We develop a new technique for proving lower bounds in property testing, by showing a strong connection between testing and communication complexity. We give a simple scheme for reducing communication problems to testing problems, thus allowing us to use known lower bounds in communication complexity to prove lower bounds in testing. This scheme is general and implies a number of new testing bounds, as well as simpler proofs of several known bounds. For the problem of testing whether a boolean function is k-linear (a parity function on k variables), we achieve a lower bound of Omega(k) queries, even for adaptive algorithms with two-sided error, thus confirming a conjecture of Goldreich (2010). The same argument behind this lower bound also implies a new proof of known lower bounds for testing related classes such as k-juntas. For some classes, such as the class of monotone functions and the class of s-sparse GF(2) polynomials, we significantly strengthen the best known bounds.

Property Testing Lower Bounds via Communication Complexity.

Reed-Muller (RM) codes are among the oldest, simplest and perhaps most ubiquitous family of codes They are used in many areas of coding theory in both electrical engineering and computer science Yet, many of their important properties are still under investigation This paper covers some of the recent developments regarding the weight enumerator and the capacity-achieving properties of RM codes, as well as some of the algorithmic developments In particular, the paper discusses the recent connections established between RM codes, thresholds of Boolean functions, polarization theory, hypercontractivity, and the techniques of approximating low weight codewords using lower degree polynomials (when codewords are viewed as evaluation vectors of degree $r$ polynomials in $m$ variables) It then overviews some of the algorithms for decoding RM codes It covers both algorithms with provable performance guarantees for every block length, as well as algorithms with state-of-the-art performances in practical regimes, which do not perform as well for large block length Finally, the paper concludes with a few open problems

Reed–Muller Codes: Theory and Algorithms

A matrix M: A × X → {−1,1} corresponds to the following learning problem: An unknown element x ∈ X is chosen uniformly at random. A learner tries to learn x from a stream of samples, (a1, b1), (a2, b2) …, where for every i, ai ∈ A is chosen uniformly at random and bi = M(ai,x). Assume that k, l, r are such that any submatrix of M of at least 2−k · |A| rows and at least 2−l · |X| columns, has a bias of at most 2−r. We show that any learning algorithm for the learning problem corresponding to M requires either a memory of size at least Ω(k · l ), or at least 2Ω(r) samples. The result holds even if the learner has an exponentially small success probability (of 2−Ω(r)). In particular, this shows that for a large class of learning problems, any learning algorithm requires either a memory of size at least Ω((log|X|) · (log|A|)) or an exponential number of samples, achieving a tight Ω((log|X|) · (log|A|)) lower bound on the size of the memory, rather than a bound of Ω(min{(log|X|)2,(log|A|)2}) obtained in previous works by Raz [FOCS’17] and Moshkovitz and Moshkovitz [ITCS’18]. Moreover, our result implies all previous memory-samples lower bounds, as well as a number of new applications. Our proof builds on the work of Raz [FOCS’17] that gave a general technique for proving memory samples lower bounds.

Extractor-based time-space lower bounds for learning

The NP-completeness of SAT is a celebrated example of the power of bounded-depth computation: the core of the argument is a depth reduction establishing that any small nondeterministic circuit - an arbitrary NP computation on an arbitrary input - can be simulated by a small non deterministic circuit of depth 2 with unbounded fan-in - a SAT instance. Many other examples permeate theoretical computer science. On the Power of Small-Depth Computation discusses a selected subset of them, and includes a few unpublished proofs. On the Power of Small-Depth Computation starts with a unified treatment of the challenge of exhibiting explicit functions that have small correlation with low-degree polynomials over {0, 1}. It goes on to describe an unpublished proof that small bounded-depth circuits (AC) have exponentially small correlation with the parity function. The proof is due to Adam Klivans and Salil Vadhan; it builds upon and simplifies previous ones. Thereafter, it looks at a depth-reduction result by Leslie Valiant, the proof of which has not before appeared in full. It concludes by presenting the result by Benny Applebaum, Yuval Ishai, and Eyal Kushilevitz that shows that, under standard complexity theoretic assumptions, many cryptographic primitives can be implemented in very restricted computational models. On the Power of Small-Depth Computation is an ideal primer for anyone with an interest in computational complexity, random structures and algorithms and theoretical computer science generally.

/pdf/on-the-power-of-small-depth-computation-2kjron63dd.pdf

On the Power of Small-Depth Computation

We generalise the existence of combinatorial designs to the setting of subset sums in lattices with coordinates indexed by labelled faces of simplicial complexes. This general framework includes the problem of decomposing hypergraphs with extra edge data, such as colours and orders, and so incorporates a wide range of variations on the basic design problem, notably Baranyai-type generalisations, such as resolvable hypergraph designs, large sets of hypergraph designs and decompositions of designs by designs. Our method also gives approximate counting results, which is new for many structures whose existence was previously known, such as high dimensional permutations or Sudoku squares.

The existence of designs II

We study the problem of how well a typical multivariate polynomial can be approximated by lower-degree polynomials over $${\mathbb F}$$. We prove that almost all degree d polynomials have only an exponentially small correlation with all polynomials of degree at most d − 1, for all degrees d up to Θ(n). That is, a random degree d polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low-degree polynomial. Recently, several results regarding the weight distribution of Reed–Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed–Muller codes.

Random low-degree polynomials are hard to approximate

We study the problem of how well a typical multivariate polynomial can be approximated by lower degree polynomials over ${\mathbb{F}_{2}}$. We prove that, with very high probability, a random degree d + 1 polynomial has only an exponentially small correlation with all polynomials of degree d , for all degrees d up to $\Theta\left(n\right)$. That is, a random degree d + 1 polynomial does not admit a good approximation of lower degree. In order to prove this, we prove far tail estimates on the distribution of the bias of a random low degree polynomial. Recently, several results regarding the weight distribution of Reed---Muller codes were obtained. Our results can be interpreted as a new large deviation bound on the weight distribution of Reed---Muller codes.

Random Low Degree Polynomials are Hard to Approximate

Moran, Naor, and Segev have asked what is the minimal r=r(n, k) for which there exists an (n,k)-monotone encoding of length r, i.e., a monotone injective function from subsets of size up to k of {1, 2,..., n} to r bits. Monotone encodings are relevant to the study of tamper-proof data structures and arise also in the design of broadcast schemes in certain communication networks. To answer this question, we develop a relaxation of k-superimposed families, which we call alpha-fraction k -multiuser tracing ((k, alpha)-FUT (fraction user-tracing) families). We show that r(n, k) = Theta(k log(n/k)) by proving tight asymptotic lower and upper bounds on the size of (k, alpha)-FUT families and by constructing an (n,k)-monotone encoding of length O(k log(n/k)). We also present an explicit construction of an (n, 2)-monotone encoding of length 2 log n+O(1), which is optimal up to an additive constant.

/pdf/optimal-monotone-encodings-4hddou9jbs.pdf

Optimal Monotone Encodings

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan et al. have recently suggested to restrict the tester to take its queries from a smaller, typically random, subset of the inputs. This model is called active testing, in resemblance of active learning. Active testing gets more dicult as the size of the set we can query from decreases, and the extreme case is when it is exactly the number of queries we perform (so the algorithm actually has no choice). This is known as passive testing, or testing from random examples. In their paper, Balcan et al. have shown that active and passive testing of dictator functions is as hard as learning them, and requires (log n) queries (unlike the classic model, in which it can be done in a constant number of queries). We extend this result to k-linear functions, proving that passive and active testing of them requires ( k logn) queries, assuming k is not too large. Other classes of functions we consider are juntas, partially symmetric functions, linear functions, and low degree polynomials. For linear functions we provide tight bounds on the query complexity in both active and passive models (which asymptotically dier). The analysis for low degree polynomials is less complete and the exact query complexity is given only for passive testing. In both these cases, the query complexity for passive testing is essentially equivalent to that of learning. For juntas and partially symmetric functions, that is, functions that depend on a small number of variables and potentially also on the Hamming weight of the input, we provide some lower and upper bounds for the dierent models. When the functions depend on a constant number of variables, our analysis for both families is asymptotically tight. Moreover, the family of partially symmetric functions is the first example for which the query complexities in all these models are asymptotically dierent. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdfis‐Rado results on -systems.

/pdf/on-active-and-passive-testing-2g8ufoprjm.pdf

On Active and Passive Testing

Given a property of Boolean functions, what is the minimum number of queries required to determine with high probability if an input function satisfies this property or is "far" from satisfying it? This is a fundamental question in Property Testing, where traditionally the testing algorithm is allowed to pick its queries among the entire set of inputs. Balcan, Blais, Blum and Yang have recently suggested to restrict the tester to take its queries from a smaller random subset of polynomial size of the inputs. This model is called active testing, and in the extreme case when the size of the set we can query from is exactly the number of queries performed it is known as passive testing. 
We prove that passive or active testing of k-linear functions (that is, sums of k variables among n over Z_2) requires Theta(k*log n) queries, assuming k is not too large. This extends the case k=1, (that is, dictator functions), analyzed by Balcan et. al. 
We also consider other classes of functions including low degree polynomials, juntas, and partially symmetric functions. Our methods combine algebraic, combinatorial, and probabilistic techniques, including the Talagrand concentration inequality and the Erdos--Rado theorem on Delta-systems.

On active and passive testing

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