16,775 research outputs found

    Approximating Hereditary Discrepancy via Small Width Ellipsoids

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    The Discrepancy of a hypergraph is the minimum attainable value, over two-colorings of its vertices, of the maximum absolute imbalance of any hyperedge. The Hereditary Discrepancy of a hypergraph, defined as the maximum discrepancy of a restriction of the hypergraph to a subset of its vertices, is a measure of its complexity. Lovasz, Spencer and Vesztergombi (1986) related the natural extension of this quantity to matrices to rounding algorithms for linear programs, and gave a determinant based lower bound on the hereditary discrepancy. Matousek (2011) showed that this bound is tight up to a polylogarithmic factor, leaving open the question of actually computing this bound. Recent work by Nikolov, Talwar and Zhang (2013) showed a polynomial time O~(log3n)\tilde{O}(\log^3 n)-approximation to hereditary discrepancy, as a by-product of their work in differential privacy. In this paper, we give a direct simple O(log3/2n)O(\log^{3/2} n)-approximation algorithm for this problem. We show that up to this approximation factor, the hereditary discrepancy of a matrix AA is characterized by the optimal value of simple geometric convex program that seeks to minimize the largest \ell_{\infty} norm of any point in a ellipsoid containing the columns of AA. This characterization promises to be a useful tool in discrepancy theory

    The Geometry of Differential Privacy: the Sparse and Approximate Cases

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    In this work, we study trade-offs between accuracy and privacy in the context of linear queries over histograms. This is a rich class of queries that includes contingency tables and range queries, and has been a focus of a long line of work. For a set of dd linear queries over a database xRNx \in \R^N, we seek to find the differentially private mechanism that has the minimum mean squared error. For pure differential privacy, an O(log2d)O(\log^2 d) approximation to the optimal mechanism is known. Our first contribution is to give an O(log2d)O(\log^2 d) approximation guarantee for the case of (\eps,\delta)-differential privacy. Our mechanism is simple, efficient and adds correlated Gaussian noise to the answers. We prove its approximation guarantee relative to the hereditary discrepancy lower bound of Muthukrishnan and Nikolov, using tools from convex geometry. We next consider this question in the case when the number of queries exceeds the number of individuals in the database, i.e. when d>nx1d > n \triangleq \|x\|_1. It is known that better mechanisms exist in this setting. Our second main contribution is to give an (\eps,\delta)-differentially private mechanism which is optimal up to a \polylog(d,N) factor for any given query set AA and any given upper bound nn on x1\|x\|_1. This approximation is achieved by coupling the Gaussian noise addition approach with a linear regression step. We give an analogous result for the \eps-differential privacy setting. We also improve on the mean squared error upper bound for answering counting queries on a database of size nn by Blum, Ligett, and Roth, and match the lower bound implied by the work of Dinur and Nissim up to logarithmic factors. The connection between hereditary discrepancy and the privacy mechanism enables us to derive the first polylogarithmic approximation to the hereditary discrepancy of a matrix AA

    On The Hereditary Discrepancy of Homogeneous Arithmetic Progressions

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    We show that the hereditary discrepancy of homogeneous arithmetic progressions is lower bounded by n1/O(loglogn)n^{1/O(\log \log n)}. This bound is tight up to the constant in the exponent. Our lower bound goes via proving an exponential lower bound on the discrepancy of set systems of subcubes of the boolean cube {0,1}d\{0, 1\}^d.Comment: To appear in the Proceedings of the American Mathematical Societ

    On largest volume simplices and sub-determinants

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    We show that the problem of finding the simplex of largest volume in the convex hull of nn points in Qd\mathbb{Q}^d can be approximated with a factor of O(logd)d/2O(\log d)^{d/2} in polynomial time. This improves upon the previously best known approximation guarantee of d(d1)/2d^{(d-1)/2} by Khachiyan. On the other hand, we show that there exists a constant c>1c>1 such that this problem cannot be approximated with a factor of cdc^d, unless P=NPP=NP. % This improves over the 1.091.09 inapproximability that was previously known. Our hardness result holds even if n=O(d)n = O(d), in which case there exists a \bar c\,^{d}-approximation algorithm that relies on recent sampling techniques, where cˉ\bar c is again a constant. We show that similar results hold for the problem of finding the largest absolute value of a subdeterminant of a d×nd\times n matrix

    Randomized Rounding for the Largest Simplex Problem

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    The maximum volume jj-simplex problem asks to compute the jj-dimensional simplex of maximum volume inside the convex hull of a given set of nn points in Qd\mathbb{Q}^d. We give a deterministic approximation algorithm for this problem which achieves an approximation ratio of ej/2+o(j)e^{j/2 + o(j)}. The problem is known to be NP\mathrm{NP}-hard to approximate within a factor of cjc^{j} for some constant c>1c > 1. Our algorithm also gives a factor ej+o(j)e^{j + o(j)} approximation for the problem of finding the principal j×jj\times j submatrix of a rank dd positive semidefinite matrix with the largest determinant. We achieve our approximation by rounding solutions to a generalization of the DD-optimal design problem, or, equivalently, the dual of an appropriate smallest enclosing ellipsoid problem. Our arguments give a short and simple proof of a restricted invertibility principle for determinants

    Towards a Constructive Version of Banaszczyk's Vector Balancing Theorem

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    An important theorem of Banaszczyk (Random Structures & Algorithms `98) states that for any sequence of vectors of 2\ell_2 norm at most 1/51/5 and any convex body KK of Gaussian measure 1/21/2 in Rn\mathbb{R}^n, there exists a signed combination of these vectors which lands inside KK. A major open problem is to devise a constructive version of Banaszczyk's vector balancing theorem, i.e. to find an efficient algorithm which constructs the signed combination. We make progress towards this goal along several fronts. As our first contribution, we show an equivalence between Banaszczyk's theorem and the existence of O(1)O(1)-subgaussian distributions over signed combinations. For the case of symmetric convex bodies, our equivalence implies the existence of a universal signing algorithm (i.e. independent of the body), which simply samples from the subgaussian sign distribution and checks to see if the associated combination lands inside the body. For asymmetric convex bodies, we provide a novel recentering procedure, which allows us to reduce to the case where the body is symmetric. As our second main contribution, we show that the above framework can be efficiently implemented when the vectors have length O(1/logn)O(1/\sqrt{\log n}), recovering Banaszczyk's results under this stronger assumption. More precisely, we use random walk techniques to produce the required O(1)O(1)-subgaussian signing distributions when the vectors have length O(1/logn)O(1/\sqrt{\log n}), and use a stochastic gradient ascent method to implement the recentering procedure for asymmetric bodies

    Nearly Optimal Private Convolution

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    We study computing the convolution of a private input xx with a public input hh, while satisfying the guarantees of (ϵ,δ)(\epsilon, \delta)-differential privacy. Convolution is a fundamental operation, intimately related to Fourier Transforms. In our setting, the private input may represent a time series of sensitive events or a histogram of a database of confidential personal information. Convolution then captures important primitives including linear filtering, which is an essential tool in time series analysis, and aggregation queries on projections of the data. We give a nearly optimal algorithm for computing convolutions while satisfying (ϵ,δ)(\epsilon, \delta)-differential privacy. Surprisingly, we follow the simple strategy of adding independent Laplacian noise to each Fourier coefficient and bounding the privacy loss using the composition theorem of Dwork, Rothblum, and Vadhan. We derive a closed form expression for the optimal noise to add to each Fourier coefficient using convex programming duality. Our algorithm is very efficient -- it is essentially no more computationally expensive than a Fast Fourier Transform. To prove near optimality, we use the recent discrepancy lowerbounds of Muthukrishnan and Nikolov and derive a spectral lower bound using a characterization of discrepancy in terms of determinants
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