Sivakanth Gopi

2-Server PIR with Sub-Polynomial Communication

A 2-server Private Information Retrieval (PIR) scheme allows a user to retrieve the ith bit of an n-bit database replicated among two non-communicating servers, while not revealing any information about i to either server. In this work we construct a 2-server PIR scheme with total communication cost nO√(log log n)/(log n). This improves over current 2-server protocols which all require Ω(n1/3) communication. Our construction circumvents the n1/3 barrier of Razborov and Yekhanin which holds for the restricted model of bilinear group-based schemes (covering all previous 2-server schemes). The improvement comes from reducing the number of servers in existing protocols, based on Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing these protocols in an algebraic way (using polynomial interpolation) and extending them using partial derivatives.

Competitive analysis of the top- K ranking problem

Motivated by applications in recommender systems, web search, social choice and crowdsourcing, we consider the problem of identifying the set of top K items from noisy pairwise comparisons. In our setting, we are non-actively given r pairwise comparisons between each pair of n items, where each comparison has noise constrained by a very general noise model called the strong stochastic transitivity (SST) model. We analyze the competitive ratio of algorithms for the top-K problem. In particular, we present a linear time algorithm for the top-K problem which has a competitive ratio of O( √ n); i.e. to solve any instance of top-K, our algorithm needs at most O( √ n) times as many samples needed as the best possible algorithm for that instance (in contrast, all previous known algorithms for the top-K problem have competitive ratios of Ω(n) or worse). We further show that this is tight: any algorithm for the top-K problem has competitive ratio at least Ω( √ n). Stern School of Business, New York University, email: xchen3@stern.nyu.edu Department of Computer Science, Princeton University, email: sgopi@cs.princeton.edu Department of Computer Science, Princeton University, email: jiemingm@cs.princeton.edu Department of Computer Science, Princeton University, email: js44@cs.princeton.edu 1

2-Server PIR with Subpolynomial Communication

A 2-server Private Information Retrieval (PIR) scheme allows a user to retrieve the ith bit of an n-bit database replicated among two noncommunicating servers, while not revealing any information about i to either server. In this work, we construct a 2-server PIR scheme with total communication cost nO(√log / log n log n). This improves over current 2-server protocols, which all require Ω(n1/3) communication. Our construction circumvents the n1/3 barrier of Razborov and Yekhanin [2007], which holds for the restricted model of bilinear group-based schemes (covering all previous 2-server schemes). The improvement comes from reducing the number of servers in existing protocols, based on Matching Vector Codes, from 3 or 4 servers to 2. This is achieved by viewing these protocols in an algebraic way (using polynomial interpolation) and extending them using partial derivatives.

Locally testable and locally correctable codes approaching the gilbert-varshamov bound

One of the most important open problems in the theory of error-correcting codes is to determine the tradeoff between the rate R and minimum distance Δ of a binary code. The best known tradeoff is the Gilbert-Varshamov bound, and says that for every Δ ∈ (0, 1/2), there are codes with minimum distance Δ and rate R = RGV (Δ) > 0 (for a certain simple function RGV(·)). In this paper we show that the Gilbert-Varshamov bound can be achieved by codes which support local error-detection and error-correction algorithms. Specifically, we show the following results. 1. Local Testing: For all Δ ∈ (0, 1/2) and all R 2. Local Correction: For all ϵ > 0, for all Δ > 1/2 sufficiently large, and all R Furthermore, these codes have an efficient randomized construction, and the local testing and local correction algorithms can be made to run in time polynomial in the query complexity. Our results on locally correctable codes also immediately give locally decodable codes with the same parameters. Our local testing result is obtained by combining Thommesens random concatenation technique and the best known locally testable codes from [KMRS16]. Our local correction result, which is significantly more involved, also uses random concatenation, along with a number of further ideas: the Guruswami-Sudan-Indyk list decoding strategy for concatenated codes, Alon-Edmonds-Luby distance amplification, and the local list-decodability, local list-recoverability and local testability of Reed-Muller codes. Curiously, our final local correction algorithms go via local list-decoding and local testing algorithms; this seems to be the first time local testability is used in the construction of a locally correctable code.

Lower Bounds for Constant Query Affine-Invariant LCCs and LTCs

Affine-invariant codes are codes whose coordinates form a vector space over a finite field and which are invariant under affine transformations of the coordinate space. They form a natural, well-studied class of codes; they include popular codes such as Reed-Muller and Reed-Solomon. A particularly appealing feature of affine-invariant codes is that they seem well suited to admit local correctors and testers. In this work, we give lower bounds on the length of locally correctable and locally testable affine-invariant codes with constant query complexity. We show that if a code C ⊂ ΣKn is an r-query affine invariant locally correctable code (LCC), where K is a finite field and Σ is a finite alphabet, then the number of codewords in C is at most exp(OK,r,|Σ|(nr−1)). Also, we show that if C ⊂ ΣKn is an r-query affine invariant locally testable code (LTC), then the number of codewords in C is at most exp(OK,r,|Σ|(nr−2)). The dependence on n in these bounds is tight for constant-query LCCs/LTCs, since Guo, Kopparty, and Sudan (ITCS’13) constructed affine-invariant codes via lifting that have the same asymptotic tradeoffs. Note that our result holds for non-linear codes, whereas previously, Ben-Sasson and Sudan (RANDOM’11) assumed linearity to derive similar results. Our analysis uses higher-order Fourier analysis. In particular, we show that the codewords corresponding to an affine-invariant LCC/LTC must be far from each other with respect to Gowers norm of an appropriate order. This then allows us to bound the number of codewords, using known decomposition theorems, which approximate any bounded function in terms of a finite number of low-degree non-classical polynomials, up to a small error in the Gowers norm.