for the target parameter and one for the nuisance parameter. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from statistical learning and machine learning literature to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can give guarantees under weaker assumptions than in previous works and accommodate the case where the target parameter belongs to a complex nonparametric class. We characterize conditions on the metric entropy such that oracle rates---rates of the same order as if we knew the nuisance parameter---are achieved. We also analyze the rates achieved by specific estimation algorithms such as variance-penalized empirical risk minimization, neural network estimation and sparse high-dimensional linear model estimation. We highlight the applicability of our results in four settings of central importance in the literature: 1) heterogeneous treatment effect estimation, 2) offline policy optimization, 3) domain adaptation, and 4) learning with missing data.

We develop an approach for estimating models described via conditional moment restrictions, with a prototypical application being non-parametric instrumental variable regression. We introduce a min-max criterion function, under which the estimation problem can be thought of as solving a zero-sum game between a modeler who is optimizing over the hypothesis space of the target model and an adversary who identifies violating moments over a test function space. We analyze the statistical estimation rate of the resulting estimator for arbitrary hypothesis spaces, with respect to an appropriate analogue of the mean squared error metric, for ill-posed inverse problems. We show that when the minimax criterion is regularized with a second moment penalty on the test function and the test function space is sufficiently rich, then the estimation rate scales with the critical radius of the hypothesis and test function spaces, a quantity which typically gives tight fast rates. Our main result follows from a novel localized Rademacher analysis of statistical learning problems defined via minimax objectives. We provide applications of our main results for several hypothesis spaces used in practice such as: reproducing kernel Hilbert spaces, high dimensional sparse linear functions, spaces defined via shape constraints, ensemble estimators such as random forests, and neural networks. For each of these applications we provide computationally efficient optimization methods for solving the corresponding minimax problem (e.g. stochastic first-order heuristics for neural networks). In several applications, we show how our modified mean squared error rate, combined with conditions that bound the ill-posedness of the inverse problem, lead to mean squared error rates. We conclude with an extensive experimental analysis of the proposed methods.

We present a general framework for designing approximately revenue-optimal mechanisms for multi-item additive auctions. Our approach adapts the duality framework of Cai, Devanur and Weinberg (STOC 2016) and applies to both truthful and non-truthful auctions. Given a (not necessarily truthful) single-item auction format 'A' satisfying certain technical conditions, we run simultaneous item auctions augmented with a personalized entry fee for each bidder that must be paid before the auction can be accessed. These entry fees depend only on the prior distribution of bidder types, and in particular are independent of realized bids. We bound the revenue of the resulting two-part tariff mechanism using a novel geometric technique that enables revenue guarantees for many common non-truthful auctions that previously had none. Our framework can be used with many common auction formats, such as simultaneous first-price, simultaneous second-price, and simultaneous all-pay auctions. Our results for first price and all-pay are the first revenue guarantees of non-truthful mechanisms in multi-dimensional environments, addressing an open question in the literature. If all-pay auctions are used, we prove that the resulting mechanism is also credible in the sense that the auctioneer cannot benefit by deviating from the stated mechanism after observing agent bids. This is the first static credible mechanism for multi-item additive auctions that achieves a constant factor of the optimal revenue. If second-price auctions are used, we obtain a truthful O(1)-approximate mechanism with fixed entry fees that are amenable to tuning via online learning techniques.

We provide excess risk guarantees for statistical learning in a setting where the population risk with respect to which we evaluate the target model depends on an unknown model that must be to be estimated from data (a “nuisance model”). We analyze a two-stage sample splitting meta-algorithm that takes as input two arbitrary estimation algorithms: one for the target model and one for the nuisance model. We show that if the population risk satisfies a condition called Neyman orthogonality, the impact of the nuisance estimation error on the excess risk bound achieved by the meta-algorithm is of second order. Our theorem is agnostic to the particular algorithms used for the target and nuisance and only makes an assumption on their individual performance. This enables the use of a plethora of existing results from statistical learning and machine learning literature to give new guarantees for learning with a nuisance component. Moreover, by focusing on excess risk rather than parameter estimation, we can give guarantees under weaker assumptions than in previous works and accommodate the case where the target parameter belongs to a complex nonparametric class. We characterize conditions on the metric entropy such that oracle rates—rates of the same order as if we knew the nuisance model—are achieved. We also analyze the rates achieved by specific estimation algorithms such as variance-penalized empirical risk minimization, neural network estimation and sparse high-dimensional linear model estimation. We highlight the applicability of our results in four settings of central importance in the literature: 1) heterogeneous treatment effect estimation, 2) offline policy optimization, 3) domain adaptation, and 4) learning with missing data.

We propose the orthogonal random forest, an algorithm that combines Neyman-orthogonality to reduce sensitivity with respect to estimation error of nuisance parameters with generalized random forests (Athey et al., 2017)—a flexible non-parametric method for statistical estimation of conditional moment models using random forests. We provide a consistency rate and establish asymptotic normality for our estimator. We show that under mild assumptions on the consistency rate of the nuisance estimator, we can achieve the same error rate as an oracle with a priori knowledge of these nuisance parameters. We show that when the nuisance functions have a locally sparse parametrization, then a local ell_1-penalized regression achieves the required rate. We apply our method to estimate heterogeneous treatment effects from observational data with discrete treatments or continuous treatments, and we show that, unlike prior work, our method provably allows to control for a high-dimensional set of variables under standard sparsity conditions. We also provide a comprehensive empirical evaluation of our algorithm on both synthetic and real data.

We consider off-policy evaluation and optimization with continuous action spaces. We focus on observational data where the data collection policy is unknown and needs to be estimated from data. We take a semi-parametric approach where the value function takes a known parametric form in the treatment, but we are agnostic on how it depends on the observed contexts. We propose a doubly robust off-policy estimate for this setting and show that off-policy optimization based on this doubly robust estimate is robust to estimation errors of the policy function or the regression model. We also show that the variance of our off-policy estimate achieves the semi-parametric efficiency bound. Our results also apply if the model does not satisfy our semi-parametric form but rather we measure regret in terms of the best projection of the true value function to this functional space. Our work extends prior approaches of policy optimization from observational data that only considered discrete actions. We provide an experimental evaluation of our method in a synthetic data example motivated by optimal personalized pricing.

We consider dynamic pricing with many products under an evolving but low-dimensional demand model. Assuming the temporal variation in cross-elasticities exhibits low-rank structure based on fixed (latent) features of the products, we show that the revenue maximization problem reduces to an online bandit convex optimization with side information given by the observed demands. We design dynamic pricing algorithms whose revenue approaches that of the best fixed price vector in hindsight, at a rate that only depends on the intrinsic rank of the demand model and not the number of products. Our approach applies a bandit convex optimization algorithm in a projected low-dimensional space spanned by the latent product features, while simultaneously learning this span via online singular value decomposition of a carefully-crafted matrix containing the observed demands.

We consider the estimation of heterogeneous treatment effects with arbitrary machine learning methods in the presence of unobserved confounders with the aid of a valid instrument. Such settings arise in A/B tests with an intent-to-treat structure, where the experimenter randomizes over which user will receive a recommendation to take an action, and we are interested in the effect of the downstream action. We develop a statistical learning approach to the estimation of heterogeneous effects, reducing the problem to the minimization of an appropriate loss function that depends on a set of auxiliary models (each corresponding to a separate prediction task). The reduction enables the use of all recent algorithmic advances (e.g. neural nets, forests). We show that the estimated effect model is robust to estimation errors in the auxiliary models, by showing that the loss satisfies a Neyman orthogonality criterion. Our approach can be used to estimate projections of the true effect model on simpler hypothesis spaces. When these spaces are parametric, then the parameter estimates are asymptotically normal, which enables construction of confidence sets. We applied our method to estimate the effect of membership on downstream webpage engagement for a major travel webpage, using as an instrument an intent-to-treat A/B test among 4 million users, where some users received an easier membership sign-up process. We also validate our method on synthetic data and on public datasets for the effects of schooling on income.

Estimators computed from adaptively collected data do not behave like their non-adaptive brethren.Rather, the sequential dependence of the collection policy can lead to severe distributional biases that persist even in the infinite data limit. We develop a general method – $\mathbf{W}$-decorrelation – for transforming the bias of adaptive linear regression estimators into variance. The method uses only coarse-grained information about the data collection policy and does not need access to propensity scores or exact knowledge of the policy.We bound the finite-sample bias and variance of the $\mathbf{W}$-estimator and develop asymptotically correct confidence intervals based on a novel martingale central limit theorem. We then demonstrate the empirical benefits of the generic $\mathbf{W}$-decorrelation procedure in two different adaptive data settings: the multi-armed bandit and the autoregressive time series.

This paper studies semiparametric contextual bandits, a generalization of the linear stochastic bandit problem where the reward for a chosen action is modeled as a linear function of known action features confounded by a non-linear action-independent term. We design new algorithms that achieve $\tilde{O}(d\sqrt{T})$ regret over $T$ rounds, when the linear function is $d$-dimensional, which matches the best known bounds for the simpler unconfounded case and improves on a recent result of Greenwald et al. (2017). Via an empirical evaluation, we show that our algorithms outperform prior approaches when there are non-linear confounding effects on the rewards. Technically, our algorithms use a new reward estimator inspired by doubly-robust approaches and our proofs require new concentration inequalities for self-normalized martingales.

We address online learning in complex auction settings, such as sponsored search auctions, where the value of the bidder is unknown to her, evolving in an arbitrary manner and observed only if the bidder wins an allocation. We leverage the structure of the utility of the bidder and the partial feedback that bidders typically receive in auctions, in order to provide algorithms with regret rates against the best fixed bid in hindsight, that are exponentially faster in convergence in terms of dependence on the action space, than what would have been derived by applying a generic bandit algorithm and almost equivalent to what would have been achieved in the full information setting. Our results are enabled by analyzing a new online learning setting with outcome-based feedback, which generalizes learning with feedback graphs. We provide an online learning algorithm for this setting, of independent interest, with regret that grows only logarithmically with the number of actions and linearly only in the number of potential outcomes (the latter being very small in most auction settings). Last but not least, we show that our algorithm outperforms the bandit approach experimentally and that this performance is robust to dropping some of our theoretical assumptions or introducing noise in the feedback that the bidder receives.

We consider a data analyst's problem of purchasing data from strategic agents to compute an unbiased estimate of a statistic of interest. Agents incur private costs to reveal their data and the costs can be arbitrarily correlated with their data. Once revealed, data are verifiable. This paper focuses on linear unbiased estimators. We design an individually rational and incentive compatible mechanism that optimizes the worst-case mean-squared error of the estimation, where the worst-case is over the unknown correlation between costs and data, subject to a budget constraint in expectation. We characterize the form of the optimal mechanism in closed-form. We further extend our results to acquiring data for estimating a parameter in regression analysis, where private costs can correlate with the values of the dependent variable but not with the values of the independent variables.

We consider the problem of optimal dynamic information acquisition from many correlated information sources. Each period, the decision-maker jointly takes an action and allocates a fixed number of observations across the available sources. His payoff depends on the actions taken and on an unknown state. In the canonical setting of jointly normal information sources, we show that the optimal dynamic information acquisition rule proceeds myopically after finitely many periods. If signals are acquired in large blocks each period, then the optimal rule turns out to be myopic from period 1. These results demonstrate the possibility of robust and "simple" optimal information acquisition, and simplify the analysis of dynamic information acquisition in a widely used informational environment.

We study a basic auction design problem with online supply. There are two unit-demand bidders and two types of items. The first item type will arrive first for sure, and the second item type may or may not arrive. The auctioneer has to decide the allocation of an item immediately after each item arrives, but is allowed to compute payments after knowing how many items arrived. For this problem we show that there is no deterministic truthful and individually rational mechanism that, even with unbounded computational resources, gets any finite approximation factor to the optimal social welfare.

Assortment optimization refers to the problem of designing a slate of products to offer potential customers, such as stocking the shelves in a convenience store. The price of each product is fixed in advance, and a probabilistic choice function describes which product a customer will choose from any given subset. We introduce the combinatorial assortment problem, where each customer may select a bundle of products. We consider a choice model in which each consumer selects a utility-maximizing bundle subject to a private valuation function, and study the complexity of the resulting optimization problem. Our main result is an exact algorithm for k -additive valuations, under a model of vertical differentiation in which customers agree on the relative value of each pair of items but differ in their absolute willingness to pay. For valuations that are vertically differentiated but not necessarily k -additive, we show how to obtain constant approximations under a “well-priced” condition, where each product’s price is sufficiently high. We further show that even for a single customer with known valuation, any sub-polynomial approximation to the problem requires exponentially many demand queries when the valuation function is XOS, and that no FPTAS exists even when the valuation is succinctly representable.

We study an optimal contest design problem where contributors abilities are private, their costs are convex as a function of their effort, and the designer seeks to maximize their total productivity. We address the design of approximately-optimal mechanisms that are robust, in that they are independent of the ability distribution and the precise form of the cost function. We show that a very simple all-pay contest where the prize is distributed equally among the top quartile of contributors is a constant-factor approximation to the optimal, for a large class of convex cost functions, when the number of contributors is larger than some constant. This result stands in contrast to contests with linear costs, where awarding a prize to a single top contributor ("winner-takes-all»») is approximately-optimal; when costs are convex, winner-takes-all is far from optimal. We validate the performance of our approximately-optimal contest designs via simulation experiments, which uncover much better empirical performance than the worst-case guarantees. Our results are enabled by novel results in the space of optimal mechanism design with convex costs, which could be of independent interest.

We consider the design of computationally efficient online learning algorithms in an adversarial setting in which the learner has access to an offline optimization oracle. We present an algorithm called Generalized Follow-the-Perturbed-Leader and provide conditions under which it is oracle-efficient while achieving vanishing regret. Our results make significant progress on an open problem raised by Hazan and Koren [31], who showed that oracle-efficient algorithms do not exist in general [30] and asked whether one can identify properties under which oracle-efficient online learning may be possible. Our auction-design framework considers an auctioneer learning an optimal auction for a sequence of adversarially selected valuations with the goal of achieving revenue that is almost as good as the optimal auction in hindsight, among a class of auctions. We give oracle-efficient learning results for: (1) VCG auctions with bidder-specific reserves in single-parameter settings, (2) envy-free item pricing in multi-item auctions, and (3) s-level auctions of Morgenstern and Roughgarden [43] for single-item settings. The last result leads to an approximation of the overall optimal Myerson auction when bidders’ valuations are drawn according to a fast-mixing Markov process, extending prior work that only gave such guarantees for the i.i.d. setting. Finally, we derive various extensions, including: (1) oracle-efficient algorithms for the contextual learning setting in which the learner has access to side information (such as bidder demographics), (2) learning with approximate oracles such as those based on Maximal-in-Range algorithms, and (3) no-regret bidding in simultaneous auctions, resolving an open problem of Daskalakis and Syrgkanis [14].

We consider robust optimization problems, where the goal is to optimize in the worst case over a class of objective functions. We develop a reduction from robust improper optimization to stochastic optimization: given an oracle that returns $\alpha$-approximate solutions for distributions over objectives, we compute a distribution over solutions that is $\alpha$-approximate in the worst case. We show that derandomizing this solution is NP-hard in general, but can be done for a broad class of statistical learning tasks. We apply our results to robust neural network training and submodular optimization. We evaluate our approach experimentally on corrupted character classification and robust influence maximization in networks.

Analysis of efficiency of outcomes in game theoretic settings has been a main item of study at the intersection of economics and computer science. The notion of the price of anarchy takes a worst-case stance to efficiency analysis, considering instance independent guarantees of efficiency. We propose a data-dependent analog of the price of anarchy that refines this worst-case assuming access to samples of strategic behavior. We focus on auction settings, where the latter is non-trivial due to the private information held by participants. Our approach to bounding the efficiency from data is robust to statistical errors and mis-specification. Unlike traditional econometrics, which seek to learn the private information of players from observed behavior and then analyze properties of the outcome, we directly quantify the inefficiency without going through the private information. We apply our approach to datasets from a sponsored search auction system and find empirical results that are a significant improvement over bounds from worst-case analysis.

We introduce a new sample complexity measure, which we refer to as split-sample growth rate. For any hypothesis $H$ and for any sample $S$ of size $m$, the split-sample growth rate $\hat{\tau}_H(m)$ counts how many different hypotheses can empirical risk minimization output on any sub-sample of $S$ of size $m/2$. We show that the expected generalization error is upper bounded by $O\left(\sqrt{\frac{\log(\hat{\tau}_H(2m))}{m}}\right)$. Our result is enabled by a strengthening of the Rademacher complexity analysis of the expected generalization error. We show that this sample complexity measure, greatly simplifies the analysis of the sample complexity of optimal auction design, for many auction classes studied in the literature. Their sample complexity can be derived solely by noticing that in these auction classes, ERM on any sample or sub-sample will pick parameters that are equal to one of the points in the sample.

A plethora of recent work has analyzed properties of outcomes in games when each player employs a no-regret learning algorithm. Many algorithms achieve regret against the best fixed action in hindisght that decays at a rate of O(1/'T), when the game is played for T iterations. The latter rate is optimal in adversarial settings. However, in a game a player's opponents are minimizing their own regret, rather than maximizing the player's regret. (Daskalakis et al. 2014) and (Rakhlin and Sridharan 2013) showed that in two player zero-sum games O(1/T) rates are achievable. In (Syrgkanis et al. 2015), we show that O(1/T3/4) rates are achievable in general multi-player games and also analyze convergence of the dynamics to approximately optimal social welfare, where we show a convergence rate of O(1/T). The latter result was subsequently generalized to a broader class of learning algorithms by (Foster et al. 2016). This is based on joint work with Alekh Agarwal, Haipeng Luo and Robert E. Schapire.

This survey outlines a general and modular theory for proving approximation guarantees for equilibria of auctions in complex settings. This theory complements traditional economic techniques, which generally focus on exact and optimal solutions and are accordingly limited to relatively stylized settings. We highlight three user-friendly analytical tools: smoothness-type inequalities, which immediately yield approximation guarantees for many auction formats of interest in the special case of complete information and deterministic strategies; extension theorems, which extend such guarantees to randomized strategies, no-regret learning outcomes, and incomplete-information settings; and composition theorems, which extend such guarantees from simpler to more complex auctions. Combining these tools yields tight worst-case approximation guarantees for the equilibria of many widely-used auction formats.

An extensive body of recent work studies the welfare guarantees of simple and prevalent combinatorial auction formats, such as selling m items via simultaneous second price auctions (SiSPAs) [1], [2], [3]. These guarantees hold even when the auctions are repeatedly executed and the players use no-regret learning algorithms to choose their actions. Unfortunately, off-the-shelf no-regret learning algorithms for these auctions are computationally inefficient as the number of actions available to the players becomes exponential. We show that this obstacle is inevitable: there are no polynomial-time no-regret learning algorithms for SiSPAs, unless RP ⊇ NP, even when the bidders are unit-demand. Our lower bound raises the question of how good outcomes polynomially-bounded bidders may discover in such auctions. To answer this question, we propose a novel concept of learning in auctions, termed "no-envy learning." This notion is founded upon Walrasian equilibrium, and we show that it is both efficiently implementable and results in approximately optimal welfare, even when the bidders have valuations from the broad class of fractionally subadditive (XOS) valuations (assuming demand oracle access to the valuations) or coverage valuations (even without demand oracles). No-envy learning outcomes are a relaxation of no-regret learning outcomes, which maintain their approximate welfare optimality while endowing them with computational tractability. Our positive and negative results extend to several auction formats that have been studied in the literature via the smoothness paradigm. Our positive results for XOS valuations are enabled by a novel Follow-The-Perturbed-Leader algorithm for settings where the number of experts and states of nature are both infinite, and the payoff function of the learner is non-linear. We show that this algorithm has applications outside of auction settings, establishing significant gains in a recent application of no-regret learning in security games. Our efficient learning result for coverage valuations is based on a novel use of convex rounding schemes and a reduction to online convex optimization.

We provide the first oracle efficient sublinear regret algorithms for adversarial versions of the contextual bandit problem. In this problem, the learner repeatedly makes an action on the basis of a context and receives reward for the chosen action, with the goal of achieving reward competitive with a large class of policies. We analyze two settings: i) in the transductive setting the learner knows the set of contexts a priori, ii) in the small separator setting, there exists a small set of contexts such that any two policies behave differently on one of the contexts in the set. Our algorithms fall into the Follow-The-Perturbed-Leader family (Kalai and Vempala, 2005) and achieve regret O(T^3/4\sqrtK\log(N)) in the transductive setting and O(T^2/3 d^3/4 K\sqrt\log(N)) in the separator setting, where T is the number of rounds, K is the number of actions, N is the number of baseline policies, and d is the size of the separator. We actually solve the more general adversarial contextual semi-bandit linear optimization problem, whilst in the full information setting we address the even more general contextual combinatorial optimization. We provide several extensions and implications of our algorithms, such as switching regret and efficient learning with predictable sequences.

We propose a new oracle-based algorithm, BISTRO+, for the adversarial contextual bandit problem, where either contexts are drawn i.i.d. or the sequence of contexts is known a priori, but where the losses are picked adversarially. Our algorithm is computationally efficient, assuming access to an offline optimization oracle, and enjoys a regret of order $O((KT)^{\frac{2}{3}}(\log N)^{\frac{1}{3}})$, where $K$ is the number of actions, $T$ is the number of iterations, and $N$ is the number of baseline policies. Our result is the first to break the $O(T^{\frac{3}{4}})$ barrier achieved by recent algorithms, which was left as a major open problem. Our analysis employs the recent relaxation framework of (Rakhlin and Sridharan, ICML'16).

We consider a ubiquitous scenario in the Internet economy when individual decision-makers (henceforth, agents) both produce and consume information as they make strategic choices in an uncertain environment. This creates a three-way trade-off between exploration (trying out insufficiently explored alternatives to help others in the future), exploitation (making optimal decisions given the information discovered by other agents), and incentives of the agents (who are myopically interested in exploitation, while preferring the others to explore). We posit a principal who controls the flow of information from agents that came before to the ones that arrive later, and strives to coordinate the agents towards a socially optimal balance between exploration and exploitation, not using any monetary transfers. The goal is to design a recommendation policy for the principal which respects agents' incentives and minimizes a suitable notion of regret.

We study the quality of outcomes in repeated games when the population of players is dynamically changing, and where participants use learning algorithms to adapt to the dynamic environment. Price of anarchy has originally been introduced to study the Nash equilibria of one-shot games. Many games studied in computer science, such as packet routing or ad-auctions, are played repeatedly. Given the computational hardness of Nash equilibria, an attractive alternative in repeated game settings is that players use no-regret learning algorithms. The price of total anarchy considers the quality of such learning outcomes, assuming a steady environment and player population, which is rarely the case in online settings. In this paper we analyze efficiency of repeated games in dynamically changing environments. An important trait of learning behavior is its versatility to changing environments, assuming that the learning method used is adaptive, i.e., doesn't rely too heavily on experience from the distant past. We show that, in large classes of games, if players choose their strategies in a way that guarantees low adaptive regret, high social welfare is ensured, even under very frequent changes. A main technical tool for our analysis is the existence of a solution to the welfare maximization problem that is both close to optimal and relatively stable over time. Such a solution serves as a benchmark in the efficiency analysis of learning outcomes. We show that such a stable and close to optimal solution exists for many problems, even in cases when the exact optimal solution can be very unstable. We further show that a sufficient condition on the existence of stable outcomes is the existence of a differentially private algorithm for the welfare maximization problem. Hence, we draw a strong connection between differential privacy and high efficiency of learning outcomes in frequently changing repeated games. We demonstrate our techniques by focusing on two classes of games as examples: independent item auctions and congestion games. In both applications we show that adaptive learning guarantees high social welfare even with surprisingly high churn in the player population.

We present an analysis framework for bounding the price of anarchy (POA) in games that have many players, as in many of the games most pertinent to computer science applications. We use this framework to demonstrate that, in many of the models in which the POA has been studied, the POA in large games is much smaller than the worst-case bound. Our framework also differentiates between mechanisms with similar worst-case performance, such as simultaneous uniform-price auctions and greedy combinatorial auctions, thereby providing new insights about which mechanisms are likely to perform well in realistic settings.

We introduce a new hierarchy over monotone set functions, that we refer to as MPH (Maximum over Positive Hypergraphs). Levels of the hierarchy correspond to the degree of complementarity in a given function. The highest level of the hierarchy, MPH-m (where m is the total number of items) captures all monotone functions. The lowest level, MPH-1, captures all monotone submodular functions, and more generally, the class of functions known as XOS. Every monotone function that has a positive hypergraph representation of rank k (in the sense defined by Abraham, Babaioff, Dughmi and Roughgarden [EC 2012]) is in MPH-k. Every monotone function that has supermodular degree k (in the sense defined by Feige and Izsak [ITCS 2013]) is in MPH-(k+1). In both cases, the converse direction does not hold, even in an approximate sense. We present additional results that demonstrate the expressiveness power of MPH-k.One can obtain good approximation ratios for some natural optimization problems, provided that functions are required to lie in low levels of the MPH hierarchy. We present two such applications. One shows that the maximum welfare problem can be approximated within a ratio of k+1 if all players hold valuation functions in MPH-k. The other is an upper bound of 2k on the price of anarchy of simultaneous first price auctions.

Recent price-of-anarchy analyses of games of complete information suggest that coarse correlated equilibria, which characterize outcomes resulting from no-regret learning dynamics, have near-optimal welfare. This work provides two main technical results that lift this conclusion to games of incomplete information, a.k.a., Bayesian games. First, near-optimal welfare in Bayesian games follows directly from the smoothness-based proof of near-optimal welfare in the same game when the private information is public. Second, no-regret learning dynamics converge to Bayesian coarse correlated equilibrium in these incomplete information games. These results are enabled by interpretation of a Bayesian game as a stochastic game of complete information.

We show that natural classes of regularized learning algorithms with a form of recency bias achieve faster convergence rates to approximate efficiency and to coarse correlated equilibria in multiplayer normal form games. When each player in a game uses an algorithm from our class, their individual regret decays at $O(T^{-3/4})$, while the sum of utilities converges to an approximate optimum at $O(T^{-1})$--an improvement upon the worst case $O(T^{-1/2})$ rates. We show a black-box reduction for any algorithm in the class to achieve $\tilde{O}(T^{-1/2})$ rates against an adversary, while maintaining the faster rates against algorithms in the class. Our results extend those of Rakhlin and Shridharan~\cite{Rakhlin2013} and Daskalakis et al.~\cite{Daskalakis2014}, who only analyzed two-player zero-sum games for specific algorithms.

We introduce single-bid auctions as a new format for combinatorial auctions. In single-bid auctions, each bidder submits a single real-valued bid for the right to buy items at a fixed price. Contrary to other simple auction formats, such as simultaneous or sequential single-item auctions, bidders can implement no-regret learning strategies for single-bid auctions in polynomial time. Price of anarchy bounds for correlated equilibria concepts in single-bid auctions therefore have more bite than their counterparts for auctions and equilibria for which learning is not known to be computationally tractable (or worse, known to be computationally intractable [Cai and Papadimitriou 2014; Dobzinski et al. 2015] this end, we show that for any subadditive valuations the social welfare at equilibrium is an O(log m)-approximation to the optimal social welfare, where $m$ is the number of items. We also provide tighter approximation results for several subclasses. Our welfare guarantees hold for Nash equilibria and no-regret learning outcomes in both Bayesian and complete information settings via the smooth-mechanism framework. Of independent interest, our techniques show that in a combinatorial auction setting, efficiency guarantees of a mechanism via smoothness for a very restricted class of cardinality valuations extend, with a small degradation, to subadditive valuations, the largest complement-free class of valuations.

We study mechanisms that use greedy allocation rules and pay-your-bid pricing to allocate resources subject to a matroid constraint. We show that all such mechanisms obtain a constant fraction of the optimal welfare at any equilibrium of bidder behavior, via a smoothness argument. This unifies numerous recent results on the price of anarchy of simple auctions. Our results extend to polymatroid and matching constraints, and we discuss extensions to more general matroid intersections.

Individual decision-makers consume information revealed by the previous decision makers, and produce information that may help in future decision makers. This phenomenon is common in a wide range of scenarios in the Internet economy, as well as elsewhere, such as medical decisions. Each decision maker when required to select an action, would individually prefer to exploit, select the highest expected reward action conditional on her information. At the same time, each decision maker would prefer previous decision makers to explore, producing information about the rewards of various actions. A social planner, by means of carefully designed information disclosure, can incentivize the agents to balance the exploration and exploitation, and maximize social welfare. We formulate this problem as a multi-arm bandit problem (and various generalizations thereof) under incentive-compatibility constraints induced by agents' Bayesian priors. We design an incentive-compatible bandit algorithm for the social planner with asymptotically optimal regret. Further, we provide a black-box reduction from an arbitrary multi-arm bandit algorithm to an incentive-compatible one, with only a constant multiplicative increase in regret. This reduction works for very general bandit settings, even ones that incorporate contexts and arbitrary partial feedback.

The main goal of this paper is to develop a theory of inference of player valuations from observed data in the generalized second price auction without relying on the Nash equilibrium assumption. Existing work in Economics on inferring agent values from data relies on the assumption that all participant strategies are best responses of the observed play of other players, i.e. they constitute a Nash equilibrium. In this paper, we show how to perform inference relying on a weaker assumption instead: assuming that players are using some form of no-regret learning. Learning outcomes emerged in recent years as an attractive alternative to Nash equilibrium in analyzing game outcomes, modeling players who haven't reached a stable equilibrium, but rather use algorithmic learning, aiming to learn the best way to play from previous observations. In this paper we show how to infer values of players who use algorithmic learning strategies. Such inference is an important first step before we move to testing any learning theoretic behavioral model on auction data. We apply our techniques to a dataset from Microsoft's sponsored search ad auction system.

We consider common-value hybrid auctions among two asymmetrically informed bidders, where the winning bidder pays his bid with some positive probability k and the losing bid otherwise. Under the assumption of discrete and affiliated signals, we give an explicit characterization of the (unique) equilibrium, based on a simple recurrence relation, which gives rise to a linear-time algorithm for explicitly computing the equilibrium. By analyzing the execution of the algorithm, we derive several insights about the equilibrium structure. First, we show that equilibrium revenue is decreasing in k, and that the limit second-price equilibrium selected as k->0 has highest revenue, in stark contrast to the revenue collapse of the second-price auction predicted by the trembling-hand equilibrium selection of Abraham et al. We further show that the Linkage Principle can fail to hold even in a pure first-price auction with binary signals: public revelation of a signal to both bidders may decrease the auctioneer's revenue. Lastly, we analyze the effects of public acquisition of additional information on bidder utilities and exhibit cases in which both bidders strictly prefer for a specific bidder to receive additional information.

Many websites encourage user participation via the use of virtual rewards like badges. While badges typically have no explicit value, they act as symbols of social status within a community. In this paper, we study how to design virtual incentive mechanisms that maximize total contributions to a website when users are motivated by social status. We consider a game-theoretic model where users exert costly effort to make contributions and, in return, are awarded with badges. The value of a badge is determined endogenously by the number of users who earn an equal or higher badge; as more users earn a particular badge, the value of that badge diminishes for all users. We show that among all possible mechanisms for assigning status-driven rewards, the optimal mechanism is a leaderboard with a cutoff: users that contribute less than a certain threshold receive nothing while the remaining are ranked by contribution. We next study the necessary features of approximately optimal mechanisms and find that approximate optimality is influenced by the the convexity of status valuations. When status valuations are concave, any approximately optimal mechanism must contain a coarse status partition, i.e. a partition of users into status classes whose size will grow as the population grows. Conversely when status valuations are convex, we prove that fine partitioning, that is a partition of users into status classes whose size stays constant as the population grows, is necessary for approximate optimality.

We study the design of Bayesian incentive compatible mechanisms in single parameter domains, for the objective of optimizing social efficiency as measured by social cost. In the problems we consider, a group of participants compete to receive service from a mechanism that can provide such services at a cost. The mechanism wishes to choose which agents to serve in order to maximize social efficiency, but is not willing to suffer an expected loss: the agents' payments should cover the cost of service in expectation.

We initiate the study of efficient mechanism design with guaranteed good properties even when players participate in multiple mechanisms simultaneously or sequentially. We define the class of smooth mechanisms, related to smooth games defined by Roughgarden, that can be thought of as mechanisms that generate approximately market clearing prices. We show that smooth mechanisms result in high quality outcome both in equilibrium and in learning outcomes in the full information setting, as well as in Bayesian equilibrium with uncertainty about participants. Our main result is to show that smooth mechanisms compose well: smoothness locally at each mechanism implies global efficiency.

Typical models of strategic interactions in computer science use simultaneous move games. However, in applications simultaneity is often hard or impossible to achieve. In this paper, we study the robustness of the Nash Equilibrium when the assumption of simultaneity is dropped. In particular we propose studying the sequential price of anarchy: the quality of outcomes of sequential versions of games whose simultaneous counterparts are prototypical in algorithmic game theory. We study different classes of games with high price of anarchy, and show that the subgame perfect equilibrium of their sequential version is a much more natural prediction, ruling out unreasonable equilibria, and leading to much better quality solutions.

In many natural settings agents participate in multiple different auctions that are not simultaneous. In such auctions, future opportunities affect strategic considerations of the players. The goal of this paper is to develop a quantitative understanding of outcomes of such sequential auctions. In earlier work (Paes Leme et al. 2012) we initiated the study of the price of anarchy in sequential auctions. We considered sequential first price auctions in the full information model, where players are aware of all future opportunities, as well as the valuation of all players. In this paper, we study efficiency in sequential auctions in the Bayesian environment, relaxing the informational assumption on the players. We focus on two environments, both studied in the full information model in Paes Leme et al. 2012, matching markets and matroid auctions. In the full information environment, a sequential first price cut auction for matroid settings is efficient. In Bayesian environments this is no longer the case, as we show using a simple example with three players. Our main result is a bound of 3 on the price of anarchy in both matroid auctions and matching markets. To bound the price of anarchy we need to consider possible deviations at an equilibrium. In a sequential Bayesian environment the effect of deviations is more complex than in one-shot games; early bids allow others to infer information about the player's value. We create effective deviations despite the presence of this difficulty by introducing a bluffing technique of independent interest.

In many settings agents participate in multiple different auctions that are not necessarily implemented simultaneously. Future opportunities affect strategic considerations of the players in each auction, introducing externalities. Motivated by this consideration, we study a setting of a market of buyers and sellers, where each seller holds one item, bidders have combinatorial valuations and sellers hold item auctions sequentially. Our results are qualitatively different from those of simultaneous auctions, proving that simultaneity is a crucial aspect of previous work. We prove that if sellers hold sequential first price auctions then for unit-demand bidders (matching market) every subgame perfect equilibrium achieves at least half of the optimal social welfare, while for submodular bidders or when second price auctions are used, the social welfare can be arbitrarily worse than the optimal. We also show that a first price sequential auction for buying or selling a base of a matroid is always efficient, and implements the VCG outcome. An important tool in our analysis is studying first and second price auctions with externalities (bidders have valuations for each possible winner outcome), which can be of independent interest. We show that a Pure Nash Equilibrium always exists in a first price auction with externalities.