We study the power of interactivity in local differential privacy. First, we focus on the difference between fully interactive and sequentially interactive protocols. Sequentially interactive protocols may query users adaptively in sequence, but they cannot return to previously queried users. The vast majority of existing lower bounds for local differential privacy apply only to sequentially interactive protocols, and before this paper it was not known whether fully interactive protocols were more powerful.We resolve this question. First, we classify locally private protocols by their compositionality, the multiplicative factor k ≥ 1 by which the sum of a protocol's single-round privacy parameters exceeds its overall privacy guarantee. We then show how to efficiently transform any fully interactive k-compositional protocol into an equivalent sequentially interactive protocol with an O(k) blowup in sample complexity. Next, we show that our reduction is tight by exhibiting a family of problems such that for any k, there is a fully interactive k-compositional protocol which solves the problem, while no sequentially interactive protocol can solve the problem without at least anΩ(k) factor more examples.We then turn our attention to hypothesis testing problems. We show that for a large class of compound hypothesis testing problems -which include all simple hypothesis testing problems as a special case -a simple noninteractive test is optimal among the class of all (possibly fully interactive) tests.

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We consider the problem of finding the k th highest element in a totally ordered set of n elements (Select), and partitioning a totally ordered set into the top k and bottom n − k elements (Partition) using pairwise comparisons. Motivated by settings like peer grading or crowdsourcing, where multiple rounds of interaction are costly and queried comparisons may be inconsistent with the ground truth, we evaluate algorithms based both on their total runtime and the number of interactive rounds in three comparison models: noiseless (where the comparisons are correct), erasure (where comparisons are erased with probability 1 − γ), and noisy (where comparisons are correct with probability 1/2 + γ/2 and incorrect otherwise). We provide numerous matching upper and lower bounds in all three models. Even our results in the noiseless model, which is quite well-studied in the TCS literature on parallel algorithms, are novel.

We consider the question of interactive communication, in which two remote parties perform a computation while their communication channel is (adversarially) noisy. We extend here the discussion into a more general and stronger class of noise, namely, we allow the channel to perform insertions and deletions of symbols. These types of errors may bring the parties "out of sync", so that there is no consensus regarding the current round of the protocol.In this more general noise model, we obtain the first interactive coding scheme that has a constant rate and tolerates noise rates of up to 1/18−ε. To this end we develop a novel primitive we name edit distance tree code. The edit distance tree code is designed to replace the Hamming distance constraints in Schulman's tree codes (STOC 93), with a stronger edit distance requirement. However, the straightforward generalization of tree codes to edit distance does not seem to yield a primitive that suffices for communication in the presence of synchronization problems. Giving the "right" definition of edit distance tree codes is a main conceptual contribution of this work.

We consider the problem of a single seller repeatedly selling a single item to a single buyer (specifically, the buyer has a value drawn fresh from known distribution D in every round). Prior work assumes that the buyer is fully rational and will perfectly reason about how their bids today affect the seller's decisions tomorrow. In this work we initiate a different direction: the buyer simply runs a no-regret learning algorithm over possible bids. We provide a fairly complete characterization of optimal auctions for the seller in this domain. Specifically:• If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), then the seller can extract expected revenue arbitrarily close to the expected welfare. This auction is independent of the buyer's valuation D, but somewhat unnatural as it is sometimes in the buyer's interest to overbid.• There exists a learning algorithm A such that if the buyer bids according to A then the optimal strategy for the seller is simply to post the Myerson reserve for D every round.• If the buyer bids according to EXP3 (or any "mean-based" learning algorithm), but the seller is restricted to "natural" auction formats where overbidding is dominated (e.g. Generalized First-Price or Generalized Second-Price), then the optimal strategy for the seller is a pay-your-bid format with decreasing reserves over time. Moreover, the seller's optimal achievable revenue is characterized by a linear program, and can be unboundedly better than the best truthful auction yet simultaneously unboundedly worse than the expected welfare.2. There exists a natural no-regret algorithm A such that when the buyer bids according to A, the seller's default strategy (charging the Myerson reserve every round) is optimal (Theorem 3.2).3. If the buyer uses a "mean-based" algorithm only over undominated strategies, the seller can extract revenue MBRev(D) using an auction where overbidding is dominated in every round. Moreover, we characterize MBRev(D) as the value of a linear program, and show it can be

We consider the following communication problem: Alice and Bob each have some valuation functions v 1 (·) and v 2 (·) over subsets of m items, and their goal is to partition the items into S,S in a way that maximizes the welfare, v 1 (S)+v 2 (S). We study both the allocation problem, which asks for a welfare-maximizing partition and the decision problem, which asks whether or not there exists a partition guaranteeing certain welfare, for binary XOS valuations. For interactive protocols with poly(m) communication, a tight 3/4-approximation is known for both [29,23].For interactive protocols, the allocation problem is provably harder than the decision problem: any solution to the allocation problem implies a solution to the decision problem with one additional round and log m additional bits of communication via a trivial reduction. Surprisingly, the allocation problem is provably easier for simultaneous protocols. Specifically, we show:• There exists a simultaneous, randomized protocol with polynomial communication that selects a partition whose expected welfare is at least 3/4 of the optimum. This matches the guarantee of the best interactive, randomized protocol with polynomial communication.• For all ε > 0, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is ≥ 1 or ≤ 3/4 − 1/108 + ε correctly with probability > 1/2 + 1/poly(m) requires exponential communication. (≤ 3/4 − 1/108) protocols with polynomial communication. In other words, this trivial reduction from decision to allocation problems provably requires the extra round of communication. We further discuss the implications of our results for the design of truthful combinatorial auctions in general, and extensions to general XOS valuations. In particular, our protocol for the allocation problem implies a new style of truthful mechanisms. IntroductionIntuitively, search problems (find the optimal solution) are considered "strictly harder" than decision problems (does a solution with quality ≥ Q exist?) for the following (formal) reason: once you find the optimal solution, you can simply evaluate it and check whether its quality is ≥ Q or not. The same intuition carries over to approximation as well: once you find a solution whose quality is within a factor α of optimal, you can distinguish between cases where solutions with quality ≥ Q exist and those where all solutions have quality ≤ αQ. The easy conclusion one then draws is that the communication (resp. runtime) required for an α-approximation to any decision problem is upper bounded by the communication (resp. runtime) required for an α-approximation to the corresponding search problem plus the communication (resp. runtime) required to evaluate the quality of a proposed solution.Note though that for communication problems, in addition to the negligible increase in communication (due to evaluating the quality of the proposed solution), this simple reduction might also require (at least) an extra round of communication (because the parties can evalua...

We study the active learning problem of top-k ranking from multi-wise comparisons under the popular multinomial logit model. Our goal is to identify the topk items with high probability by adaptively querying sets for comparisons and observing the noisy output of the most preferred item from each comparison. To achieve this goal, we design a new active ranking algorithm without using any information about the underlying items' preference scores. We also establish a matching lower bound on the sample complexity even when the set of preference scores is given to the algorithm. These two results together show that the proposed algorithm is nearly instance optimal (similar to instance optimal [12], but up to polylog factors). Our work extends the existing literature on rank aggregation in three directions. First, instead of studying a static problem with fixed data, we investigate the top-k ranking problem in an active learning setting. Second, we show our algorithm is nearly instance optimal, which is a much stronger theoretical guarantee. Finally, we extend the pairwise comparison to the multi-wise comparison, which has not been fully explored in ranking literature.

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top K items from noisy pairwise comparisons. In our setting, we are non-actively given r pairwise comparisons between each pair of n items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity (SST) model. We analyze the competitive ratio of algorithms for the top-K problem. In particular, we present a linear time algorithm for the top-K problem which has a competitive ratio ofÕ( √ n); i.e. to solve any instance of top-K, our algorithm needs at mostÕ( √ n) times as many samples needed as the best possible algorithm for that instance (in contrast, all previous known algorithms for the top-K problem have competitive ratios ofΩ(n) or worse). We further show that this is tight: any algorithm for the top-K problem has competitive ratio at leastΩ( √ n). * Full version of this paper can be found at https://arxiv.org/abs/1605.03933.