David Xiao

Lower Bounds on Information Complexity via Zero-Communication Protocols and Applications

We show that almost all known lower bound methods for communication complexity are also lower bounds for the information complexity. In particular, we define a relaxed version of the partition bound of Jain and Klauck [Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity, 2010, pp. 247--258] and prove that it lower bounds the information complexity of any function. Our relaxed partition bound subsumes all norm-based methods (e.g., the

Derandomizing the Ahlswede-Winter matrix-valued Chernoff bound using pessimistic estimators, and applications

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Protocols and lower bounds for failure localization in the internet

method) and rectangle-based methods (e.g., the rectangle/corruption bound, the smooth rectangle bound, and the discrepancy bound), except the partition bound. Our result uses a new connection between rectangles and zero-communication protocols, where the players can either output a value or abort. We prove, using a sampling protocol designed by Braverman and Weinstein [in Approximation, Randomization, and Combinatorial Optimization, Lecture Notes in Comput. Sci. 7408, Springer, Heidelberg, 2012, pp. 459--470], the following compression l...

A randomness-efficient sampler for matrix-valued functions and applications

Ahlswede and Winter (IEEE Trans. Inf. Th. 2002) introduced a Chernoff bound for matrix-valued random variables, which is a non-trivial generalization of the usual Chernoff bound for real-valued random variables. We present an efficient derandomization of their bound using the method of pessimistic estimators (see Raghavan (JCSS 1988)). As a consequence, we derandomize an efficient construction by Alon and Roichman (RSA 1994) of an expanding Cayley graph of logarithmic degree on any (possibly non-abelian) group. This gives an optimal solution to the homomorphism testing problem of Shpilka and Wigderson (STOC 2004). We also apply these pessimistic estimators to the problem of solving semidefinite covering problems, thus giving a deterministic algorithm for the quantum hypergraph cover problem of Ahslwede and Winter.

Errata to (nearly) round-optimal black-box constructions of commitments secure against selective opening attacks

A secure failure-localization path-quality-monitoring (FLPQM) protocols allows a sender to localize faulty links on a single path through a network to a receiver, even when intermediate nodes on the path behave adversarially. Such protocols were proposed as tools that enable Internet service providers to select high-performance paths through the Internet, or to enforce contractual obligations. We give the first formal definitions of security for FL-PQM protocols and construct: 1. A simple FL-PQM protocol that can localize a faulty link every time a packet is not correctly delivered. This protocols communication overhead is O(1) additional messages of length O(n) per packet (where n is the security parameter). 2. A more efficient FL-PQM protocol that can localize a faulty link when a noticeable fraction of the packets sent during some time period are not correctly delivered. The number of additional messages is an arbitrarily small fraction of the total number of packets. We also prove lower bounds for such protocols: 1. Every secure FL-PQM protocol requires each intermediate node on the path to have some shared secret information (e.g. keys). 2. If secure FL-PQM protocols exist then so do one-way functions. 3. Every black-box construction of a FL-PQM protocol from a random oracle that securely localizes every packet and adds at most O(log n) messages overhead per packet requires each intermediate node to invoke the oracle. These results show that implementing FL-PQM requires active cooperation (i.e. maintaining keys and agreeing on, and performing, cryptographic protocols) from all of the intermediate nodes along the path. This may be problematic in the Internet, where links operate at extremely high speeds, and intermediate nodes are owned by competing business entities with little incentive to cooperate.

Sample Complexity Bounds on Differentially Private Learning via Communication Complexity

In this paper we give a randomness-efficient sampler for matrix-valued functions. Specifically, we show that a random walk on an expander approximates the recent Chernoff-like bound for matrix-valued functions of Ahlswede and Winter [2002], in a manner which depends optimally on the spectral gap. The proof uses perturbation theory, and is a generalization of Gillmans and Lezauds analyses of the Ajtai-Komlos-Szemeredi sampler for real-valued functions [Gillman, 1993]. Derandomizing our sampler gives a few applications, yielding deterministic polynomial time algorithms for problems in which derandomizing independent sampling gives only quasi-polynomial time deterministic algorithms. The first (which was our original motivation) is to a polynomial-time derandomization of the Alon-Roichman theorem [Alon and Roichman, 1994]: given a group of size n, find O(log n) elements which generate it as an expander. This implies a second application - efficiently constructing a randomness-optimal homo-morphism tester, significantly improving the previous result of Shpilka and Wigderson [2004]. A third application, which derandomizes a generalization of the set cover problem, is deferred to the full version of this paper.

Path-quality monitoring in the presence of adversaries: the secure sketch protocols

Several proofs initially presented by the author [2] were shown to be incorrect in a recent work of Ostrovsky et al [1]. In this notice we summarize the errors and summarize the current state of the art after taking into account the errors and subsequent work.

Path-quality monitoring in the presence of adversaries

In this work we analyze the sample complexity of classification by differentially private algorithms. Differential privacy is a strong and well-studied notion of privacy introduced by Dwork et al. (2006) that ensures that the output of an algorithm leaks little information about the data point provided by any of the participating individuals. Sample complexity of private PAC and agnostic learning was studied in a number of prior works starting with (Kasiviswanathan et al., 2008) but a number of basic questions still remain open, most notably whether learning with privacy requires more samples than learning without privacy. We show that the sample complexity of learning with (pure) differential privacy can be arbitrarily higher than the sample complexity of learning without the privacy constraint or the sample complexity of learning with approximate differential privacy. Our second contribution and the main tool is an equivalence between the sample complexity of (pure) differentially private learning of a concept class

On Basing Lower-Bounds for Learning on Worst-Case Assumptions

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Is privacy compatible with truthfulness

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