We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node q, the algorithm learns either that q is the target, or is given an edge out of q that lies on a shortest path from q to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability p > 1 2 (a known constant), and an (adversarial) incorrect one with probability 1 − p.Our main positive result is that when p = 1 (i.e., all answers are correct), log 2 n queries are always sufficient. For general p, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than (1−δ) log 2 n 1−H(p) +o(log n)+O(log 2 (1/δ)) queries, and identifies the target correctly with probability at least 1−δ. Here, H(p) = −(p log p+(1−p) log(1−p)) denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm.Even for p = 1, we show several hardness results for the problem of determining whether a target can be found using K queries. Our upper bound of log 2 n implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semiadaptive version, in which one may query r nodes each in k rounds, we show membership in Σ 2k−1 in the polynomial hierarchy, and hardness for Σ 2k−5 .

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We present novel, computationally efficient, and differentially private algorithms for two fundamental high-dimensional learning problems: learning a multivariate Gaussian and learning a product distribution over the Boolean hypercube in total variation distance. The sample complexity of our algorithms nearly matches the sample complexity of the optimal non-private learners for these tasks in a wide range of parameters, showing that privacy comes essentially for free for these problems. In particular, in contrast to previous approaches, our algorithm for learning Gaussians does not require strong a priori bounds on the range of the parameters. Our algorithms introduce a novel technical approach to reducing the sensitivity of the estimation procedure that we call recursive private preconditioning.

We give the first polynomial-time, polynomial-sample, differentially private estimator for the mean and covariance of an arbitrary Gaussian distribution N ( , Σ) in R . All previous estimators are either nonconstructive, with unbounded running time, or require the user to specify a priori bounds on the parameters and Σ. The primary new technical tool in our algorithm is a new differentially private preconditioner that takes samples from an arbitrary Gaussian N (0, Σ) and returns a matrix such that Σ has constant condition number.

We prove new lower bounds for statistical estimation tasks under the constraint of pε, δqdifferential privacy. First, we provide tight lower bounds for private covariance estimation of Gaussian distributions. We show that estimating the covariance matrix in Frobenius norm requires Ω `d2 ˘samples, and in spectral norm requires Ω ´d 3 2 ¯samples, both matching upper bounds up to logarithmic factors. We prove these bounds via our main technical contribution, a broad generalization of the fingerprinting method [BUV14] to exponential families. Additionally, using the private Assouad method of Acharya, Sun, and Zhang [ASZ21], we show a tight Ω `d α 2 ε ˘lower bound for estimating the mean of a distribution with bounded covariance to α-error in ℓ 2 -distance. Prior known lower bounds for all these problems were either polynomially weaker or held under the stricter condition of pε, 0q-differential privacy.