Regularity, Boosting, and Efficiently Simulating Every High-Entropy Distribution

Abstract

We show that every high-entropy distribution is indistinguishable from an efficiently samplable distribution of the same entropy. Specifically, we prove that if D is a distribution over {0, 1}n of min-entropy at least n − k, then for every S and there is a circuit C of size at most S · poly( −1 , 2 k ) that samples a distribution of entropy at least n − k that is -indistinguishable from D by circuits of size S.Stated in a more abstract form (where we refer to indistinguishability by arbitrary families of distinguishers rather than bounded-size circuits), our result implies (a) the Weak Szemerédi Regularity Lemma of Frieze and Kannan (b) a constructive version of the Dense Model Theorem of Green, Tao and Ziegler with better quantitative parameters (polynomial rather than exponential in the distinguishing probability ), and (c) the Impagliazzo Hardcore Set Lemma. It appears to be the general result underlying the known connections between "regularity" results in graph theory, "decomposition" results in additive combinatorics, and the Hardcore Lemma in complexity theory.We present two proofs of our result, one in the spirit of Nisan's proof of the Hardcore Lemma via duality of linear programming, and one similar to Impagliazzo's "boosting" proof. A third proof by iterative partitioning, which gives the complexity of the sampler to be exponential in

How to Fake Auxiliary Input

“…In [25], this result is used to prove that every high-entropy distribution is indistinguishable from an efficiently samplable distribution of the same entropy. Moreover, it is shown that many fundamental results including the Dense Model Theorem [23,14,21,10,24], Impagliazzo's hardcore lemma [18] and a version of Szémeredi's Regularity Lemma [11] follow from this theorem.…”

“…2 A dependency on ε −1 is also necessary. Indeed, Trevisan et al [25,Rem. 1.6] show that such a dependency is necessary for the simulator h in (1).…”

“…We then prove that (1) holds for some simulator h : Z → [0, 1] of complexity ε −2 relative to F . Unfortunately, we cannot use the result from [25] in a black-box way at this point as we need the simulator h : Z → [0, 1] to define a probability distribution in the sense that h(x, b) ≥ 0 for all (x, b) and b∈{0,1} ℓ h(x, b) = 1 for all x. Ensuring these conditions is the most delicate part of the proof.…”

“…In [25], two different proofs for (1) are given. The first proof uses duality of linear programming in the form of the min-max theorem for two-player zero-sum games.…”

“…We do not know if one can directly prove an implication in the other direction, so we prove (2) from scratch. Similarly to [25], the core of our proof uses boosting with the same energy function as the one used in [25].…”