We study the fundamental problem of high-dimensional mean estimation in a robust model where a constant fraction of the samples are adversarially corrupted. Recent work gave the first polynomial time algorithms for this problem with dimension-independent error guarantees for several families of structured distributions.In this work, we give the first nearly-linear time algorithms for high-dimensional robust mean estimation. Specifically, we focus on distributions with (i) known covariance and sub-gaussian tails, and (ii) unknown bounded covariance. Given N samples on R d , an ǫ-fraction of which may be arbitrarily corrupted, our algorithms run in time O(N d)/poly(ǫ) and approximate the true mean within the information-theoretically optimal error, up to constant factors. Previous robust algorithms with comparable error guarantees have running times Ω(N d 2 ), for ǫ = Ω(1).Our algorithms rely on a natural family of SDPs parameterized by our current guess ν for the unknown mean µ ⋆ . We give a win-win analysis establishing the following: either a nearoptimal solution to the primal SDP yields a good candidate for µ ⋆ -independent of our current guess ν -or a near-optimal solution to the dual SDP yields a new guess ν ′ whose distance from µ ⋆ is smaller by a constant factor. We exploit the special structure of the corresponding SDPs to show that they are approximately solvable in nearly-linear time. Our approach is quite general, and we believe it can also be applied to obtain nearly-linear time algorithms for other high-dimensional robust learning problems.

We pose and study a fundamental algorithmic problem which we term mixture selection, arising as a building block in a number of game-theoretic applications: Given a function g from the n-dimensional hypercube to the bounded interval [−1, 1], and an n × m matrix A with bounded entries, maximize g(Ax) over x in the m-dimensional simplex. This problem arises naturally when one seeks to design a lottery over items for sale in an auction, or craft the posterior beliefs for agents in a Bayesian game through the provision of information (a.k.a. signaling).We present an approximation algorithm for this problem when g simultaneously satisfies two "smoothness" properties: Lipschitz continuity with respect to the L ∞ norm, and noise stability. The latter notion, which we define and cater to our setting, controls the degree to which lowprobability -and possibly correlated -errors in the inputs of g can impact its output. The approximation guarantee of our algorithm degrades gracefully as a function of the Lipschitz continuity and noise stability of g. In particular, when g is both O(1)-Lipschitz continuous and O(1)-stable, we obtain an (additive) polynomial-time approximation scheme (PTAS) for mixture selection. We also show that neither assumption suffices by itself for an additive PTAS, and both assumptions together do not suffice for an additive fully polynomial-time approximation scheme (FPTAS).We apply our algorithm for mixture selection to a number of different game-theoretic applications, focusing on problems from mechanism design and optimal signaling. In particular, we make progress on a number of open problems suggested in prior work by easily reducing them to mixture selection: we resolve an important special case of the small-menu lottery design problem posed by Dughmi, Han, and Nisan [DHN14]; we resolve the problem of revenuemaximizing signaling in Bayesian second-price auctions posed by Emek et al. [EFG + 12] and Miltersen and Sheffet [BMS12]; we design a quasipolynomial-time approximation scheme for the optimal signaling problem in normal form games suggested by Dughmi [Dug14]; and we design an approximation algorithm for the optimal signaling problem in the voting model of Alonso and Câmara [AC14].

We study the optimization problem faced by a perfectly informed principal in a Bayesian game, who reveals information to the players about the state of nature to obtain a desirable equilibrium. This signaling problem is the natural design question motivated by uncertainty in games and has attracted much recent attention. We present new hardness results for signaling problems in (a) Bayesian two-player zero-sum games, and (b) Bayesian network routing games.For Bayesian zero-sum games, when the principal seeks to maximize the equilibrium utility of a player, we show that it is NP-hard to obtain an additive FPTAS. Our hardness proof exploits duality and the equivalence of separation and optimization in a novel way. Further, we rule out an additive PTAS assuming planted clique hardness, which states that no polynomial time algorithm can recover a planted clique from an Erdős-Rényi random graph. Complementing these, we obtain a PTAS for a structured class of zero-sum games (where obtaining an FPTAS is still NP-hard) when the payoff matrices obey a Lipschitz condition. Previous results ruled out an FPTAS assuming planted-clique hardness, and a PTAS only for implicit games with quasi-polynomial-size strategy sets.For Bayesian network routing games, wherein the principal seeks to minimize the average latency of the Nash flow, we show that it is NP-hard to obtain a (multiplicative) 4 3 − ǫ -approximation, even for linear latency functions. This is the optimal inapproximability result for linear latencies, since we show that full revelation achieves a 4 3 -approximation for linear latencies.

In this article, we study fairness in committee selection problems. We consider a general notion of fairness via stability: A committee is stable if no coalition of voters can deviate and choose a committee of proportional size, so that all these voters strictly prefer the new committee to the existing one. Our main contribution is to extend this definition to stability of a distribution (or lottery) over committees. We consider two canonical voter preference models: the A pproval S et setting where each voter approves a set of candidates and prefers committees with larger intersection with this set; and the R anking R epresentative setting where each voter ranks committees based on how much she likes her favorite candidate in a committee. Our main result is to show that stable lotteries always exist for these canonical preference models. Interestingly, given preferences of voters over committees, the procedure for computing an approximately stable lottery is the same for both models and therefore extends to the setting where some voters have the former preference structure and others have the latter. Our existence proof uses the probabilistic method and a new large deviation inequality that may be of independent interest.

In this paper, we study fairness in committee selection problems. We consider a general notion of fairness via stability: A committee is stable if no coalition of voters can deviate and choose a committee of proportional size, so that all these voters strictly prefer the new committee to the existing one. Our main contribution is to extend this definition to stability of a distribution (or lottery) over committees. We consider two canonical voter preference models: the Approval Set setting where each voter approves a set of candidates and prefers committees with larger intersection with this set; and the Ranking setting where each voter ranks committees based on how much she likes her favorite candidate in a committee. Our main result is to show that stable lotteries always exist for these canonical preference models. Interestingly, given preferences of voters over committees, the procedure for computing an approximately stable lottery is the same for both models and therefore extends to the setting where some voters have the former preference structure and others have the latter. Our existence proof uses the probabilistic method and a new large deviation inequality that may be of independent interest. item is used by several long-running applications [16,13]. Each data item can be treated as a candidate, and each application as a voter whose utility for an item corresponds to the speedup obtained by caching that item. In this context, we need a fair caching policy that provides proportional speedup to all applications.In this paper, we propose a new notion of proportionality in committee selection that generalizes several previously considered notions. Our main contribution is to show that stable solutions always exist for the stability notion that we propose, and such solutions can be computed efficiently. In contrast, stable solutions may not exist for some of the previously studied notions, and for the notions where stable solutions do exist, we do not know how to compute them efficiently (see Section 1.5 for a detailed discussion). Preference ModelsBefore proceeding further, we define the preference model of the voters for committees. We consider two canonical ordinal preference models over committees, both of which have been extensively studied in social choice literature.Approval Set. In this model [5,2,23,3], each voter v specifies an approval set A v ⊆ C of candidates. Given two committees S 1 and S 2 ,, the voter strictly prefers committees in which she has more approved candidates.Ranking. In this model [8], each voter v has a preference order over candidates in C. In this case, S 1 ≻ v S 2 iff v's favorite candidate in S 1 is ranked higher (in her preference ordering) than her favorite candidate in S 2 . The Chamberlin-Courant voting rule [6] for committee selection finds the social optimum assuming a cardinal preference function of this form.These models have been extensively studied because it is relatively simple to elicit an approval vote or a ranking over candidates. Viewed in terms of underlying...