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33
Testing and Reconstruction of Lipschitz Functions with Applications to Data Privacy
 ELECTRONIC COLLOQUIUM ON COMPUTATIONAL COMPLEXITY, REPORT NO. 57 (2011)
, 2011
"... A function f: D → R has Lipschitz constant c if dR(f(x), f(y)) ≤ c · dD(x, y) for all x, y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts ..."
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Cited by 21 (4 self)
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A function f: D → R has Lipschitz constant c if dR(f(x), f(y)) ≤ c · dD(x, y) for all x, y in D, where dR and dD denote the distance functions on the range and domain of f, respectively. We say a function is Lipschitz if it has Lipschitz constant 1. (Note that rescaling by a factor of 1/c converts a function with a Lipschitz constant c into a Lipschitz function.) In other words, Lipschitz functions are not very sensitive to small changes in the input. We initiate the study of testing and local reconstruction of the Lipschitz property of functions. A property tester has to distinguish functions with the property (in this case, Lipschitz) from functions that are ɛfar from having the property, that is, differ from every function with the property on at least an ɛ fraction of the domain. A local filter reconstructs an arbitrary function f to ensure that the reconstructed function g has the desired property (in this case, is Lipschitz), changing f only when necessary. A local filter is given a function f and a query x and, after looking up the value of f on a small number of points, it has to output g(x) for some function g, which has the desired property and does not depend on x. If f has the property, g must be equal to f. We consider functions over domains {0, 1} d, {1,..., n} and {1,..., n} d, equipped with ℓ1 distance.
An optimal lower bound on the communication complexity of GapHammingDistance
 In Proc. 43rd Annual ACM Symposium on the Theory of Computing
, 2011
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Is Submodularity Testable?
 ALGORITHMICA
, 2012
"... We initiate the study of property testing of submodularity on the boolean hypercube. Submodular functions come up in a variety of applications in combinatorial optimization. For a vast range of algorithms, the existence of an oracle to a submodular function is assumed. But how does one check if thi ..."
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Cited by 16 (0 self)
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We initiate the study of property testing of submodularity on the boolean hypercube. Submodular functions come up in a variety of applications in combinatorial optimization. For a vast range of algorithms, the existence of an oracle to a submodular function is assumed. But how does one check if this oracle indeed represents a submodular function? Consider a function f:{0, 1} n → R. The distance to submodularity is the minimum fraction of values of f that need to be modified to make f submodular. If this distance is more than ɛ>0, then we say that f is ɛfar from being submodular. The aim is to have an efficient procedure that, given input f that is ɛfar from being submodular, certifies that f is not submodular. We analyze a natural tester for this problem, and prove that it runs in subexponential time. This gives the first nontrivial tester for submodularity. On the other hand, we prove an interesting lower bound (that is, unfortunately, quite far from the upper bound) suggesting that this tester cannot be efficient in terms of ɛ. This involves nontrivial examples of functions which are far from submodular and yet do not exhibit too many local violations. We also provide some constructions indicating the difficulty in designing a tester for submodularity. We construct a partial function defined on exponentially many
Optimal bounds for monotonicity and Lipschitz testing over the hypercube and hypergrids
, 2012
"... The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f: [k]n 7 → R (for some ordered range R). The hypergrid/cube has a natural partial order given by coordinatewise ordering, denoted by ..."
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Cited by 11 (5 self)
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The problem of monotonicity testing over the hypergrid and its special case, the hypercube, is a classic question in property testing. We are given query access to f: [k]n 7 → R (for some ordered range R). The hypergrid/cube has a natural partial order given by coordinatewise ordering, denoted by ≺. A function is monotone if for all pairs x ≺ y, f(x) ≤ f(y). The distance to monotonicity, εf, is the minimum fraction of values of f that need to be changed to make f monotone. For k = 2 (the boolean hypercube), the usual tester is the edge tester, which checks monotonicity on adjacent pairs of domain points. It is known that the edge tester using O(ε−1n log R) samples can distinguish a monotone function from one where εf> ε. On the other hand, the best lower bound for monotonicity testing over general R is Ω(n). We resolve this long standing open problem and prove that O(n/ε) samples suffice for the edge tester. For hypergrids, known testers require O(ε−1n log k log R) samples, while the best known (nonadaptive) lower bound is Ω(ε−1n log k). We give a (nonadaptive) monotonicity tester for hypergrids running in O(ε−1n log k) time. Our techniques lead to optimal property testers (with the same running time) for the natural Lipschitz property on hypercubes and hypergrids. (A cLipschitz function is one where f(x) − f(y)  ≤ c‖x − y‖1.) In fact, we give a general unified proof for O(ε−1n log k)query testers for a class of “boundedderivative ” properties, a class containing both monotonicity and Lipschitz. Categories and Subject Descriptors F.2.2 [Analysis of algorithms and problem complex
Lower Bounds for Testing Properties of Functions on Hypergrid Domains
"... Abstract. We introduce strong, and in many cases optimal, lower bounds for the number of queries required to nonadaptively test three fundamental properties of functions f: [n] d → R on the hypergrid: monotonicity, convexity, and the Lipschitz property. Our lower bounds also apply to the more restri ..."
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Abstract. We introduce strong, and in many cases optimal, lower bounds for the number of queries required to nonadaptively test three fundamental properties of functions f: [n] d → R on the hypergrid: monotonicity, convexity, and the Lipschitz property. Our lower bounds also apply to the more restricted setting of functions f: [n] → R on the line (i.e., to hypergrids with d = 1), where they give optimal lower bounds for all three properties. The lower bound for testing convexity is the first lower bound for that property, and the lower bound for the Lipschitz property is new for tests with 2sided error. We obtain our lower bounds via the connection to communication complexity established by Blais, Brody, and Matulef (2012). Our results are the first to apply this method to functions with nonhypercube domains. A key ingredient in this generalization is the set of Walsh functions, an orthonormal basis of the set of functions f: [n] d → R. 1
Learning pseudoBoolean kDNF and submodular functions
, 2013
"... We prove that any submodular function f: {0, 1} n → {0, 1,..., k} can be represented as a pseudoBoolean 2kDNF formula. PseudoBoolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show t ..."
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We prove that any submodular function f: {0, 1} n → {0, 1,..., k} can be represented as a pseudoBoolean 2kDNF formula. PseudoBoolean DNFs are a natural generalization of DNF representation for functions with integer range. Each term in such a formula has an associated integral constant. We show that an analog of H˚astad’s switching lemma holds for pseudoBoolean kDNFs if all constants associated with the terms of the formula are bounded. This allows us to generalize Mansour’s PAClearning algorithm for kDNFs to pseudoBoolean kDNFs, and hence gives a PAClearning algorithm with membership queries under the uniform distribution for submodular functions of the form f: {0, 1} n → {0, 1,..., k}. Our algorithm runs in time polynomial in n, k O(k log k/ɛ) and log(1/δ) and works even in the agnostic setting. The line of previous work on learning submodular functions [Balcan, Harvey (STOC ’11), Gupta, Hardt, Roth, Ullman; (STOC ’11), Cheraghchi, Klivans, Kothari, Lee
Lower bounds for testing computability by small width OBDDs
 IN PROC. 8TH ANNUAL THEORY AND APPLICATIONS OF MODELS OF COMPUTATION
, 2011
"... We consider the problem of testing whether a function f: {0, 1} n → {0, 1} is computable by a readonce, width2 ordered binary decision diagram (OBDD), also known as a branching program. This problem has two variants: one where the variables must occur in a fixed, known order, and one where the v ..."
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Cited by 6 (1 self)
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We consider the problem of testing whether a function f: {0, 1} n → {0, 1} is computable by a readonce, width2 ordered binary decision diagram (OBDD), also known as a branching program. This problem has two variants: one where the variables must occur in a fixed, known order, and one where the variables are allowed to occur in an arbitrary order. We show that for both variants, any nonadaptive testing algorithm must make Ω(n) queries, and thus any adaptive testing algorithm must make Ω(log n) queries. We also consider the more general problem of testing computability by widthw OBDDs where the variables occur in a fixed order. We show that for any constant w ≥ 4, Ω(n) queries are required, resolving a conjecture of Goldreich [15]. We prove all of our lower bounds using a new technique of Blais, Brody, and Matulef [6], giving simple reductions from known hard problems in communication complexity to the testing problems at hand. Our result for width2 OBDDs provides the first example of the power of this technique for proving strong nonadaptive bounds.
The nonadaptive query complexity of testing kparities
, 2012
"... We prove tight bounds of Θ(k log k) queries for nonadaptively testing whether a function f: {0, 1} n → {0, 1} is a kparity or far from any kparity. Both upper and lower bounds are new. The lower bound combines a recent method of Blais, Brody and Matulef [BBM11] to get testing lower bounds from co ..."
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Cited by 5 (1 self)
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We prove tight bounds of Θ(k log k) queries for nonadaptively testing whether a function f: {0, 1} n → {0, 1} is a kparity or far from any kparity. Both upper and lower bounds are new. The lower bound combines a recent method of Blais, Brody and Matulef [BBM11] to get testing lower bounds from communication complexity, with a new Θ(k log k) bound for the oneway communication complexity of kdisjointness. 1
An optimal lower bound for monotonicity testing over hypergrids
 In APPROXRANDOM
, 2013
"... Abstract. For positive integers n, d, consider the hypergrid [n]d with the coordinatewise product partial ordering denoted by ≺. A function f: [n]d → N is monotone if ∀x ≺ y, f(x) ≤ f(y). A function f is εfar from monotone if at least an εfraction of values must be changed to make f monotone. Gi ..."
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Cited by 4 (2 self)
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Abstract. For positive integers n, d, consider the hypergrid [n]d with the coordinatewise product partial ordering denoted by ≺. A function f: [n]d → N is monotone if ∀x ≺ y, f(x) ≤ f(y). A function f is εfar from monotone if at least an εfraction of values must be changed to make f monotone. Given a parameter ε, a monotonicity tester must distinguish with high probability a monotone function from one that is εfar. We prove that any (adaptive, twosided) monotonicity tester for functions f: [n]d → N must make Ω(ε−1d logn−ε−1 log ε−1) queries. Recent upper bounds show the existence of O(ε−1d logn) query monotonicity testers for hypergrids. This closes the question of monotonicity testing for hypergrids over arbitrary ranges. The previous best lower bound for general hypergrids was a nonadaptive bound of Ω(d logn).
Certifying equality with limited interaction
"... The equality problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing o ..."
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The equality problem is usually one’s first encounter with communication complexity and is one of the most fundamental problems in the field. Although its deterministic and randomized communication complexity were settled decades ago, we find several new things to say about the problem by focusing on three subtle aspects. The first is to consider the expected communication cost (at a worstcase input) for a protocol that uses limited interaction—i.e., a bounded number of rounds of communication—and whose error probability is zero or close to it. The second is to treat the false negative error rate separately from the false positive error rate. The third is to consider the information cost of such protocols. We obtain asymptotically optimal roundsversuscost tradeoffs for equality: both expected communication cost and information cost scale as Θ(log log · · · logn), with r − 1 logs, where r is the number of rounds. These bounds hold even when the false negative rate approaches 1. For the case of zeroerror communication cost, we obtain essentially matching bounds, up to a tiny additive constant. We also provide some applications.