Constructive Proofs of Concentration Bounds

Impagliazzo, Russell; Kabanets, Valentine

doi:10.1007/978-3-642-15369-3_46

Cited by 60 publications

(77 citation statements)

References 45 publications

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“…[Ung09] showed how to derive threshold DPTs from XOR lemmas, and recent work of [IK10] gave a way to derive threshold DPTs from sufficiently strong DPTs; see also the earlier works cited in [Ung09,IK10]. However, the results of [IK10] do not apply for our purposes, and the threshold DPT we prove is more general than we'd get by applying the results of [Ung09] to our XOR lemma. In any case the proof of our threshold DPT is, we feel, quite natural, and actually forms the basis for the proof of our XOR lemma.…”

Section: Our Resultsmentioning

confidence: 90%

“…An XOR lemma is a result which upper-bounds the success probability p ′ achievable by algorithms for f ⊕k using T ′ resources, under the assumption that any algorithm using T resources has success probability at most p. 1 An obvious difference from DPTs is that in an XOR lemma, p ′ must always be at least 1/2, since f ⊕k is Boolean and the algorithm could simply guess a random bit. The hope is that (p ′ − 1/2) decays exponentially with k. Research on XOR lemmas has proceeded in parallel with research on direct product theorems; the known results are of similar strength (with some exceptions), and in some cases there are reductions known from XOR lemmas to DPTs or vice versa; see [Ung09,IK10] for an overview and recent results of this type.…”

Section: Direct Product Theoremsmentioning

confidence: 99%

“…It is at least known that, under this limitation, a DPT with p ′ = p k can be shown using [Imp95,Hol05], as remarked in [IK10].…”

Section: Direct Product Theoremsmentioning

confidence: 99%

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Improved direct product theorems for randomized query complexity

Drucker

2012

comput. complex.

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The "direct product problem" is a fundamental question in complexity theory which seeks to understand how the difficulty of computing a function on each of k independent inputs scales with k. We prove the following direct product theorem (DPT) for query complexity: if every T -query algorithm has success probability at most 1 − ε in computing the Boolean function f on input distribution µ, then for α ≤ 1, every αεT k-query algorithm has success probability at most (2 αε (1 − ε)) k in computing the k-fold direct product f ⊗k correctly on k independent inputs from µ. In light of examples due to Shaltiel, this statement gives an essentially optimal tradeoff between the query bound and the error probability. Using this DPT, we show that for an absolute constant α > 0, the worst-case success probability of any αR 2 (f )k-query randomized algorithm for f ⊗k falls exponentially with k. The best previous statement of this type, due to Klauck,Špalek, and de Wolf, required a query bound of O(bs(f )k).Our proof technique involves defining and analyzing a collection of martingales associated with an algorithm attempting to solve f ⊗k . Our method is quite general and yields a new XOR lemma and threshold DPT for the query model, as well as DPTs for the query complexity of learning tasks, search problems, and tasks involving interaction with dynamic entities. We also give a version of our DPT in which decision tree size is the resource of interest.

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Section: Our Resultsmentioning

confidence: 90%

Section: Direct Product Theoremsmentioning

confidence: 99%

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Improved direct product theorems for randomized query complexity

Drucker

2012

comput. complex.

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“…Throughout, we use the following version of Chernoff bound for negatively correlated random variables first proved by [26]; see, also [13,19]. …”

Section: Preliminariesmentioning

confidence: 99%

Combinatorial Auctions Do Need Modest Interaction

Assadi

2017

Proceedings of the 2017 ACM Conference on Economics and Computation

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We study the necessity of interaction for obtaining efficient allocations in combinatorial auctions with subadditive bidders. This problem was originally introduced by Dobzinski, Nisan, and Oren [9] as the following simple market scenario: m items are to be allocated among n bidders in a distributed setting where bidders valuations are private and hence communication is needed to obtain an efficient allocation. The communication happens in rounds: in each round, each bidder, simultaneously with others, broadcasts a message to all parties involved. At the end, the central planner computes an allocation solely based on the communicated messages. Dobzinski et al. [9] showed that (at least some) interaction is necessary for obtaining any efficient allocation: no non-interactive (1-round) protocol with polynomial communication (in the number of items and bidders) can achieve approximation ratio better than Ω(m 1/4 ), while for any r ≥ 1, there exists r-round protocols that achieve O(r · m 1/r+1 ) approximation with polynomial communication; in particular, O(log m) rounds of interaction suffice to obtain an (almost) efficient allocation, i.e., a polylog(m)-approximation.A natural question at this point is to identify the "right" level of interaction (i.e., number of rounds) necessary to obtain an efficient allocation. In this paper, we resolve this question by providing an almost tight round-approximation tradeoff for this problem: we show that for any r ≥ 1, any r-round protocol that uses poly(m, n) bits of communication can only approximate the social welfare up to a factor of Ω(). This in particular implies that Ω( log m log log m ) rounds of interaction are necessary for obtaining any efficient allocation (i.e., a constant or even a polylog(m)-approximation) in these markets. Our work builds on the recent multi-party round-elimination technique of Alon, Nisan, Raz, and Weinstein [2] -used to prove similar-inspirit lower bounds for round-approximation tradeoff in unit-demand (matching) markets -and settles an open question posed by Dobzinski et al. [9] and Alon et al. [2].

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“…For example, one can use a bound by Panconesi and Srinivasan [PS97], with a compact proof by Impagliazzo and Kabanets [IK10]. Letting Z i be the indicator variable of a stage being bad, the claim guarantees that for any…”

Section: Set Uniformly At Random the Variables In W T Let F : {0 1}mentioning

confidence: 99%

Extractors for Circuit Sources

Viola¹

2014

SIAM J. Comput.

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We obtain the first deterministic extractors for sources generated (or sampled) by small circuits of bounded depth. Our main results are:(1) We extract k(k/nd) O(1) bits with exponentially small error from n-bit sources of min-entropy k that are generated by functions f : {0, 1} → {0, 1} n where each output bit depends on ≤ d input bits. In particular, we extract from NC 0 sources,(2) We extract k(k/n 1+γ ) O(1) bits with super-polynomially small error from n-bit sources of min-entropy k that are generated by poly(n)-size AC 0 circuits, for any γ > 0.As our starting point, we revisit the connection by Trevisan and Vadhan (FOCS 2000) between circuit lower bounds and extractors for sources generated by circuits. We note that such extractors (with very weak parameters) are equivalent to lower bounds for generating distributions (FOCS 2010; with Lovett, CCC 2011). Building on those bounds, we prove that the sources in (1) and (2) are (close to) a convex combination of high-entropy "bit-block" sources. Introduced here, such sources are a special case of affine ones. As extractors for (1) and (2) one can use the extractor for low-weight affine sources by Rao (CCC 2009).Along the way, we exhibit an explicit boolean function b : {0, 1} n → {0, 1} such that poly(n)-size AC 0 circuits cannot generate the distribution (Y, b(Y )), solving a problem about the complexity of distributions.Independently, De and Watson (RANDOM 2011) obtain a result similar to (1) in the special case d = o(lg n).

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Constructive Proofs of Concentration Bounds

Cited by 60 publications

References 45 publications

Improved direct product theorems for randomized query complexity

Improved direct product theorems for randomized query complexity

Combinatorial Auctions Do Need Modest Interaction

Extractors for Circuit Sources

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