arXiv subjects

Rocco A. Servedio

Publications and source records attributed to Rocco A. Servedio.

At least 91 records · Page 5Linked to original sources

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}

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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.

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arXiv2012

Learning mixtures of structured distributions over discrete domains

Let $\mathfrak{C}$ be a class of probability distributions over the discrete domain $[n] = \{1,...,n\}.$ We show that if $\mathfrak{C}$ satisfies a rather general condition -- essentially, that each distribution in $\mathfrak{C}$ can be well-approximated by a variable-width histogram with few bins -- then there is a highly efficient (both in terms of running time and sample complexity) algorithm that can learn any mixture of $k$ unknown distributions from $\mathfrak{C}.$ We analyze several natural types of distributions over $[n]$, including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated by a histogram with few bins. Applying our general algorithm, we obtain near-optimally efficient algorithms for all these mixture learning problems.

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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.

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arXiv2012

On the Distribution of the Fourier Spectrum of Halfspaces

Bourgain showed that any noise stable Boolean function $f$ can be well-approximated by a junta. In this note we give an exponential sharpening of the parameters of Bourgain's result under the additional assumption that $f$ is a halfspace.

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arXiv2011

Testing $k$-Modal Distributions: Optimal Algorithms via Reductions

We give highly efficient algorithms, and almost matching lower bounds, for a range of basic statistical problems that involve testing and estimating the L_1 distance between two k-modal distributions $p$ and $q$ over the discrete domain $\{1,\dots,n\}$. More precisely, we consider the following four problems: given sample access to an unknown k-modal distribution $p$, Testing identity to a known or unknown distribution: 1. Determine whether $p = q$ (for an explicitly given k-modal distribution $q$) versus $p$ is $\eps$-far from $q$; 2. Determine whether $p=q$ (where $q$ is available via sample access) versus $p$ is $\eps$-far from $q$; Estimating $L_1$ distance ("tolerant testing'') against a known or unknown distribution: 3. Approximate $d_{TV}(p,q)$ to within additive $\eps$ where $q$ is an explicitly given k-modal distribution $q$; 4. Approximate $d_{TV}(p,q)$ to within additive $\eps$ where $q$ is available via sample access. For each of these four problems we give sub-logarithmic sample algorithms, that we show are tight up to additive $\poly(k)$ and multiplicative $\polylog\log n+\polylog k$ factors. Thus our bounds significantly improve the previous results of \cite{BKR:04}, which were for testing identity of distributions (items (1) and (2) above) in the special cases k=0 (monotone distributions) and k=1 (unimodal distributions) and required $O((\log n)^3)$ samples. As our main conceptual contribution, we introduce a new reduction-based approach for distribution-testing problems that lets us obtain all the above results in a unified way. Roughly speaking, this approach enables us to transform various distribution testing problems for k-modal distributions over $\{1,\dots,n\}$ to the corresponding distribution testing problems for unrestricted distributions over a much smaller domain $\{1,\dots,\ell\}$ where $\ell = O(k \log n).$

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arXiv2011

Private Data Release via Learning Thresholds

This work considers computationally efficient privacy-preserving data release. We study the task of analyzing a database containing sensitive information about individual participants. Given a set of statistical queries on the data, we want to release approximate answers to the queries while also guaranteeing differential privacy---protecting each participant's sensitive data. Our focus is on computationally efficient data release algorithms; we seek algorithms whose running time is polynomial, or at least sub-exponential, in the data dimensionality. Our primary contribution is a computationally efficient reduction from differentially private data release for a class of counting queries, to learning thresholded sums of predicates from a related class. We instantiate this general reduction with a variety of algorithms for learning thresholds. These instantiations yield several new results for differentially private data release. As two examples, taking {0,1}^d to be the data domain (of dimension d), we obtain differentially private algorithms for: (*) Releasing all k-way conjunctions. For any given k, the resulting data release algorithm has bounded error as long as the database is of size at least d^{O(\sqrt{k\log(k\log d)})}. The running time is polynomial in the database size. (*) Releasing a (1-γ)-fraction of all parity queries. For any γ\geq \poly(1/d), the algorithm has bounded error as long as the database is of size at least \poly(d). The running time is polynomial in the database size. Several other instantiations yield further results for privacy-preserving data release. Of the two results highlighted above, the first learning algorithm uses techniques for representing thresholded sums of predicates as low-degree polynomial threshold functions. The second learning algorithm is based on Jackson's Harmonic Sieve algorithm [Jackson 1997].

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arXiv2011

Learning transformed product distributions

We consider the problem of learning an unknown product distribution $X$ over $\{0,1\}^n$ using samples $f(X)$ where $f$ is a \emph{known} transformation function. Each choice of a transformation function $f$ specifies a learning problem in this framework. Information-theoretic arguments show that for every transformation function $f$ the corresponding learning problem can be solved to accuracy $\eps$, using $\tilde{O}(n/\eps^2)$ examples, by a generic algorithm whose running time may be exponential in $n.$ We show that this learning problem can be computationally intractable even for constant $\eps$ and rather simple transformation functions. Moreover, the above sample complexity bound is nearly optimal for the general problem, as we give a simple explicit linear transformation function $f(x)=w \cdot x$ with integer weights $w_i \leq n$ and prove that the corresponding learning problem requires $Ω(n)$ samples. As our main positive result we give a highly efficient algorithm for learning a sum of independent unknown Bernoulli random variables, corresponding to the transformation function $f(x)= \sum_{i=1}^n x_i$. Our algorithm learns to $\eps$-accuracy in poly$(n)$ time, using a surprising poly$(1/\eps)$ number of samples that is independent of $n.$ We also give an efficient algorithm that uses $\log n \cdot \poly(1/\eps)$ samples but has running time that is only $\poly(\log n, 1/\eps).$

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arXiv2010

Hardness Results for Agnostically Learning Low-Degree Polynomial Threshold Functions

Hardness results for maximum agreement problems have close connections to hardness results for proper learning in computational learning theory. In this paper we prove two hardness results for the problem of finding a low degree polynomial threshold function (PTF) which has the maximum possible agreement with a given set of labeled examples in $\R^n \times \{-1,1\}.$ We prove that for any constants $d\geq 1, \eps > 0$, {itemize} Assuming the Unique Games Conjecture, no polynomial-time algorithm can find a degree-$d$ PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labeled examples in $\R^n \times \{-1,1\}$, even if there exists a degree-$d$ PTF that is consistent with a $1-\eps$ fraction of the examples. It is $\NP$-hard to find a degree-2 PTF that is consistent with a $(\half + \eps)$ fraction of a given set of labeled examples in $\R^n \times \{-1,1\}$, even if there exists a halfspace (degree-1 PTF) that is consistent with a $1 - \eps$ fraction of the examples. {itemize} These results immediately imply the following hardness of learning results: (i) Assuming the Unique Games Conjecture, there is no better-than-trivial proper learning algorithm that agnostically learns degree-$d$ PTFs under arbitrary distributions; (ii) There is no better-than-trivial learning algorithm that outputs degree-2 PTFs and agnostically learns halfspaces (i.e. degree-1 PTFs) under arbitrary distributions.

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arXiv2010

A regularity lemma, and low-weight approximators, for low-degree polynomial threshold functions

We give a "regularity lemma" for degree-d polynomial threshold functions (PTFs) over the Boolean cube {-1,1}^n. This result shows that every degree-d PTF can be decomposed into a constant number of subfunctions such that almost all of the subfunctions are close to being regular PTFs. Here a "regular PTF is a PTF sign(p(x)) where the influence of each variable on the polynomial p(x) is a small fraction of the total influence of p. As an application of this regularity lemma, we prove that for any constants d \geq 1, \eps \geq 0, every degree-d PTF over n variables has can be approximated to accuracy eps by a constant-degree PTF that has integer weights of total magnitude O(n^d). This weight bound is shown to be optimal up to constant factors.

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arXiv2009

Average sensitivity and noise sensitivity of polynomial threshold functions

We give the first non-trivial upper bounds on the average sensitivity and noise sensitivity of degree-$d$ polynomial threshold functions (PTFs). These bounds hold both for PTFs over the Boolean hypercube and for PTFs over $\R^n$ under the standard $n$-dimensional Gaussian distribution. Our bound on the Boolean average sensitivity of PTFs represents progress towards the resolution of a conjecture of Gotsman and Linial \cite{GL:94}, which states that the symmetric function slicing the middle $d$ layers of the Boolean hypercube has the highest average sensitivity of all degree-$d$ PTFs. Via the $L_1$ polynomial regression algorithm of Kalai et al. \cite{KKMS:08}, our bounds on Gaussian and Boolean noise sensitivity yield polynomial-time agnostic learning algorithms for the broad class of constant-degree PTFs under these input distributions. The main ingredients used to obtain our bounds on both average and noise sensitivity of PTFs in the Gaussian setting are tail bounds and anti-concentration bounds on low-degree polynomials in Gaussian random variables \cite{Janson:97,CW:01}. To obtain our bound on the Boolean average sensitivity of PTFs, we generalize the ``critical-index'' machinery of \cite{Servedio:07cc} (which in that work applies to halfspaces, i.e. degree-1 PTFs) to general PTFs. Together with the "invariance principle" of \cite{MOO:05}, this lets us extend our techniques from the Gaussian setting to the Boolean setting. Our bound on Boolean noise sensitivity is achieved via a simple reduction from upper bounds on average sensitivity of Boolean PTFs to corresponding bounds on noise sensitivity.

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arXiv2009

Improved Approximation of Linear Threshold Functions

We prove two main results on how arbitrary linear threshold functions $f(x) = \sign(w\cdot x - θ)$ over the $n$-dimensional Boolean hypercube can be approximated by simple threshold functions. Our first result shows that every $n$-variable threshold function $f$ is $\eps$-close to a threshold function depending only on $\Inf(f)^2 \cdot \poly(1/\eps)$ many variables, where $\Inf(f)$ denotes the total influence or average sensitivity of $f.$ This is an exponential sharpening of Friedgut's well-known theorem \cite{Friedgut:98}, which states that every Boolean function $f$ is $\eps$-close to a function depending only on $2^{O(\Inf(f)/\eps)}$ many variables, for the case of threshold functions. We complement this upper bound by showing that $Ω(\Inf(f)^2 + 1/ε^2)$ many variables are required for $ε$-approximating threshold functions. Our second result is a proof that every $n$-variable threshold function is $\eps$-close to a threshold function with integer weights at most $\poly(n) \cdot 2^{\tilde{O}(1/\eps^{2/3})}.$ This is a significant improvement, in the dependence on the error parameter $\eps$, on an earlier result of \cite{Servedio:07cc} which gave a $\poly(n) \cdot 2^{\tilde{O}(1/\eps^{2})}$ bound. Our improvement is obtained via a new proof technique that uses strong anti-concentration bounds from probability theory. The new technique also gives a simple and modular proof of the original \cite{Servedio:07cc} result, and extends to give low-weight approximators for threshold functions under a range of probability distributions beyond just the uniform distribution.

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arXiv2008

Efficiently Testing Sparse GF(2) Polynomials

We give the first algorithm that is both query-efficient and time-efficient for testing whether an unknown function $f: \{0,1\}^n \to \{0,1\}$ is an $s$-sparse GF(2) polynomial versus $\eps$-far from every such polynomial. Our algorithm makes $\poly(s,1/\eps)$ black-box queries to $f$ and runs in time $n \cdot \poly(s,1/\eps)$. The only previous algorithm for this testing problem \cite{DLM+:07} used poly$(s,1/\eps)$ queries, but had running time exponential in $s$ and super-polynomial in $1/\eps$. Our approach significantly extends the ``testing by implicit learning'' methodology of \cite{DLM+:07}. The learning component of that earlier work was a brute-force exhaustive search over a concept class to find a hypothesis consistent with a sample of random examples. In this work, the learning component is a sophisticated exact learning algorithm for sparse GF(2) polynomials due to Schapire and Sellie \cite{SchapireSellie:96}. A crucial element of this work, which enables us to simulate the membership queries required by \cite{SchapireSellie:96}, is an analysis establishing new properties of how sparse GF(2) polynomials simplify under certain restrictions of ``low-influence'' sets of variables.

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arXiv2007

Quantum Algorithms for Learning and Testing Juntas

In this article we develop quantum algorithms for learning and testing juntas, i.e. Boolean functions which depend only on an unknown set of k out of n input variables. Our aim is to develop efficient algorithms: - whose sample complexity has no dependence on n, the dimension of the domain the Boolean functions are defined over; - with no access to any classical or quantum membership ("black-box") queries. Instead, our algorithms use only classical examples generated uniformly at random and fixed quantum superpositions of such classical examples; - which require only a few quantum examples but possibly many classical random examples (which are considered quite "cheap" relative to quantum examples). Our quantum algorithms are based on a subroutine FS which enables sampling according to the Fourier spectrum of f; the FS subroutine was used in earlier work of Bshouty and Jackson on quantum learning. Our results are as follows: - We give an algorithm for testing k-juntas to accuracy $ε$ that uses $O(k/ε)$ quantum examples. This improves on the number of examples used by the best known classical algorithm. - We establish the following lower bound: any FS-based k-junta testing algorithm requires $Ω(\sqrt{k})$ queries. - We give an algorithm for learning $k$-juntas to accuracy $ε$ that uses $O(ε^{-1} k\log k)$ quantum examples and $O(2^k \log(1/ε))$ random examples. We show that this learning algorithms is close to optimal by giving a related lower bound.

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arXiv2007

Learning Unions of $ω(1)$-Dimensional Rectangles

We consider the problem of learning unions of rectangles over the domain $[b]^n$, in the uniform distribution membership query learning setting, where both b and n are "large". We obtain poly$(n, \log b)$-time algorithms for the following classes: - poly$(n \log b)$-way Majority of $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles. - Union of poly$(\log(n \log b))$ many $O(\frac{\log^2 (n \log b)} {(\log \log(n \log b) \log \log \log (n \log b))^2})$-dimensional rectangles. - poly$(n \log b)$-way Majority of poly$(n \log b)$-Or of disjoint $O(\frac{\log(n \log b)} {\log \log(n \log b)})$-dimensional rectangles. Our main algorithmic tool is an extension of Jackson's boosting- and Fourier-based Harmonic Sieve algorithm [Jackson 1997] to the domain $[b]^n$, building on work of [Akavia, Goldwasser, Safra 2003]. Other ingredients used to obtain the results stated above are techniques from exact learning [Beimel, Kushilevitz 1998] and ideas from recent work on learning augmented $AC^{0}$ circuits [Jackson, Klivans, Servedio 2002] and on representing Boolean functions as thresholds of parities [Klivans, Servedio 2001].

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arXiv2006

PAC Learning Mixtures of Axis-Aligned Gaussians with No Separation Assumption

We propose and analyze a new vantage point for the learning of mixtures of Gaussians: namely, the PAC-style model of learning probability distributions introduced by Kearns et al. Here the task is to construct a hypothesis mixture of Gaussians that is statistically indistinguishable from the actual mixture generating the data; specifically, the KL-divergence should be at most epsilon. In this scenario, we give a poly(n/epsilon)-time algorithm that learns the class of mixtures of any constant number of axis-aligned Gaussians in n-dimensional Euclidean space. Our algorithm makes no assumptions about the separation between the means of the Gaussians, nor does it have any dependence on the minimum mixing weight. This is in contrast to learning results known in the ``clustering'' model, where such assumptions are unavoidable. Our algorithm relies on the method of moments, and a subalgorithm developed in previous work by the authors (FOCS 2005) for a discrete mixture-learning problem.

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arXiv2005

Every decision tree has an influential variable

We prove that for any decision tree calculating a boolean function $f:\{-1,1\}^n\to\{-1,1\}$, \[ \Var[f] \le \sum_{i=1}^n δ_i \Inf_i(f), \] where $δ_i$ is the probability that the $i$th input variable is read and $\Inf_i(f)$ is the influence of the $i$th variable on $f$. The variance, influence and probability are taken with respect to an arbitrary product measure on $\{-1,1\}^n$. It follows that the minimum depth of a decision tree calculating a given balanced function is at least the reciprocal of the largest influence of any input variable. Likewise, any balanced boolean function with a decision tree of depth $d$ has a variable with influence at least $\frac{1}{d}$. The only previous nontrivial lower bound known was $Ω(d 2^{-d})$. Our inequality has many generalizations, allowing us to prove influence lower bounds for randomized decision trees, decision trees on arbitrary product probability spaces, and decision trees with non-boolean outputs. As an application of our results we give a very easy proof that the randomized query complexity of nontrivial monotone graph properties is at least $Ω(v^{4/3}/p^{1/3})$, where $v$ is the number of vertices and $p \leq \half$ is the critical threshold probability. This supersedes the milestone $Ω(v^{4/3})$ bound of Hajnal and is sometimes superior to the best known lower bounds of Chakrabarti-Khot and Friedgut-Kahn-Wigderson.

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