arXiv subjects

Anindya De

Publications and source records attributed to Anindya De.

At least 55 records · Page 3Linked to original sources

arXiv2017

Sharp bounds for population recovery

The population recovery problem is a basic problem in noisy unsupervised learning that has attracted significant research attention in recent years [WY12,DRWY12, MS13, BIMP13, LZ15,DST16]. A number of different variants of this problem have been studied, often under assumptions on the unknown distribution (such as that it has restricted support size). In this work we study the sample complexity and algorithmic complexity of the most general version of the problem, under both bit-flip noise and erasure noise model. We give essentially matching upper and lower sample complexity bounds for both noise models, and efficient algorithms matching these sample complexity bounds up to polynomial factors.

cs.DS↗

arXiv2017

Noise Stability is computable and low dimensional

Questions of noise stability play an important role in hardness of approximation in computer science as well as in the theory of voting. In many applications, the goal is to find an optimizer of noise stability among all possible partitions of $\mathbb{R}^n$ for $n \geq 1$ to $k$ parts with given Gaussian measures $μ_1,\ldots,μ_k$. We call a partition $ε$-optimal, if its noise stability is optimal up to an additive $ε$. In this paper, we give an explicit, computable function $n(ε)$ such that an $ε$-optimal partition exists in $\mathbb{R}^{n(ε)}$. This result has implications for the computability of certain problems in non-interactive simulation, which are addressed in a subsequent work.

math.PR↗

arXiv2017

Non interactive simulation of correlated distributions is decidable

A basic problem in information theory is the following: Let $\mathbf{P} = (\mathbf{X}, \mathbf{Y})$ be an arbitrary distribution where the marginals $\mathbf{X}$ and $\mathbf{Y}$ are (potentially) correlated. Let Alice and Bob be two players where Alice gets samples $\{x_i\}_{i \ge 1}$ and Bob gets samples $\{y_i\}_{i \ge 1}$ and for all $i$, $(x_i, y_i) \sim \mathbf{P}$. What joint distributions $\mathbf{Q}$ can be simulated by Alice and Bob without any interaction? Classical works in information theory by G{á}cs-K{ö}rner and Wyner answer this question when at least one of $\mathbf{P}$ or $\mathbf{Q}$ is the distribution on $\{0,1\} \times \{0,1\}$ where each marginal is unbiased and identical. However, other than this special case, the answer to this question is understood in very few cases. Recently, Ghazi, Kamath and Sudan showed that this problem is decidable for $\mathbf{Q}$ supported on $\{0,1\} \times \{0,1\}$. We extend their result to $\mathbf{Q}$ supported on any finite alphabet. We rely on recent results in Gaussian geometry (by the authors) as well as a new \emph{smoothing argument} inspired by the method of \emph{boosting} from learning theory and potential function arguments from complexity theory and additive combinatorics.

cs.CC↗

arXiv2016

Optimal mean-based algorithms for trace reconstruction

In the (deletion-channel) trace reconstruction problem, there is an unknown $n$-bit source string $x$. An algorithm is given access to independent traces of $x$, where a trace is formed by deleting each bit of~$x$ independently with probability~$δ$. The goal of the algorithm is to recover~$x$ exactly (with high probability), while minimizing samples (number of traces) and running time. Previously, the best known algorithm for the trace reconstruction problem was due to Holenstein~et~al.; it uses $\exp(\tilde{O}(n^{1/2}))$ samples and running time for any fixed $0 < δ< 1$. It is also what we call a "mean-based algorithm", meaning that it only uses the empirical means of the individual bits of the traces. Holenstein~et~al.~also gave a lower bound, showing that any mean-based algorithm must use at least $n^{\tildeΩ(\log n)}$ samples. In this paper we improve both of these results, obtaining matching upper and lower bounds for mean-based trace reconstruction. For any constant deletion rate $0 < δ< 1$, we give a mean-based algorithm that uses $\exp(O(n^{1/3}))$ time and traces; we also prove that any mean-based algorithm must use at least $\exp(Ω(n^{1/3}))$ traces. In fact, we obtain matching upper and lower bounds even for $δ$ subconstant and $ρ:= 1-δ$ subconstant: when $(\log^3 n)/n \ll δ\leq 1/2$ the bound is $\exp(-Θ(δn)^{1/3})$, and when $1/\sqrt{n} \ll ρ\leq 1/2$ the bound is $\exp(-Θ(n/ρ)^{1/3})$. Our proofs involve estimates for the maxima of Littlewood polynomials on complex disks. We show that these techniques can also be used to perform trace reconstruction with random insertions and bit-flips in addition to deletions. We also find a surprising result: for deletion probabilities $δ> 1/2$, the presence of insertions can actually help with trace reconstruction.

cs.CC↗

arXiv2016

A Size-Free CLT for Poisson Multinomials and its Applications

An $(n,k)$-Poisson Multinomial Distribution (PMD) is the distribution of the sum of $n$ independent random vectors supported on the set ${\cal B}_k=\{e_1,\ldots,e_k\}$ of standard basis vectors in $\mathbb{R}^k$. We show that any $(n,k)$-PMD is ${\rm poly}\left({k\over σ}\right)$-close in total variation distance to the (appropriately discretized) multi-dimensional Gaussian with the same first two moments, removing the dependence on $n$ from the Central Limit Theorem of Valiant and Valiant. Interestingly, our CLT is obtained by bootstrapping the Valiant-Valiant CLT itself through the structural characterization of PMDs shown in recent work by Daskalakis, Kamath, and Tzamos. In turn, our stronger CLT can be leveraged to obtain an efficient PTAS for approximate Nash equilibria in anonymous games, significantly improving the state of the art, and matching qualitatively the running time dependence on $n$ and $1/\varepsilon$ of the best known algorithm for two-strategy anonymous games. Our new CLT also enables the construction of covers for the set of $(n,k)$-PMDs, which are proper and whose size is shown to be essentially optimal. Our cover construction combines our CLT with the Shapley-Folkman theorem and recent sparsification results for Laplacian matrices by Batson, Spielman, and Srivastava. Our cover size lower bound is based on an algebraic geometric construction. Finally, leveraging the structural properties of the Fourier spectrum of PMDs we show that these distributions can be learned from $O_k(1/\varepsilon^2)$ samples in ${\rm poly}_k(1/\varepsilon)$-time, removing the quasi-polynomial dependence of the running time on $1/\varepsilon$ from the algorithm of Daskalakis, Kamath, and Tzamos.

cs.DS↗

arXiv2016

Noisy population recovery in polynomial time

In the noisy population recovery problem of Dvir et al., the goal is to learn an unknown distribution $f$ on binary strings of length $n$ from noisy samples. For some parameter $μ\in [0,1]$, a noisy sample is generated by flipping each coordinate of a sample from $f$ independently with probability $(1-μ)/2$. We assume an upper bound $k$ on the size of the support of the distribution, and the goal is to estimate the probability of any string to within some given error $\varepsilon$. It is known that the algorithmic complexity and sample complexity of this problem are polynomially related to each other. We show that for $μ> 0$, the sample complexity (and hence the algorithmic complexity) is bounded by a polynomial in $k$, $n$ and $1/\varepsilon$ improving upon the previous best result of $\mathsf{poly}(k^{\log\log k},n,1/\varepsilon)$ due to Lovett and Zhang. Our proof combines ideas from Lovett and Zhang with a \emph{noise attenuated} version of Möbius inversion. In turn, the latter crucially uses the construction of \emph{robust local inverse} due to Moitra and Saks.

cs.CC↗

arXiv2014

Boolean function monotonicity testing requires (almost) $n^{1/2}$ non-adaptive queries

We prove a lower bound of $Ω(n^{1/2 - c})$, for all $c>0$, on the query complexity of (two-sided error) non-adaptive algorithms for testing whether an $n$-variable Boolean function is monotone versus constant-far from monotone. This improves a $\tildeΩ(n^{1/5})$ lower bound for the same problem that was recently given in [CST14] and is very close to $Ω(n^{1/2})$, which we conjecture is the optimal lower bound for this model.

cs.CC↗

arXiv2013

Efficient deterministic approximate counting for low-degree polynomial threshold functions

We give a deterministic algorithm for approximately counting satisfying assignments of a degree-$d$ polynomial threshold function (PTF). Given a degree-$d$ input polynomial $p(x_1,\dots,x_n)$ over $R^n$ and a parameter $ε> 0$, our algorithm approximates $\Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0]$ to within an additive $\pm ε$ in time $O_{d,ε}(1)\cdot \mathop{poly}(n^d)$. (Any sort of efficient multiplicative approximation is impossible even for randomized algorithms assuming $NP\not=RP$.) Note that the running time of our algorithm (as a function of $n^d$, the number of coefficients of a degree-$d$ PTF) is a \emph{fixed} polynomial. The fastest previous algorithm for this problem (due to Kane), based on constructions of unconditional pseudorandom generators for degree-$d$ PTFs, runs in time $n^{O_{d,c}(1) \cdot ε^{-c}}$ for all $c > 0$. The key novel contributions of this work are: A new multivariate central limit theorem, proved using tools from Malliavin calculus and Stein's Method. This new CLT shows that any collection of Gaussian polynomials with small eigenvalues must have a joint distribution which is very close to a multidimensional Gaussian distribution. A new decomposition of low-degree multilinear polynomials over Gaussian inputs. Roughly speaking we show that (up to some small error) any such polynomial can be decomposed into a bounded number of multilinear polynomials all of which have extremely small eigenvalues. We use these new ingredients to give a deterministic algorithm for a Gaussian-space version of the approximate counting problem, and then employ standard techniques for working with low-degree PTFs (invariance principles and regularity lemmas) to reduce the original approximate counting problem over the Boolean hypercube to the Gaussian version.

cs.CC↗

arXiv2013

Deterministic Approximate Counting for Degree-$2$ Polynomial Threshold Functions

We give a {\em deterministic} algorithm for approximately computing the fraction of Boolean assignments that satisfy a degree-$2$ polynomial threshold function. Given a degree-2 input polynomial $p(x_1,\dots,x_n)$ and a parameter $\eps > 0$, the algorithm approximates \[ \Pr_{x \sim \{-1,1\}^n}[p(x) \geq 0] \] to within an additive $\pm \eps$ in time $\poly(n,2^{\poly(1/\eps)})$. Note that it is NP-hard to determine whether the above probability is nonzero, so any sort of multiplicative approximation is almost certainly impossible even for efficient randomized algorithms. This is the first deterministic algorithm for this counting problem in which the running time is polynomial in $n$ for $\eps= o(1)$. For "regular" polynomials $p$ (those in which no individual variable's influence is large compared to the sum of all $n$ variable influences) our algorithm runs in $\poly(n,1/\eps)$ time. The algorithm also runs in $\poly(n,1/\eps)$ time to approximate $\Pr_{x \sim N(0,1)^n}[p(x) \geq 0]$ to within an additive $\pm \eps$, for any degree-2 polynomial $p$. As an application of our counting result, we give a deterministic FPT multiplicative $(1 \pm \eps)$-approximation algorithm to approximate the $k$-th absolute moment $\E_{x \sim \{-1,1\}^n}[|p(x)^k|]$ of a degree-2 polynomial. The algorithm's running time is of the form $\poly(n) \cdot f(k,1/\eps)$.

cs.CC↗

arXiv2013

Deterministic Approximate Counting for Juntas of Degree-$2$ Polynomial Threshold Functions

Let $g: \{-1,1\}^k \to \{-1,1\}$ be any Boolean function and $q_1,\dots,q_k$ be any degree-2 polynomials over $\{-1,1\}^n.$ We give a \emph{deterministic} algorithm which, given as input explicit descriptions of $g,q_1,\dots,q_k$ and an accuracy parameter $\eps>0$, approximates \[\Pr_{x \sim \{-1,1\}^n}[g(\sign(q_1(x)),\dots,\sign(q_k(x)))=1]\] to within an additive $\pm \eps$. For any constant $\eps > 0$ and $k \geq 1$ the running time of our algorithm is a fixed polynomial in $n$. This is the first fixed polynomial-time algorithm that can deterministically approximately count satisfying assignments of a natural class of depth-3 Boolean circuits. Our algorithm extends a recent result \cite{DDS13:deg2count} which gave a deterministic approximate counting algorithm for a single degree-2 polynomial threshold function $\sign(q(x)),$ corresponding to the $k=1$ case of our result. Our algorithm and analysis requires several novel technical ingredients that go significantly beyond the tools required to handle the $k=1$ case in \cite{DDS13:deg2count}. One of these is a new multidimensional central limit theorem for degree-2 polynomials in Gaussian random variables which builds on recent Malliavin-calculus-based results from probability theory. We use this CLT as the basis of a new decomposition technique for $k$-tuples of degree-2 Gaussian polynomials and thus obtain an efficient deterministic approximate counting algorithm for the Gaussian distribution. Finally, a third new ingredient is a "regularity lemma" for \emph{$k$-tuples} of degree-$d$ polynomial threshold functions. This generalizes both the regularity lemmas of \cite{DSTW:10,HKM:09} and the regularity lemma of Gopalan et al \cite{GOWZ10}. Our new regularity lemma lets us extend our deterministic approximate counting results from the Gaussian to the Boolean domain.

cs.CC↗

arXiv2013

Explicit Optimal Hardness via Gaussian stability results

The results of Raghavendra (2008) show that assuming Khot's Unique Games Conjecture (2002), for every constraint satisfaction problem there exists a generic semi-definite program that achieves the optimal approximation factor. This result is existential as it does not provide an explicit optimal rounding procedure nor does it allow to calculate exactly the Unique Games hardness of the problem. Obtaining an explicit optimal approximation scheme and the corresponding approximation factor is a difficult challenge for each specific approximation problem. An approach for determining the exact approximation factor and the corresponding optimal rounding was established in the analysis of MAX-CUT (KKMO 2004) and the use of the Invariance Principle (MOO 2005). However, this approach crucially relies on results explicitly proving optimal partitions in Gaussian space. Until recently, Borell's result (Borell 1985) was the only non-trivial Gaussian partition result known. In this paper we derive the first explicit optimal approximation algorithm and the corresponding approximation factor using a new result on Gaussian partitions due to Isaksson and Mossel (2012). This Gaussian result allows us to determine exactly the Unique Games Hardness of MAX-3-EQUAL. In particular, our results show that Zwick algorithm for this problem achieves the optimal approximation factor and prove that the approximation achieved by the algorithm is $\approx 0.796$ as conjectured by Zwick. We further use the previously known optimal Gaussian partitions results to obtain a new Unique Games Hardness factor for MAX-k-CSP : Using the well known fact that jointly normal pairwise independent random variables are fully independent, we show that the the UGC hardness of Max-k-CSP is $\frac{\lceil (k+1)/2 \rceil}{2^{k-1}}$, improving on results of Austrin and Mossel (2009).

cs.CC↗

arXiv2013

A Polynomial-time Approximation Scheme for Fault-tolerant Distributed Storage

We consider a problem which has received considerable attention in systems literature because of its applications to routing in delay tolerant networks and replica placement in distributed storage systems. In abstract terms the problem can be stated as follows: Given a random variable $X$ generated by a known product distribution over $\{0,1\}^n$ and a target value $0 \leq θ\leq 1$, output a non-negative vector $w$, with $\|w\|_1 \le 1$, which maximizes the probability of the event $w \cdot X \ge θ$. This is a challenging non-convex optimization problem for which even computing the value $\Pr[w \cdot X \ge θ]$ of a proposed solution vector $w$ is #P-hard. We provide an additive EPTAS for this problem which, for constant-bounded product distributions, runs in $ \poly(n) \cdot 2^{\poly(1/\eps)}$ time and outputs an $\eps$-approximately optimal solution vector $w$ for this problem. Our approach is inspired by, and extends, recent structural results from the complexity-theoretic study of linear threshold functions. Furthermore, in spite of the objective function being non-smooth, we give a \emph{unicriterion} PTAS while previous work for such objective functions has typically led to a \emph{bicriterion} PTAS. We believe our techniques may be applicable to get unicriterion PTAS for other non-smooth objective functions.

cs.DS↗

arXiv2013

A robust Khintchine inequality, and algorithms for computing optimal constants in Fourier analysis and high-dimensional geometry

This paper makes two contributions towards determining some well-studied optimal constants in Fourier analysis \newa{of Boolean functions} and high-dimensional geometry. \begin{enumerate} \item It has been known since 1994 \cite{GL:94} that every linear threshold function has squared Fourier mass at least 1/2 on its degree-0 and degree-1 coefficients. Denote the minimum such Fourier mass by $\w^{\leq 1}[\ltf]$, where the minimum is taken over all $n$-variable linear threshold functions and all $n \ge 0$. Benjamini, Kalai and Schramm \cite{BKS:99} have conjectured that the true value of $\w^{\leq 1}[\ltf]$ is $2/π$. We make progress on this conjecture by proving that $\w^{\leq 1}[\ltf] \geq 1/2 + c$ for some absolute constant $c>0$. The key ingredient in our proof is a "robust" version of the well-known Khintchine inequality in functional analysis, which we believe may be of independent interest. \item We give an algorithm with the following property: given any $η> 0$, the algorithm runs in time $2^{\poly(1/η)}$ and determines the value of $\w^{\leq 1}[\ltf]$ up to an additive error of $\pmη$. We give a similar $2^{\poly(1/η)}$-time algorithm to determine \emph{Tomaszewski's constant} to within an additive error of $\pm η$; this is the minimum (over all origin-centered hyperplanes $H$) fraction of points in $\{-1,1\}^n$ that lie within Euclidean distance 1 of $H$. Tomaszewski's constant is conjectured to be 1/2; lower bounds on it have been given by Holzman and Kleitman \cite{HK92} and independently by Ben-Tal, Nemirovski and Roos \cite{BNR02}. Our algorithms combine tools from anti-concentration of sums of independent random variables, Fourier analysis, and Hermite analysis of linear threshold functions. \end{enumerate}

cs.CC↗

arXiv2012

The Inverse Shapley Value Problem

For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ 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$(n)$ 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 $\eps=n^{-1/8}.$

cs.GT↗

arXiv2012

Inverse problems in approximate uniform generation

We initiate the study of \emph{inverse} problems in approximate uniform generation, focusing on uniform generation of satisfying assignments of various types of Boolean functions. In such an inverse problem, the algorithm is given uniform random satisfying assignments of an unknown function $f$ belonging to a class $\C$ of Boolean functions, and the goal is to output a probability distribution $D$ which is $ε$-close, in total variation distance, to the uniform distribution over $f^{-1}(1)$. Positive results: We prove a general positive result establishing sufficient conditions for efficient inverse approximate uniform generation for a class $\C$. We define a new type of algorithm called a \emph{densifier} for $\C$, and show (roughly speaking) how to combine (i) a densifier, (ii) an approximate counting / uniform generation algorithm, and (iii) a Statistical Query learning algorithm, to obtain an inverse approximate uniform generation algorithm. We apply this general result to obtain a poly$(n,1/\eps)$-time algorithm for the class of halfspaces; and a quasipoly$(n,1/\eps)$-time algorithm for the class of $\poly(n)$-size DNF formulas. Negative results: We prove a general negative result establishing that the existence of certain types of signature schemes in cryptography implies the hardness of certain inverse approximate uniform generation problems. This implies that there are no {subexponential}-time inverse approximate uniform generation algorithms for 3-CNF formulas; for intersections of two halfspaces; for degree-2 polynomial threshold functions; and for monotone 2-CNF formulas. Finally, we show that there is no general relationship between the complexity of the "forward" approximate uniform generation problem and the complexity of the inverse problem for a class $\C$ -- it is possible for either one to be easy while the other is hard.

cs.CC↗

arXiv2012

Majority is Stablest : Discrete and SoS

The Majority is Stablest Theorem has numerous applications in hardness of approximation and social choice theory. We give a new proof of the Majority is Stablest Theorem by induction on the dimension of the discrete cube. Unlike the previous proof, it uses neither the "invariance principle" nor Borell's result in Gaussian space. The new proof is general enough to include all previous variants of majority is stablest such as "it ain't over until it's over" and "Majority is most predictable". Moreover, the new proof allows us to derive a proof of Majority is Stablest in a constant level of the Sum of Squares hierarchy.This implies in particular that Khot-Vishnoi instance of Max-Cut does not provide a gap instance for the Lasserre hierarchy.

cs.CC↗

arXiv2012

Trevisan's extractor in the presence of quantum side information

Randomness extraction involves the processing of purely classical information and is therefore usually studied in the framework of classical probability theory. However, such a classical treatment is generally too restrictive for applications, where side information about the values taken by classical random variables may be represented by the state of a quantum system. This is particularly relevant in the context of cryptography, where an adversary may make use of quantum devices. Here, we show that the well known construction paradigm for extractors proposed by Trevisan is sound in the presence of quantum side information. We exploit the modularity of this paradigm to give several concrete extractor constructions, which, e.g, extract all the conditional (smooth) min-entropy of the source using a seed of length poly-logarithmic in the input, or only require the seed to be weakly random.

quant-ph↗

arXiv2012

Nearly optimal solutions for the Chow Parameters Problem and low-weight approximation of halfspaces

The \emph{Chow parameters} of a Boolean function $f: \{-1,1\}^n \to \{-1,1\}$ are its $n+1$ degree-0 and degree-1 Fourier coefficients. It has been known since 1961 (Chow, Tannenbaum) that the (exact values of the) Chow parameters of any linear threshold function $f$ uniquely specify $f$ within the space of all Boolean functions, but until recently (O'Donnell and Servedio) nothing was known about efficient algorithms for \emph{reconstructing} $f$ (exactly or approximately) from exact or approximate values of its Chow parameters. We refer to this reconstruction problem as the \emph{Chow Parameters Problem.} Our main result is a new algorithm for the Chow Parameters Problem which, given (sufficiently accurate approximations to) the Chow parameters of any linear threshold function $f$, runs in time $\tilde{O}(n^2)\cdot (1/\eps)^{O(\log^2(1/\eps))}$ and with high probability outputs a representation of an LTF $f'$ that is $\eps$-close to $f$. The only previous algorithm (O'Donnell and Servedio) had running time $\poly(n) \cdot 2^{2^{\tilde{O}(1/\eps^2)}}.$ As a byproduct of our approach, we show that for any linear threshold function $f$ over $\{-1,1\}^n$, there is a linear threshold function $f'$ which is $\eps$-close to $f$ and has all weights that are integers at most $\sqrt{n} \cdot (1/\eps)^{O(\log^2(1/\eps))}$. This significantly improves the best previous result of Diakonikolas and Servedio which gave a $\poly(n) \cdot 2^{\tilde{O}(1/\eps^{2/3})}$ weight bound, and is close to the known lower bound of $\max\{\sqrt{n},$ $(1/\eps)^{Ω(\log \log (1/\eps))}\}$ (Goldberg, Servedio). Our techniques also yield improved algorithms for related problems in learning theory.

cs.CC↗