arXiv subjects

Avrim Blum

Publications and source records attributed to Avrim Blum.

81 records · Page 5Linked to original sources

arXiv2011

Fast Private Data Release Algorithms for Sparse Queries

We revisit the problem of accurately answering large classes of statistical queries while preserving differential privacy. Previous approaches to this problem have either been very general but have not had run-time polynomial in the size of the database, have applied only to very limited classes of queries, or have relaxed the notion of worst-case error guarantees. In this paper we consider the large class of sparse queries, which take non-zero values on only polynomially many universe elements. We give efficient query release algorithms for this class, in both the interactive and the non-interactive setting. Our algorithms also achieve better accuracy bounds than previous general techniques do when applied to sparse queries: our bounds are independent of the universe size. In fact, even the runtime of our interactive mechanism is independent of the universe size, and so can be implemented in the "infinite universe" model in which no finite universe need be specified by the data curator.

cs.DS↗

arXiv2011

Welfare and Profit Maximization with Production Costs

Combinatorial Auctions are a central problem in Algorithmic Mechanism Design: pricing and allocating goods to buyers with complex preferences in order to maximize some desired objective (e.g., social welfare, revenue, or profit). The problem has been well-studied in the case of limited supply (one copy of each item), and in the case of digital goods (the seller can produce additional copies at no cost). Yet in the case of resources---oil, labor, computing cycles, etc.---neither of these abstractions is just right: additional supplies of these resources can be found, but at increasing difficulty (marginal cost) as resources are depleted. In this work, we initiate the study of the algorithmic mechanism design problem of combinatorial pricing under increasing marginal cost. The goal is to sell these goods to buyers with unknown and arbitrary combinatorial valuation functions to maximize either the social welfare, or the seller's profit; specifically we focus on the setting of \emph{posted item prices} with buyers arriving online. We give algorithms that achieve {\em constant factor} approximations for a class of natural cost functions---linear, low-degree polynomial, logarithmic---and that give logarithmic approximations for more general increasing marginal cost functions (along with a necessary additive loss). We show that these bounds are essentially best possible for these settings.

cs.GT↗

arXiv2011

A Learning Theory Approach to Non-Interactive Database Privacy

In this paper we demonstrate that, ignoring computational constraints, it is possible to privately release synthetic databases that are useful for large classes of queries -- much larger in size than the database itself. Specifically, we give a mechanism that privately releases synthetic data for a class of queries over a discrete domain with error that grows as a function of the size of the smallest net approximately representing the answers to that class of queries. We show that this in particular implies a mechanism for counting queries that gives error guarantees that grow only with the VC-dimension of the class of queries, which itself grows only logarithmically with the size of the query class. We also show that it is not possible to privately release even simple classes of queries (such as intervals and their generalizations) over continuous domains. Despite this, we give a privacy-preserving polynomial time algorithm that releases information useful for all halfspace queries, given a slight relaxation of the utility guarantee. This algorithm does not release synthetic data, but instead another data structure capable of representing an answer for each query. We also give an efficient algorithm for releasing synthetic data for the class of interval queries and axis-aligned rectangles of constant dimension. Finally, inspired by learning theory, we introduce a new notion of data privacy, which we call distributional privacy, and show that it is strictly stronger than the prevailing privacy notion, differential privacy.

cs.DS↗

arXiv2011

Center-based Clustering under Perturbation Stability

Clustering under most popular objective functions is NP-hard, even to approximate well, and so unlikely to be efficiently solvable in the worst case. Recently, Bilu and Linial \cite{Bilu09} suggested an approach aimed at bypassing this computational barrier by using properties of instances one might hope to hold in practice. In particular, they argue that instances in practice should be stable to small perturbations in the metric space and give an efficient algorithm for clustering instances of the Max-Cut problem that are stable to perturbations of size $O(n^{1/2})$. In addition, they conjecture that instances stable to as little as O(1) perturbations should be solvable in polynomial time. In this paper we prove that this conjecture is true for any center-based clustering objective (such as $k$-median, $k$-means, and $k$-center). Specifically, we show we can efficiently find the optimal clustering assuming only stability to factor-3 perturbations of the underlying metric in spaces without Steiner points, and stability to factor $2+\sqrt{3}$ perturbations for general metrics. In particular, we show for such instances that the popular Single-Linkage algorithm combined with dynamic programming will find the optimal clustering. We also present NP-hardness results under a weaker but related condition.

cs.DS↗

arXiv2009

Separating populations with wide data: A spectral analysis

In this paper, we consider the problem of partitioning a small data sample drawn from a mixture of $k$ product distributions. We are interested in the case that individual features are of low average quality $γ$, and we want to use as few of them as possible to correctly partition the sample. We analyze a spectral technique that is able to approximately optimize the total data size--the product of number of data points $n$ and the number of features $K$--needed to correctly perform this partitioning as a function of $1/γ$ for $K>n$. Our goal is motivated by an application in clustering individuals according to their population of origin using markers, when the divergence between any two of the populations is small.

stat.ML↗

arXiv2003

On Statistical Query Sampling and NMR Quantum Computing

We introduce a ``Statistical Query Sampling'' model, in which the goal of an algorithm is to produce an element in a hidden set $Ssubseteqbit^n$ with reasonable probability. The algorithm gains information about $S$ through oracle calls (statistical queries), where the algorithm submits a query function $g(cdot)$ and receives an approximation to $Pr_{x in S}[g(x)=1]$. We show how this model is related to NMR quantum computing, in which only statistical properties of an ensemble of quantum systems can be measured, and in particular to the question of whether one can translate standard quantum algorithms to the NMR setting without putting all of their classical post-processing into the quantum system. Using Fourier analysis techniques developed in the related context of {em statistical query learning}, we prove a number of lower bounds (both information-theoretic and cryptographic) on the ability of algorithms to produces an $xin S$, even when the set $S$ is fairly simple. These lower bounds point out a difficulty in efficiently applying NMR quantum computing to algorithms such as Shor's and Simon's algorithm that involve significant classical post-processing. We also explicitly relate the notion of statistical query sampling to that of statistical query learning. An extended abstract appeared in the 18th Aunnual IEEE Conference of Computational Complexity (CCC 2003), 2003. Keywords: statistical query, NMR quantum computing, lower bound

quant-ph↗

arXiv2000

Noise-Tolerant Learning, the Parity Problem, and the Statistical Query Model

We describe a slightly sub-exponential time algorithm for learning parity functions in the presence of random classification noise. This results in a polynomial-time algorithm for the case of parity functions that depend on only the first O(log n log log n) bits of input. This is the first known instance of an efficient noise-tolerant algorithm for a concept class that is provably not learnable in the Statistical Query model of Kearns. Thus, we demonstrate that the set of problems learnable in the statistical query model is a strict subset of those problems learnable in the presence of noise in the PAC model. In coding-theory terms, what we give is a poly(n)-time algorithm for decoding linear k by n codes in the presence of random noise for the case of k = c log n loglog n for some c > 0. (The case of k = O(log n) is trivial since one can just individually check each of the 2^k possible messages and choose the one that yields the closest codeword.) A natural extension of the statistical query model is to allow queries about statistical properties that involve t-tuples of examples (as opposed to single examples). The second result of this paper is to show that any class of functions learnable (strongly or weakly) with t-wise queries for t = O(log n) is also weakly learnable with standard unary queries. Hence this natural extension to the statistical query model does not increase the set of weakly learnable functions.

cs.LG↗

arXiv1994

On the minimum latency problem

We are given a set of points $p_1,\ldots , p_n$ and a symmetric distance matrix $(d_{ij})$ giving the distance between $p_i$ and $p_j$. We wish to construct a tour that minimizes $\sum_{i=1}^n \ell(i)$, where $\ell(i)$ is the {\em latency} of $p_i$, defined to be the distance traveled before first visiting $p_i$. This problem is also known in the literature as the {\em deliveryman problem} or the {\em traveling repairman problem}. It arises in a number of applications including disk-head scheduling, and turns out to be surprisingly different from the traveling salesman problem in character. We give exact and approximate solutions to a number of cases, including a constant-factor approximation algorithm whenever the distance matrix satisfies the triangle inequality.

math.CO↗

arXiv1994

An on-line algorithm for improving performance in navigation

Recent papers have shown optimally-competitive on-line strategies for a robot traveling from a point $s$ to a point $t$ in certain unknown geometric environments. We consider the question: Having gained some partial information about the scene on its first trip from $s$ to $t$, can the robot improve its performance on subsequent trips it might make? This is a type of on-line problem where a strategy must exploit {\em partial information \/} about the future (e.g., about obstacles that lie ahead). For scenes with axis-parallel rectangular obstacles where the Euclidean distance between $s$ and $t$ is $n$, we present a deterministic algorithm whose {\em average\/} trip length after $k$ trips, $k \leq n$, is $O(\rootnbyk)$ times the length of the shortest $s$-$t$ path in the scene. We also show that this is the best a deterministic strategy can do. This algorithm can be thought of as performing an optimal tradeoff between search effort and the goodness of the path found. We improve this algorithm so that for {\em every\/} $i \leq n$, the robot's $i$th trip length is $O(\rootnbyi)$ times the shortest $s$-$t$ path length. A key idea of the paper is that a {\em tree\/} structure can be defined in the scene, where the nodes are portions of certain obstacles and the edges are ``short'' paths from a node to its children. The core of our algorithms is an on-line strategy for traversing this tree optimally.

math.CO↗