We consider a natural setting where network parameters are estimated from noisy and incomplete information about the network. More specifically, we investigate how we can efficiently estimate the number of small subgraphs (e.g., edges, triangles, etc.) based on full access to one or two noisy and incomplete samples of a large underlying network and on few queries revealing the neighborhood of carefully selected vertices. After specifying a random generator which removes edges from the underlying graph, we present estimators with strong provable performance guarantees, which exploit information from the noisy network samples and query a constant number of the most important vertices for the estimation. Our experimental evaluation shows that, in practice, a single noisy network sample and a couple of hundreds neighborhood queries suffice for accurately estimating the number of triangles in networks with millions of vertices and edges.
We study the problem of list-decodable linear regression, where an adversary can corrupt a majority of the examples. Specifically, we are given a set T of labeled examples (x, y) ∈ R d × R and a parameter 0 < α < 1/2 such that an α-fraction of the points in T are i.i.d. samples from a linear regression model with Gaussian covariates, and the remaining (1−α)-fraction of the points are drawn from an arbitrary noise distribution. The goal is to output a small list of hypothesis vectors such that at least one of them is close to the target regression vector. Our main result is a Statistical Query (SQ) lower bound of d poly(1/α) for this problem. Our SQ lower bound qualitatively matches the performance of previously developed algorithms, providing evidence that current upper bounds for this task are nearly best possible.
We study high-dimensional robust statistics tasks in the streaming model. A recent line of work obtained computationally efficient algorithms for a range of high-dimensional robust estimation tasks. Unfortunately, all previous algorithms require storing the entire dataset, incurring memory at least quadratic in the dimension. In this work, we develop the first efficient streaming algorithms for high-dimensional robust statistics with near-optimal memory requirements (up to logarithmic factors). Our main result is for the task of high-dimensional robust mean estimation in (a strengthening of) Huber's contamination model. We give an efficient single-pass streaming algorithm for this task with near-optimal error guarantees and space complexity nearly-linear in the dimension. As a corollary, we obtain streaming algorithms with near-optimal space complexity for several more complex tasks, including robust covariance estimation, robust regression, and more generally robust stochastic optimization.
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