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Nader H. Bshouty

Publications and source records attributed to Nader H. Bshouty.

At least 37 records · Page 2Linked to original sources
arXiv2020

Optimal Deterministic Group Testing Algorithms to Estimate the Number of Defectives

We study the problem of estimating the number of defective items $d$ within a pile of $n$ elements up to a multiplicative factor of $Δ>1$, using deterministic group testing algorithms. We bring lower and upper bounds on the number of tests required in both the adaptive and the non-adaptive deterministic settings given an upper bound $D$ on the defectives number. For the adaptive deterministic settings, our results show that, any algorithm for estimating the defectives number up to a multiplicative factor of $Δ$ must make at least $Ω\left((D/Δ^2)\log (n/D) \right )$ tests. This extends the same lower bound achieved in \cite{ALA17} for non-adaptive algorithms. Moreover, we give a polynomial time adaptive algorithm that shows that our bound is tight up to a small additive term. For non-adaptive algorithms, an upper bound of $O((D/Δ^2)$ $(\log (n/D)+\log Δ) )$ is achieved by means of non-constructive proof. This improves the lower bound $O((\log D)/(\logΔ))D\log n)$ from \cite{ALA17} and matches the lower bound up to a small additive term. In addition, we study polynomial time constructive algorithms. We use existing polynomial time constructible \emph{expander regular bipartite graphs}, \emph{extractors} and \emph{condensers} to construct two polynomial time algorithms. The first algorithm makes $O((D^{1+o(1)}/Δ^2)\cdot \log n)$ tests, and the second makes $(D/Δ^2)\cdot quazipoly$ $(\log n)$ tests. This is the first explicit construction with an almost optimal test complexity.

cs.IT↗
arXiv2020

Almost Optimal Distribution-free Junta Testing

We consider the problem of testing whether an unknown $n$-variable Boolean function is a $k$-junta in the distribution-free property testing model, where the distance between function is measured with respect to an arbitrary and unknown probability distribution over $\{0,1\}^n$. Chen, Liu, Servedio, Sheng and Xie showed that the distribution-free $k$-junta testing can be performed, with one-sided error, by an adaptive algorithm that makes $\tilde O(k^2)/ε$ queries. In this paper, we give a simple two-sided error adaptive algorithm that makes $\tilde O(k/ε)$ queries.

cs.DS↗
arXiv2020

An Optimal Tester for $k$-Linear

A Boolean function $f:\{0,1\}^n\to \{0,1\}$ is $k$-linear if it returns the sum (over the binary field $F_2$) of $k$ coordinates of the input. In this paper, we study property testing of the classes $k$-Linear, the class of all $k$-linear functions, and $k$-Linear$^*$, the class $\cup_{j=0}^kj$-Linear. We give a non-adaptive distribution-free two-sided $ε$-tester for $k$-Linear that makes $$O\left(k\log k+\frac{1}ε\right)$$ queries. This matches the lower bound known from the literature. We then give a non-adaptive distribution-free one-sided $ε$-tester for $k$-Linear$^*$ that makes the same number of queries and show that any non-adaptive uniform-distribution one-sided $ε$-tester for $k$-Linear must make at least $ \tildeΩ(k)\log n+Ω(1/ε)$ queries. The latter bound, almost matches the upper bound $O(k\log n+1/ε)$ known from the literature. We then show that any adaptive uniform-distribution one-sided $ε$-tester for $k$-Linear must make at least $\tildeΩ(\sqrt{k})\log n+Ω(1/ε)$ queries.

cs.CC↗
arXiv2019

Bounds for the Number of Tests in Non-Adaptive Randomized Algorithms for Group Testing

We study the group testing problem with non-adaptive randomized algorithms. Several models have been discussed in the literature to determine how to randomly choose the tests. For a model ${\cal M}$, let $m_{\cal M}(n,d)$ be the minimum number of tests required to detect at most $d$ defectives within $n$ items, with success probability at least $1-δ$, for some constant $δ$. In this paper, we study the measures $$c_{\cal M}(d)=\lim_{n\to \infty} \frac{m_{\cal M}(n,d)}{\ln n} \mbox{ and } c_{\cal M}=\lim_{d\to \infty} \frac{c_{\cal M}(d)}{d}.$$ In the literature, the analyses of such models only give upper bounds for $c_{\cal M}(d)$ and $c_{\cal M}$, and for some of them, the bounds are not tight. We give new analyses that yield tight bounds for $c_{\cal M}(d)$ and $c_{\cal M}$ for all the known models~${\cal M}$.

cs.LG↗
arXiv2019

Adaptive Exact Learning of Decision Trees from Membership Queries

In this paper we study the adaptive learnability of decision trees of depth at most $d$ from membership queries. This has many applications in automated scientific discovery such as drugs development and software update problem. Feldman solves the problem in a randomized polynomial time algorithm that asks $\tilde O(2^{2d})\log n$ queries and Kushilevitz-Mansour in a deterministic polynomial time algorithm that asks $ 2^{18d+o(d)}\log n$ queries. We improve the query complexity of both algorithms. We give a randomized polynomial time algorithm that asks $\tilde O(2^{2d}) + 2^{d}\log n$ queries and a deterministic polynomial time algorithm that asks $2^{5.83d}+2^{2d+o(d)}\log n$ queries.

cs.LG↗
arXiv2018

Elementary Proofs of Some Stirling Bounds

We give elementary proofs of several Stirling's precise bounds. We first improve all the precise bounds from the literature and give new precise bounds. In particular, we show that for all $n\ge 8$ $$\sqrt{2πn}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n}-\frac{1}{360n^3+103n}} \ge n!\ge \sqrt{2πn}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n}-\frac{1}{360n^3+102n}}$$ and for all $n\ge 3$ $$\sqrt{2πn}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n+\frac{2}{5n}-\frac{1.1}{10n^3}}} \ge n!\ge \sqrt{2πn}\left(\frac{n}{e}\right)^n e^{\frac{1}{12n+\frac{2}{5n}-\frac{0.9}{10n^3}}}.$$

math.FA↗
arXiv2018

On Learning Graphs with Edge-Detecting Queries

We consider the problem of learning a general graph $G=(V,E)$ using edge-detecting queries, where the number of vertices $|V|=n$ is given to the learner. The information theoretic lower bound gives $m\log n$ for the number of queries, where $m=|E|$ is the number of edges. In case the number of edges $m$ is also given to the learner, Angluin-Chen's Las Vegas algorithm \cite{AC08} runs in $4$ rounds and detects the edges in $O(m\log n)$ queries. In the other harder case where the number of edges $m$ is unknown, their algorithm runs in $5$ rounds and asks $O(m\log n+\sqrt{m}\log^2 n)$ queries. There have been two open problems: \emph{(i)} can the number of queries be reduced to $O(m\log n)$ in the second case, and, \emph{(ii)} can the number of rounds be reduced without substantially increasing the number of queries (in both cases). For the first open problem (when $m$ is unknown) we give two algorithms. The first is an $O(1)$-round Las Vegas algorithm that asks $m\log n+\sqrt{m}(\log^{[k]}n)\log n$ queries for any constant $k$ where $\log^{[k]}n=\log \stackrel{k}{\cdots} \log n$. The second is an $O(\log^*n)$-round Las Vegas algorithm that asks $O(m\log n)$ queries. This solves the first open problem for any practical $n$, for example, $n<2^{65536}$. We also show that no deterministic algorithm can solve this problem in a constant number of rounds. To solve the second problem we study the case when $m$ is known. We first show that any non-adaptive Monte Carlo algorithm (one-round) must ask at least $Ω(m^2\log n)$ queries, and any two-round Las Vegas algorithm must ask at least $m^{4/3-o(1)}\log n$ queries on average. We then give two two-round Monte Carlo algorithms, the first asks $O(m^{4/3}\log n)$ queries for any $n$ and $m$, and the second asks $O(m\log n)$ queries when $n>2^m$. Finally, we give a $3$-round Monte Carlo algorithm that asks $O(m\log n)$ queries for any $n$ and $m$.

cs.LG↗
arXiv2018

On Polynomial time Constructions of Minimum Height Decision Tree

In this paper we study a polynomial time algorithms that for an input $A\subseteq {B_m}$ outputs a decision tree for $A$ of minimum depth. This problem has many applications that include, to name a few, computer vision, group testing, exact learning from membership queries and game theory. Arkin et al. and Moshkov gave a polynomial time $(\ln |A|)$- approximation algorithm (for the depth). The result of Dinur and Steurer for set cover implies that this problem cannot be approximated with ratio $(1-o(1))\cdot \ln |A|$, unless P=NP. Moskov the combinatorial measure of extended teaching dimension of $A$, $ETD(A)$. He showed that $ETD(A)$ is a lower bound for the depth of the decision tree for $A$ and then gave an {\it exponential time} $ETD(A)/\log(ETD(A))$-approximation algorithm. In this paper we further study the $ETD(A)$ measure and a new combinatorial measure, $DEN(A)$, that we call the density of the set $A$. We show that $DEN(A)\le ETD(A)+1$. We then give two results. The first result is that the lower bound $ETD(A)$ of Moshkov for the depth of the decision tree for $A$ is greater than the bounds that are obtained by the classical technique used in the literature. The second result is a polynomial time $(\ln 2) DEN(A)$-approximation (and therefore $(\ln 2) ETD(A)$-approximation) algorithm for the depth of the decision tree of $A$. We also show that a better approximation ratio implies P=NP. We then apply the above results to learning the class of disjunctions of predicates from membership queries. We show that the $ETD$ of this class is bounded from above by the degree $d$ of its Hasse diagram. We then show that Moshkov algorithm can be run in polynomial time and is $(d/\log d)$-approximation algorithm. This gives optimal algorithms when the degree is constant. For example, learning axis parallel rays over constant dimension space.

cs.DS↗
arXiv2017

Non-Adaptive Randomized Algorithm for Group Testing

We study the problem of group testing with a non-adaptive randomized algorithm in the random incidence design (RID) model where each entry in the test is chosen randomly independently from $\{0,1\}$ with a fixed probability $p$. The property that is sufficient and necessary for a unique decoding is the separability of the tests, but unfortunately no linear time algorithm is known for such tests. In order to achieve linear-time decodable tests, the algorithms in the literature use the disjunction property that gives almost optimal number of tests. We define a new property for the tests which we call semi-disjunction property. We show that there is a linear time decoding for such test and for $d\to \infty$ the number of tests converges to the number of tests with the separability property and is therefore optimal (in the RID model). Our analysis shows that, in the RID model, the number of tests in our algorithm is better than the one with the disjunction property even for small $d$.

cs.LG↗
arXiv2017

The Maximum Cosine Framework for Deriving Perceptron Based Linear Classifiers

In this work, we introduce a mathematical framework, called the Maximum Cosine Framework or MCF, for deriving new linear classifiers. The method is based on selecting an appropriate bound on the cosine of the angle between the target function and the algorithm's. To justify its correctness, we use the MCF to show how to regenerate the update rule of Aggressive ROMMA. Moreover, we construct a cosine bound from which we build the Maximum Cosine Perceptron algorithm or, for short, the MCP algorithm. We prove that the MCP shares the same mistake bound like the Perceptron. In addition, we demonstrate the promising performance of the MCP on a real dataset. Our experiments show that, under the restriction of single pass learning, the MCP algorithm outperforms PA and Aggressive ROMMA.

cs.LG↗
arXiv2017

Exact Learning of Juntas from Membership Queries

In this paper, we study adaptive and non-adaptive exact learning of Juntas from membership queries. We use new techniques to find new bounds, narrow some of the gaps between the lower bounds and upper bounds and find new deterministic and randomized algorithms with small query and time complexities. Some of the bounds are tight in the sense that finding better ones either gives a breakthrough result in some long-standing combinatorial open problem or needs a new technique that is beyond the existing ones.

cs.LG↗
arXiv2017

Learning Disjunctions of Predicates

Let $F$ be a set of boolean functions. We present an algorithm for learning $F_\vee := \{\vee_{f\in S} f \mid S \subseteq F\}$ from membership queries. Our algorithm asks at most $|F| \cdot OPT(F_\vee)$ membership queries where $OPT(F_\vee)$ is the minimum worst case number of membership queries for learning $F_\vee$. When $F$ is a set of halfspaces over a constant dimension space or a set of variable inequalities, our algorithm runs in polynomial time. The problem we address has practical importance in the field of program synthesis, where the goal is to synthesize a program that meets some requirements. Program synthesis has become popular especially in settings aiming to help end users. In such settings, the requirements are not provided upfront and the synthesizer can only learn them by posing membership queries to the end user. Our work enables such synthesizers to learn the exact requirements while bounding the number of membership queries.

cs.LG↗
arXiv2017

Exact Learning from an Honest Teacher That Answers Membership Queries

Given a teacher that holds a function $f:X\to R$ from some class of functions $C$. The teacher can receive from the learner an element~$d$ in the domain $X$ (a query) and returns the value of the function in $d$, $f(d)\in R$. The learner goal is to find $f$ with a minimum number of queries, optimal time complexity, and optimal resources. In this survey, we present some of the results known from the literature, different techniques used, some new problems, and open problems.

cs.LG↗
arXiv2016

Enumerating all the Irreducible Polynomials over Finite Field

In this paper we give a detailed analysis of deterministic and randomized algorithms that enumerate any number of irreducible polynomials of degree $n$ over a finite field and their roots in the extension field in quasilinear where $N=n^2$ is the size of the output.} time cost per element. Our algorithm is based on an improved algorithm for enumerating all the Lyndon words of length $n$ in linear delay time and the known reduction of Lyndon words to irreducible polynomials.

cs.DM↗
arXiv2016

Derandomizing Chernoff Bound with Union Bound with an Application to $k$-wise Independent Sets

Derandomization of Chernoff bound with union bound is already proven in many papers. We here give another explicit version of it that obtains a construction of size that is arbitrary close to the probabilistic nonconstructive size. We apply this to give a new simple polynomial time constructions of almost $k$-wise independent sets. We also give almost tight lower bounds for the size of $k$-wise independent sets.

cs.DM↗
arXiv2015

Almost Optimal Cover-Free Families

Roughly speaking, an $(n,(r,s))$-Cover Free Family (CFF) is a small set of $n$-bit strings such that: "in any $d:=r+s$ indices we see all patterns of weight $r$". CFFs have been of interest for a long time both in discrete mathematics as part of block design theory, and in theoretical computer science where they have found a variety of applications, for example, in parametrized algorithms where they were introduced in the recent breakthrough work of Fomin, Lokshtanov and Saurabh under the name `lopsided universal sets'. In this paper we give the first explicit construction of cover-free families of optimal size up to lower order multiplicative terms, {for any $r$ and $s$}. In fact, our construction time is almost linear in the size of the family. Before our work, such a result existed only for $r=d^{o(1)}$. and $r= ω(d/(\log\log d\log\log\log d))$. As a sample application, we improve the running times of parameterized algorithms from the recent work of Gabizon, Lokshtanov and Pilipczuk.

cs.DM↗