1,769 research outputs found

    Learning circuits with few negations

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    Monotone Boolean functions, and the monotone Boolean circuits that compute them, have been intensively studied in complexity theory. In this paper we study the structure of Boolean functions in terms of the minimum number of negations in any circuit computing them, a complexity measure that interpolates between monotone functions and the class of all functions. We study this generalization of monotonicity from the vantage point of learning theory, giving near-matching upper and lower bounds on the uniform-distribution learnability of circuits in terms of the number of negations they contain. Our upper bounds are based on a new structural characterization of negation-limited circuits that extends a classical result of A. A. Markov. Our lower bounds, which employ Fourier-analytic tools from hardness amplification, give new results even for circuits with no negations (i.e. monotone functions)

    The Inverse Shapley Value Problem

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    For ff a weighted voting scheme used by nn voters to choose between two candidates, the nn \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of ff provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the "influence" of voters. The \emph{Inverse Shapley Value Problem} is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant \eps > 0 our algorithm runs in fixed poly(n)(n) time (the degree of the polynomial is independent of \eps) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error \eps. If there is a "reasonable" voting scheme in which all voting weights are integers at most \poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time \poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error $\eps=n^{-1/8}.

    Learning DNF Expressions from Fourier Spectrum

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    Since its introduction by Valiant in 1984, PAC learning of DNF expressions remains one of the central problems in learning theory. We consider this problem in the setting where the underlying distribution is uniform, or more generally, a product distribution. Kalai, Samorodnitsky and Teng (2009) showed that in this setting a DNF expression can be efficiently approximated from its "heavy" low-degree Fourier coefficients alone. This is in contrast to previous approaches where boosting was used and thus Fourier coefficients of the target function modified by various distributions were needed. This property is crucial for learning of DNF expressions over smoothed product distributions, a learning model introduced by Kalai et al. (2009) and inspired by the seminal smoothed analysis model of Spielman and Teng (2001). We introduce a new approach to learning (or approximating) a polynomial threshold functions which is based on creating a function with range [-1,1] that approximately agrees with the unknown function on low-degree Fourier coefficients. We then describe conditions under which this is sufficient for learning polynomial threshold functions. Our approach yields a new, simple algorithm for approximating any polynomial-size DNF expression from its "heavy" low-degree Fourier coefficients alone. Our algorithm greatly simplifies the proof of learnability of DNF expressions over smoothed product distributions. We also describe an application of our algorithm to learning monotone DNF expressions over product distributions. Building on the work of Servedio (2001), we give an algorithm that runs in time \poly((s \cdot \log{(s/\eps)})^{\log{(s/\eps)}}, n), where ss is the size of the target DNF expression and \eps is the accuracy. This improves on \poly((s \cdot \log{(ns/\eps)})^{\log{(s/\eps)} \cdot \log{(1/\eps)}}, n) bound of Servedio (2001).Comment: Appears in Conference on Learning Theory (COLT) 201

    Learning Coverage Functions and Private Release of Marginals

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    We study the problem of approximating and learning coverage functions. A function c:2[n]R+c: 2^{[n]} \rightarrow \mathbf{R}^{+} is a coverage function, if there exists a universe UU with non-negative weights w(u)w(u) for each uUu \in U and subsets A1,A2,,AnA_1, A_2, \ldots, A_n of UU such that c(S)=uiSAiw(u)c(S) = \sum_{u \in \cup_{i \in S} A_i} w(u). Alternatively, coverage functions can be described as non-negative linear combinations of monotone disjunctions. They are a natural subclass of submodular functions and arise in a number of applications. We give an algorithm that for any γ,δ>0\gamma,\delta>0, given random and uniform examples of an unknown coverage function cc, finds a function hh that approximates cc within factor 1+γ1+\gamma on all but δ\delta-fraction of the points in time poly(n,1/γ,1/δ)poly(n,1/\gamma,1/\delta). This is the first fully-polynomial algorithm for learning an interesting class of functions in the demanding PMAC model of Balcan and Harvey (2011). Our algorithms are based on several new structural properties of coverage functions. Using the results in (Feldman and Kothari, 2014), we also show that coverage functions are learnable agnostically with excess 1\ell_1-error ϵ\epsilon over all product and symmetric distributions in time nlog(1/ϵ)n^{\log(1/\epsilon)}. In contrast, we show that, without assumptions on the distribution, learning coverage functions is at least as hard as learning polynomial-size disjoint DNF formulas, a class of functions for which the best known algorithm runs in time 2O~(n1/3)2^{\tilde{O}(n^{1/3})} (Klivans and Servedio, 2004). As an application of our learning results, we give simple differentially-private algorithms for releasing monotone conjunction counting queries with low average error. In particular, for any knk \leq n, we obtain private release of kk-way marginals with average error αˉ\bar{\alpha} in time nO(log(1/αˉ))n^{O(\log(1/\bar{\alpha}))}

    Uncertainty in Crowd Data Sourcing under Structural Constraints

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    Applications extracting data from crowdsourcing platforms must deal with the uncertainty of crowd answers in two different ways: first, by deriving estimates of the correct value from the answers; second, by choosing crowd questions whose answers are expected to minimize this uncertainty relative to the overall data collection goal. Such problems are already challenging when we assume that questions are unrelated and answers are independent, but they are even more complicated when we assume that the unknown values follow hard structural constraints (such as monotonicity). In this vision paper, we examine how to formally address this issue with an approach inspired by [Amsterdamer et al., 2013]. We describe a generalized setting where we model constraints as linear inequalities, and use them to guide the choice of crowd questions and the processing of answers. We present the main challenges arising in this setting, and propose directions to solve them.Comment: 8 pages, vision paper. To appear at UnCrowd 201
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