Group testing is the process of pooling arbitrary subsets from a set of n items so as to identify, with a minimal number of tests, a "small" subset of d defective items. In "classical" non-adaptive group testing, it is known that when d is substantially smaller than n, Θ(d log(n)) tests are both information-theoretically necessary and sufficient to guarantee recovery with high probability. Group testing schemes in the literature meeting this bound require most items to be tested Ω(log(n)) times, and most tests to incorporate Ω(n/d) items.Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be "sparse". Specifically, we consider (separately) scenarios in which (a) items are finitely divisible and hence may participate in at most γ ∈ o(log(n)) tests; or (b) tests are size-constrained to pool no more than ρ ∈ o(n/d)items per test. For both scenarios we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In particular, one of our main results shows that γ-finite divisibility of items forces any non-adaptive group testing algorithm with probability of recovery error at most ǫ to perform at least γd(n/d) (1−5ǫ)/γ tests. Analogously, for ρ-sized constrained tests, we show an information-theoretic lower bound of Ω(n/ρ) tests -hence in both settings the number of tests required grow dramatically (relative to the classical setting) as a function of n. In both scenarios we provide both randomized constructions (under both ǫ-error and zero-error reconstruction guarantees) and explicit constructions of designs with computationally efficient reconstruction algorithms that require a number of tests that are optimal up to constant or small polynomial factors in some regimes of n, d, γ and ρ. The randomized design/reconstruction algorithm in the ρ-sized test scenario is universal -independent of the value of d, as long as ρ ∈ o(n/d). We also investigate the effect of unreliability/noise in test outcomes. * A preliminary version [24] of this paper appeared in the
Group testing is the process of pooling arbitrary subsets from a set of n items so as to identify, with a minimal number of disjunctive tests, a "small" subset of d defective items. In "classical" non-adaptive group testing, it is known that when d = o(n 1−δ ) for any δ > 0, θ(d log(n)) tests are both information-theoretically necessary, and sufficient to guarantee recovery with high probability. Group testing schemes in the literature meeting this bound require most items to be tested Ω(log(n)) times, and most tests to incorporate Ω(n/d) items.Motivated by physical considerations, we study group testing models in which the testing procedure is constrained to be "sparse". Specifically, we consider (separately) scenarios in which (a) items are finitely divisible and hence may participate in at most γ tests; and (b) tests are size-constrained to pool no more than ρ items per test. For both scenarios we provide information-theoretic lower bounds on the number of tests required to guarantee high probability recovery. In particular, one of our main results shows that γ-finite divisibility of items forces any group testing algorithm with probability of recovery error at most ǫ to perform at least Ω(γd(n/d) (1−2ǫ)/((1+2ǫ)γ) ) tests. Analogously, for ρ-sized constrained tests, we show an information-theoretic lower bound of Ω(n log(n/d)/(ρ log(n/ρd))). In both scenarios we provide both randomized constructions (under both ǫ-error and zero-error reconstruction guarantees) and explicit constructions of computationally efficient group-testing algorithms (under ǫ-error reconstruction guarantees) that require a number of tests that are optimal up to constant factors in some regimes of n, d, γ and ρ. We also investigate the effect of unreliability/noise in test outcomes.
In this work, we present a family of vector quantization schemes vqSGD (Vector-antized Stochastic Gradient Descent) that provide asymptotic reduction in the communication cost with convergence guarantees in distributed computation and learning se ings. In particular, we consider a randomized scheme, based on convex hull of a point set, that returns an unbiased estimator of a d-dimensional gradient vector with bounded variance. We provide multiple e cient instances of our scheme that require only O(log d) bits of communication. Further, we show that vqSGD also provides strong privacy guarantees. Experimentally, we show vqSGD performs equally well compared to other state-of-the-art quantization schemes, while substantially reducing the communication cost.
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