We take an unorthodox view of voting by expanding the design space to include both the elicitation rule, whereby voters map their (cardinal) preferences to votes, and the aggregation rule, which transforms the reported votes into collective decisions. Intuitively, there is a tradeoff between the communication requirements of the elicitation rule (i.e., the number of bits of information that voters need to provide about their preferences) and the efficiency of the outcome of the aggregation rule, which we measure through distortion (i.e., how well the utilitarian social welfare of the outcome approximates the maximum social welfare in the worst case). Our results chart the Pareto frontier of the communication-distortion tradeoff.

Recent breakthroughs in AI for multi-agent games like Go, Poker, and Dota, have seen great strides in recent years. Yet none of these games address the real-life challenge of cooperation in the presence of unknown and uncertain teammates. This challenge is a key game mechanism in hidden role games. Here we develop the DeepRole algorithm, a multi-agent reinforcement learning agent that we test on "The Resistance: Avalon", the most popular hidden role game. DeepRole combines counterfactual regret minimization (CFR) with deep value networks trained through self-play. Our algorithm integrates deductive reasoning into vector-form CFR to reason about joint beliefs and deduce partially observable actions. We augment deep value networks with constraints that yield interpretable representations of win probabilities. These innovations enable DeepRole to scale to the full Avalon game. Empirical game-theoretic methods show that DeepRole outperforms other hand-crafted and learned agents in five-player Avalon. DeepRole played with and against human players on the web in hybrid human-agent teams. We find that DeepRole outperforms human players as both a cooperator and a competitor.

Self-play methods based on regret minimization have become the state of the art for computing Nash equilibria in large two-players zero-sum extensive-form games. These methods fundamentally rely on the hierarchical structure of the players' sequential strategy spaces to construct a regret minimizer that recursively minimizes regret at each decision point in the game tree. In this paper, we introduce the first efficient regret minimization algorithm for computing extensive-form correlated equilibria in large two-player general-sum games with no chance moves. Designing such an algorithm is significantly more challenging than designing one for the Nash equilibrium counterpart, as the constraints that define the space of correlation plans lack the hierarchical structure and might even form cycles. We show that some of the constraints are redundant and can be excluded from consideration, and present an efficient algorithm that generates the space of extensive-form correlation plans incrementally from the remaining constraints. This structural decomposition is achieved via a special convexity-preserving operation that we coin scaled extension. We show that a regret minimizer can be designed for a scaled extension of any two convex sets, and that from the decomposition we then obtain a global regret minimizer. Our algorithm produces feasible iterates. Experiments show that it significantly outperforms prior approaches and for larger problems it is the only viable option.

We study a wide class of non-convex non-concave min-max games that generalizes over standard bilinear zero-sum games. In this class, players control the inputs of a smooth function whose output is being applied to a bilinear zero-sum game. This class of games is motivated by the indirect nature of the competition in Generative Adversarial Networks, where players control the parameters of a neural network while the actual competition happens between the distributions that the generator and discriminator capture. We establish theoretically, that depending on the specific instance of the problem gradient-descent-ascent dynamics can exhibit a variety of behaviors antithetical to convergence to the game theoretically meaningful min-max solution. Specifically, different forms of recurrent behavior (including periodicity and Poincar\'{e} recurrence) are possible as well as convergence to spurious (non-min-max) equilibria for a positive measure of initial conditions. At the technical level, our analysis combines tools from optimization theory, game theory and dynamical systems.

This paper investigates the evaluation of learned multiagent strategies in the incomplete information setting, which plays a critical role in ranking and training of agents. Traditionally, researchers have relied on Elo ratings for this purpose, with recent works also using methods based on Nash equilibria. Unfortunately, Elo is unable to handle intransitive agent interactions, and other techniques are restricted to zero-sum, two-player settings or are limited by the fact that the Nash equilibrium is intractable to compute. Recently, a ranking method called $\alpha$-Rank, relying on a new graph-based game-theoretic solution concept, was shown to tractably apply to general games. However, evaluations based on Elo or $\alpha$-Rank typically assume noise-free game outcomes, despite the data often being collected from noisy simulations, making this assumption unrealistic in practice. This paper investigates multiagent evaluation in the incomplete information regime, involving general-sum many-player games with noisy outcomes. We derive sample complexity guarantees required to confidently rank agents in this setting. We propose adaptive algorithms for accurate ranking, provide correctness and sample complexity guarantees, then introduce a means of connecting uncertainties in noisy match outcomes to uncertainties in rankings. We evaluate the performance of these approaches in several domains, including Bernoulli games, a soccer meta-game, and Kuhn poker.

We study the problem of {\em distribution-independent} PAC learning of halfspaces in the presence of Massart noise. Specifically, we are given a set of labeled examples $(\bx, y)$ drawn from a distribution $\D$ on $\R^{d+1}$ such that the marginal distribution on the unlabeled points $\bx$ is arbitrary and the labels $y$ are generated by an unknown halfspace corrupted with Massart noise at noise rate $\eta<1/2$. The goal is to find a hypothesis $h$ that minimizes the misclassification error $\pr_{(\bx, y) \sim \D} \left[ h(\bx) \neq y \right]$. We give a $\poly\left(d, 1/\eps\right)$ time algorithm for this problem with misclassification error $\eta+\eps$. We also provide evidence that improving on the error guarantee of our algorithm might be computationally hard. Prior to our work, no efficient weak (distribution-independent) learner was known in this model, even for the class of disjunctions. The existence of such an algorithm for halfspaces (or even disjunctions) has been posed as an open question in various works, starting with Sloan (1988), Cohen (1997), and was most recently highlighted in Avrim Blum's FOCS 2003 tutorial.

A basic question in learning theory is to identify if two distributions are identical when we have access only to examples sampled from the distributions. This basic task is considered, for example, in the context of Generative Adversarial Networks (GANs), where a discriminator is trained to distinguish between a real-life distribution and a synthetic distribution. Classically, we use a hypothesis class $H$ and claim that the two distributions are distinct if for some $h\in H$ the expected value on the two distributions is (significantly) different. Our starting point is the following fundamental problem: "is having the hypothesis dependent on more than a single random example beneficial". To address this challenge we define $k$-ary based discriminators, which have a family of Boolean $k$-ary functions $\G$. Each function $g\in \G$ naturally defines a hyper-graph, indicating whether a given hyper-edge exists. A function $g\in \G$ distinguishes between two distributions, if the expected value of $g$, on a $k$-tuple of i.i.d examples, on the two distributions is (significantly) different. We study the expressiveness of families of $k$-ary functions, compared to the classical hypothesis class $H$, which is $k=1$. We show a separation in expressiveness of $k+1$-ary versus $k$-ary functions. This demonstrate the great benefit of having $k\geq 2$ as distinguishers. For $k\geq 2$ we introduce a notion similar to the VC-dimension, and show that it controls the sample complexity. We proceed and provide upper and lower bounds as a function of our extended notion of VC-dimension.

We study the problem of {\em properly} learning large margin halfspaces in the agnostic PAC model. In more detail, we study the complexity of properly learning $d$-dimensional halfspaces on the unit ball within misclassification error $\alpha \cdot \opt_{\gamma} + \eps$, where $\opt_{\gamma}$ is the optimal $\gamma$-margin error rate and $\alpha \geq 1$ is the approximation ratio. We give learning algorithms and computational hardness results for this problem, for all values of the approximation ratio $\alpha \geq 1$, that are nearly-matching for a range of parameters. Specifically, for the natural setting that $\alpha$ is any constant bigger than one, we provide an essentially tight complexity characterization. On the positive side, we give an $\alpha = 1.01$-approximate proper learner that uses $O(1/(\eps^2\gamma^2))$ samples (which is optimal) and runs in time $\poly(d/\eps) \cdot 2^{\tilde{O}(1/\gamma^2)}$. On the negative side, we show that {\em any} constant factor approximate proper learner has runtime $\poly(d/\eps) \cdot 2^{(1/\gamma)^{2-o(1)}}$, assuming the Exponential Time Hypothesis.

A crucial assumption in most statistical learning theory is that samples are independently and identically distributed (i.i.d.). However, for many real applications, the i.i.d. assumption does not hold. We consider learning problems in which examples are dependent and their dependency relation is characterized by a graph. To establish algorithm-dependent generalization theory for learning with non-i.i.d. data, we first prove novel McDiarmid-type concentration inequalities for Lipschitz functions of graph-dependent random variables. We show that concentration relies on the forest complexity of the graph, which characterizes the strength of the dependency. We demonstrate that for many types of dependent data, the forest complexity is small and thus implies good concentration. Based on our new inequalities we are able to build stability bounds for learning from graph-dependent data.

We study three fundamental statistical-learning problems: distribution estimation, property estimation, and property testing. We establish the profile maximum likelihood (PML) estimator as the first unified sample-optimal approach to a wide range of learning tasks. In particular, for every alphabet size $k$ and desired accuracy $\varepsilon$: \textbf{Distribution estimation} Under $\ell_1$ distance, PML yields optimal $\Theta(k/(\varepsilon^2\log k))$ sample complexity for sorted-distribution estimation, and a PML-based estimator empirically outperforms the Good-Turing estimator on the actual distribution; \textbf{Additive property estimation} For a broad class of additive properties, the PML plug-in estimator uses just four times the sample size required by the best estimator to achieve roughly twice its error, with exponentially higher confidence; \textbf{$\alpha$-R\'enyi entropy estimation} For an integer $\alpha>1$, the PML plug-in estimator has optimal $k^{1-1/\alpha}$ sample complexity; for non-integer $\alpha>3/4$, the PML plug-in estimator has sample complexity lower than the state of the art; \textbf{Identity testing} In testing whether an unknown distribution is equal to or at least $\varepsilon$ far from a given distribution in $\ell_1$ distance, a PML-based tester achieves the optimal sample complexity up to logarithmic factors of $k$. With minor modifications, most of these results also hold for a near-linear-time computable variant of PML.