Abstract. Differential Privacy (DP) has emerged as a formal, flexible framework for privacy protection, with a guarantee that is agnostic to auxiliary information and that admits simple rules for composition. Benefits notwithstanding, a major drawback of DP is that it provides noisy 1 responses to queries, making it unsuitable for many applications. We propose a new notion called Noiseless Privacy that provides exact answers to queries, without adding any noise whatsoever. While the form of our guarantee is similar to DP, where the privacy comes from is very different, based on statistical assumptions on the data and on restrictions to the auxiliary information available to the adversary. We present a first set of results for Noiseless Privacy of arbitrary Boolean-function queries and of linear Real-function queries, when data are drawn independently, from nearly-uniform and Gaussian distributions respectively. We also derive simple rules for composition under models of dynamically changing data.

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Let f be a polynomial of degree d in n variables over a finite field F. The polynomial is said to be unbiased if the distribution of f (x) for a uniform input x ∈ F n is close to the uniform distribution over F, and is called biased otherwise. The polynomial is said to have low rank if it can be expressed as a composition of a few lower degree polynomials. Green and Tao [Contrib. Discrete Math 2009] and Kaufman and Lovett [FOCS 2008] showed that bias implies low rank for fixed degree polynomials over fixed prime fields. This lies at the heart of many tools in higher order Fourier analysis. In this work, we extend this result to all prime fields (of size possibly growing with n). We also provide a generalization to nonprime fields in the large characteristic case. However, we state all our applications in the prime field setting for the sake of simplicity of presentation.As an immediate application, we obtain improved bounds for a suite of problems in effective algebraic geometry, including Hilbert nullstellensatz, radical membership and counting rational points in low degree varieties.Using the above generalization to large fields as a starting point, we are also able to settle the list decoding radius of fixed degree Reed-Muller codes over growing fields. The case of fixed size fields was solved by Bhowmick and Lovett [STOC 2015], which resolved a conjecture of Gopalan-Klivans-Zuckerman [STOC 2008]. Here, we show that the list decoding radius is equal the minimum distance of the code for all fixed degrees, even when the field size is possibly growing with n.

We describe the design of our federated task processing system. Originally, the system was created to support two specific federated tasks: evaluation and tuning of on-device ML systems, primarily for the purpose of personalizing these systems. In recent years, support for an additional federated task has been added: federated learning (FL) of deep neural networks. To our knowledge, only one other system has been described in literature that supports FL at scale. We include comparisons to that system to help discuss design decisions and attached trade-offs. Finally, we describe two specific large scale personalization use cases in detail to showcase the applicability of federated tuning to on-device personalization and to highlight application specific solutions.

We consider the question of decoding Reed-Muller codes over F n 2 beyond their list-decoding radius. Since, by definition, in this regime one cannot demand an efficient exact list-decoder, we seek an approximate decoder: Given a word F and radii r > r > 0, the goal is to output a codeword within radius r of F , if there exists a codeword within distance r. As opposed to the list decoding problem, it suffices here to output any codeword with this property, since the list may be too large if r exceeds the list decoding radius. Prior to our work, such decoders were known for Reed-Muller codes of degree 2, due to works of Wolf and the second author [FOCS 2011]. In this work we make the first progress on this problem for the degree 3 where the list decoding radius is 1/8. We show that there is a constant δ = 1/2− 1/8 > 1/8 and an efficient approximate decoder, that given query access to a function F : F n 2 → F 2 , such that F is within distance r = δ − ε from a cubic polynomial, runs in time polynomial in message length and outputs with high probability a cubic polynomial which is at distance at most r = 1/2 − ε from F , where ε is a quasi polynomial function of ε.

A Matching Vector (MV) family modulo m is a pair of ordered lists U = (u 1 , . . . , u t ) and V = (v 1 , . . . , v t ) where u i , v j ∈ Z n m with the following inner product pattern: for any i, u i , v i = 0, and for any i = j, u i , v j = 0. A MV family is called q-restricted if inner products u i , v j take at most q different values.Our interest in MV families stems from their recent application in the construction of subexponential locally decodable codes (LDCs). There, q-restricted MV families are used to construct LDCs with q queries, and there is special interest in the regime where q is constant. When m is a prime it is known that such constructions yield codes with exponential block length. However, for composite m the behaviour is dramatically different. A recent work by Efremenko In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When q is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus m is constant (as it is in the construction of Efremenko [Efr09]) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a well-known conjecture in additive combinatorics, the polynomial Freiman-Ruzsa conjecture over

A Matching Vector (MV) family modulo m is a pair of ordered lists U = (u1, . . . , ut) and V = (v1, . . . , vt) where ui, vj ∈ Z n m with the following inner product pattern: for any i, ui, vi = 0, and for any i = j, ui, vj = 0. A MV family is called q-restricted if inner products ui, vj take at most q different values.Our interest in MV families stems from their recent application in the construction of sub-exponential locally decodable codes (LDCs). There, q-restricted MV families are used to construct LDCs with q queries, and there is special interest in the regime where q is constant. When m is a prime it is known that such constructions yield codes with exponential block length. However, for composite m the behaviour is dramatically different. A recent work by Efremenko [8] (based on an approach initiated by Yekhanin [24]) gives the first sub-exponential LDC with constant queries. It is based on a construction of a MV family of super-polynomial size by Grolmusz [10] modulo composite m.In this work, we prove two lower bounds on the block length of LDCs which are based on black box construction using MV families. When q is constant (or sufficiently small), we prove that such LDCs must have a quadratic block length. When the modulus m is constant (as it is in the construction of Efremenko [8]) we prove a super-polynomial lower bound on the block-length of the LDCs, assuming a * A full version of this paper is available at

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