With a typically large number of participants in local algorithms ( n in the millions), this reduction in time complexity, in particular at the user side, is crucial for the use of such algorithms in practice. We implemented Algorithm TreeHist to verify our theoretical analysis and compared its performance with the performance of Google's RAPPOR code.

With a typically large number of participants in local algorithms (n in the millions), this reduction in time complexity, in particular at the user side, is crucial for the use of such algorithms in practice. We implemented Algorithm TreeHist to verify our theoretical analysis and compared its performance with the performance of Google's RAPPOR code.

Recently, Valiant and Valiant [1, 2] showed that a class of distributional properties, which includes such practically relevant properties as entropy, the number of distinct elements, and distance metrics between pairs of distributions, can be estimated given a sublinear sized sample. Specifically, given a sample consisting of independent draws from any distribution over at most n distinct elements, these properties can be estimated accurately using a sample of size O(n/ log n). We propose a novel modification of this approach and show: 1) theoretically, this estimator is optimal (to constant factors, over worst-case instances), and 2) in practice, it performs exceptionally well for a variety of estimation tasks, on a variety of natural distributions, for a wide range of parameters. Perhaps unsurprisingly, the key step in our approach is to first use the sample to characterize the "unseen" portion of the distribution. This goes beyond such tools as the Good-Turing frequency estimation scheme, which estimates the total probability mass of the unobserved portion of the distribution: we seek to estimate the shape of the unobserved portion of the distribution. This approach is robust, general, and theoretically principled; we expect that it may be fruitfully used as a component within larger machine learning and data analysis systems.

We analyze how symmetries can be used to compress structures (also known as interpretations) onto a smaller domain without loss of information. This analysis suggests the possibility to solve satisfiability problems in the compressed domain for better performance. Thus, we propose a 2-step novel method: (i) the sentence to be satisfied is automatically translated into an equisatisfiable sentence over a ``lifted'' vocabulary that allows domain compression; (ii) satisfiability of the lifted sentence is checked by growing the (initially unknown) compressed domain until a satisfying structure is found. The key issue is to ensure that this satisfying structure can always be expanded into an uncompressed structure that satisfies the original sentence to be satisfied. We present an adequate translation for sentences in typed first-order logic extended with aggregates. Our experimental evaluation shows large speedups for generative configuration problems. The method also has applications in the verification of software operating on complex data structures. Our results justify further research in automatic translation of sentences for symmetry reduction.

Standpoint EL is a multi-modal extension of the popular description logic EL that allows for the integrated representation of domain knowledge relative to diverse standpoints or perspectives. Advantageously, its satisfiability problem has recently been shown to be in PTime, making it a promising framework for large-scale knowledge integration. In this paper, we show that we can further push the expressivity of this formalism, arriving at an extended logic, called Standpoint EL+, which allows for axiom negation, role chain axioms, self-loops, and other features, while maintaining tractability. This is achieved by designing a satisfiability-checking deduction calculus, which at the same time addresses the need for practical algorithms. We demonstrate the feasibility of our calculus by presenting a prototypical Datalog implementation of its deduction rules.

We consider the task of weighted first-order model counting (WFOMC) used for probabilistic inference in the area of statistical relational learning. Given a formula $\phi$, domain size $n$ and a pair of weight functions, what is the weighted sum of all models of $\phi$ over a domain of size $n$? It was shown that computing WFOMC of any logical sentence with at most two logical variables can be done in time polynomial in $n$. However, it was also shown that the task is $\texttt{#}P_1$-complete once we add the third variable, which inspired the search for extensions of the two-variable fragment that would still permit a running time polynomial in $n$. One of such extension is the two-variable fragment with counting quantifiers. In this paper, we prove that adding a linear order axiom (which forces one of the predicates in $\phi$ to introduce a linear ordering of the domain elements in each model of $\phi$) on top of the counting quantifiers still permits a computation time polynomial in the domain size. We present a new dynamic programming-based algorithm which can compute WFOMC with linear order in time polynomial in $n$, thus proving our primary claim.

Weighted knowledge bases for description logics with typicality have been recently considered under a "concept-wise" multipreference semantics (in both the two-valued and fuzzy case), as the basis of a logical semantics of Multilayer Perceptrons. In this paper we consider weighted conditional EL^bot knowledge bases in the two-valued case, and exploit ASP and asprin for encoding concept-wise multipreference entailment for weighted KBs with integer weights.

In this paper, we aim to study how the performance of modern answer set programming (ASP) solvers is influenced by the treewidth of the input program and to investigate the consequences of this relationship. We first perform an experimental evaluation that shows that the solving performance is heavily influenced by treewidth, given ground input programs that are otherwise uniform, both in size and construction. This observation leads to an important question for ASP, namely, how to design encodings such that the treewidth of the resulting ground program remains small. To this end, we study two classes of disjunctive programs, namely guarded and connection-guarded programs. In order to investigate these classes, we formalize the grounding process using MSO transductions. Our main results show that both classes guarantee that the treewidth of the program after grounding only depends on the treewidth (and the maximum degree, in case of connection-guarded programs) of the input instance. In terms of parameterized complexity, our findings yield corresponding FPT results for answer-set existence for bounded treewidth (and also degree, for connection-guarded programs) of the input instance. We further show that bounding treewidth alone leads to NP-hardness in the data complexity for connection-guarded programs, which indicates that the two classes are fundamentally different. Finally, we show that for both classes, the data complexity remains as hard as in the general case of ASP.

Local differential privacy (LDP) is a strong notion of privacy for individual users that often comes at the expense of a significant drop in utility. The classical definition of LDP assumes that all elements in the data domain are equally sensitive. However, in many applications, some symbols are more sensitive than others. This work proposes a context-aware framework of local differential privacy that allows a privacy designer to incorporate the application's context into the privacy definition. For binary data domains, we provide a universally optimal privatization scheme and highlight its connections to Warner's randomized response (RR) and Mangat's improved response. Motivated by geolocation and web search applications, for $k$-ary data domains, we consider two special cases of context-aware LDP: block-structured LDP and high-low LDP. We study discrete distribution estimation and provide communication-efficient, sample-optimal schemes and information-theoretic lower bounds for both models. We show that using contextual information can require fewer samples than classical LDP to achieve the same accuracy.

Estimating symmetric properties of a distribution, e.g. support size, coverage, entropy, distance to uniformity, are among the most fundamental problems in algorithmic statistics. While each of these properties have been studied extensively and separate optimal estimators are known for each, in striking recent work, Acharya et al. 2016 showed that there is a single estimator that is competitive for all symmetric properties. This work proved that computing the distribution that approximately maximizes \emph{profile likelihood (PML)}, i.e. the probability of observed frequency of frequencies, and returning the value of the property on this distribution is sample competitive with respect to a broad class of estimators of symmetric properties. Further, they showed that even computing an approximation of the PML suffices to achieve such a universal plug-in estimator. Unfortunately, prior to this work there was no known polynomial time algorithm to compute an approximate PML and it was open to obtain a polynomial time universal plug-in estimator through the use of approximate PML. In this paper we provide a algorithm (in number of samples) that, given $n$ samples from a distribution, computes an approximate PML distribution up to a multiplicative error of $\exp(n^{2/3} \mathrm{poly} \log(n))$ in time nearly linear in $n$. Generalizing work of Acharya et al. 2016 on the utility of approximate PML we show that our algorithm provides a nearly linear time universal plug-in estimator for all symmetric functions up to accuracy $\epsilon = \Omega(n^{-0.166})$. Further, we show how to extend our work to provide efficient polynomial-time algorithms for computing a $d$-dimensional generalization of PML (for constant $d$) that allows for universal plug-in estimation of symmetric relationships between distributions.