We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1 - \epsilon)-probability by communicating at most \epsilon bits of information, then one can win n copies with probability at least 2^{-O(\epsilon n)}. This gives an intuitive explanation why Raz's counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length $m$ by communicating O(m^{-2})-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1-\epsilon)^n, then one can win one copy with at least (1 - O(\epsilon))-probability by communicating O(\epsilon) bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker's Inequality.
We introduce a generalization of the standard framework for studying the difficulty of two-prover games. Specifically, we study the model where Alice and Bob are allowed to communicate (with information constraints) - in contrast to the usual two-prover game where they are not allowed to communicate after receiving their respective input. We study the trade-off between the information cost of the protocol and the achieved value of the game after the protocol. In particular, we show the connection of this trade-off and the amortized behavior of the game (i.e. repeated value of the game). We show that if one can win the game with at least (1 - \epsilon)-probability by communicating at most \epsilon bits of information, then one can win n copies with probability at least 2^{-O(\epsilon n)}. This gives an intuitive explanation why Raz's counter-example to strong parallel repetition [Raz2008] (the odd cycle game) is a counter-example to strong parallel repetition - one can win the odd-cycle game on a cycle of length m by communicating O(m^{-2})-bits where m is the number of vertices. Conversely, for projection games, we show that if one can win n copies with probability larger than (1-\epsilon)^n, then one can win one copy with at least (1 - O(\epsilon))-probability by communicating O(\epsilon) bits of information. By showing the equivalence between information value and amortized value, we give an alternative direction for further works in studying amortized behavior of the two-prover games. The main technical tool is the "Chi-Squared Lemma" which bounds the information cost of the protocol in terms of Chi-Squared distance, instead of usual divergence. This avoids the square loss from using Pinsker's Inequality.
We consider one-round games between a classical verifier and two provers. One of the main questions in this area is the parallel repetition question: If the game is played H times in parallel, does the maximum winning probability decay exponentially in ℓ? In the classical setting, this question was answered in the affirmative by Raz. More recently the question arose whether the decay is of the form (1 - ⊖ (ε))ℓwhere 1 - ε is the value of the game and H is the number of repetitions. This question is known as the strong parallel repetition question and was motivated by its connections to the unique games conjecture. It was resolved by Raz who showed that strong parallel repetition does not hold, even in the very special case of games known as XOR games. This opens the question whether strong parallel repetition holds in the case when the provers share entanglement. Evidence for this is provided by the behavior of XOR games, which have strong (in fact perfect) parallel repetition, and by the recently proved strong parallel repetition of linear unique games. A similar question was open for games with so-called non-signaling provers. Here the best known parallel repetition theorem is due to Holenstein, and is of the form (1 - ⊖ (ε2))ℓ. We show that strong parallel repetition holds neither with entangled provers nor with non-signaling provers. In particular we obtain that Holenstein's bound is tight. Along the way we also provide a tight characterization of the asymptotic behavior of the entangled value under parallel repetition of unique games in terms of a semidefinite program.
The parallel repetition theorem states that, for any two-prover game with value 1−ϵ (for, say, ϵ≤1/2), the value of the game repeated in parallel n times is at most (1−ϵc)Ω(n/s), where s is the answer's length (of the original game) and c is a universal constant [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803]. Several researchers asked whether this bound could be improved to (1−ϵ)Ω(n/s); this question is usually referred to as the strong parallel repetition problem. We show that the answer to this question is negative. More precisely, we consider the odd cycle game of size m, a two-prover game with value 1−1/2m. We show that the value of the odd cycle game repeated in parallel n times is at least 1−(1/m)⋅O(n). This implies that, for large enough n (say, n≥Ω(m2)), the value of the odd cycle game repeated in parallel n times is at least (1−1/4m2)O(n). Thus the following hold. 1. For parallel repetition of general games, the bounds of (1−ϵc)Ω(n/s) given in [R. Raz, SIAM J. Comput., 27 (1998), pp. 763–803; T. Holenstein, in Proceedings of STOC 2002, ACM, New York, 2002, pp. 767–775] are of the right form, up to determining the exact value of the constant c≥2. 2. For parallel repetition of XOR games, unique games, and projection games, the bounds of (1−ϵ2)Ω(n) given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] (for XOR games) and in [A. Rao, in Proceedings of STOC 2008, ACM, New York, 2008, pp. 1–10] (for unique and projection games) are tight. 3. For parallel repetition of the odd cycle game, the bound of 1−(1/m)⋅Ω~(n) given in [U. Feige, G. Kindler, and R. O'Donnell, in Proceedings of CCC 2007, IEEE Computer Society, Washington, DC, 2007, pp. 179–192] is almost tight. A major motivation for the recent interest in the strong parallel repetition problem is that a strong parallel repetition theorem would have implied that the unique game conjecture is equivalent to the NP hardness of distinguishing between instances of Max-Cut that are at least 1−ϵ2 satisfiable from instances that are at most 1−(2/π)⋅ϵ satisfiable. Our results suggest that this cannot be proved just by improving the known bounds on parallel repetition.
In this thesis, we answer several questions about the behaviour of prover-verifier interactions under parallel repetition when quantum information is allowed, and the verifier acts independently in them. \n \nWe first consider the case in which a value is associated with each of the possible outcomes of an interaction. We prove that it is not possible for the prover to improve on the optimum average value per repetition by repeating the protocol multiple times in parallel. \n \nWe look then at games in which the outcomes are classified into two types, winning outcomes and losing outcomes. We ask what is the optimal probability for the prover of winning at least k times out of n parallel repetitions, given that the optimal probability of winning when only one repetition is considered is p. A reasonable conjecture for the answer would be the answer when it is optimal for the prover to act independently. This is known to be the correct answer when k=n. We will show how this cannot be extended to the general case, presenting an example of an interaction with k=1,n=2 in which p is approximately 0.85, but it is possible to always win at least once. We will then give some upper bounds on the optimal probability for the prover of winning k times out of n parallel repetitions. These bounds are expressed as a function of p. \n \nFinally, we connect our results to the study of error reduction for quantum interactive proofs using parallel repetition.
In this thesis, we answer several questions about the behaviour of prover-verifier interactions under parallel repetition when quantum information is allowed, and the verifier acts independently in them. We first consider the case in which a value is associated with each of the possible outcomes of an interaction. We prove that it is not possible for the prover to improve on the optimum average value per repetition by repeating the protocol multiple times in parallel. We look then at games in which the outcomes are classified into two types, winning outcomes and losing outcomes. We ask what is the optimal probability for the prover of winning at least k times out of n parallel repetitions, given that the optimal probability of winning when only one repetition is considered is p. A reasonable conjecture for the answer would be \sum_{m \geq k} {n \choose m} p^m (1-p)^{n-m}, as that is the answer when it is optimal for the prover to act independently. This is known to be the correct answer when k=n, and also in the classical case. It is also correct in some generalizations of the classical case that we will discuss later. We will show how this cannot be extended to all cases, presenting an example of an interaction with k=1,n=2 in which p\approx 0.85, but it is possible to always win at least once. We will then give some upper bounds on the optimal probability for the prover of winning k times out of n parallel repetitions. These bounds are expressed as a function of p. Finally, we will connect our results to the study of error reduction for quantum interactive proofs using parallel repetition.
Consider a game where a refereed chooses (x,y) according to a publiclyknown distribution PXY, sends x to Alice, and y to Bob. Withoutcommunicating with each other, Alice responds with a value "a" and Bobresponds with a value "b". Alice and Bob jointly win if a publiclyknown predicate Q(x,y,a,b) holds. Let such a game be given and assume that the maximum probabilitythat Alice and Bob can win is v<1. Raz (SIAM J. Comput. 27, 1998)shows that if the game is repeated n times in parallel, then the probability that Alice and Bob win all games simultaneously is at most v'(n/log(s)), where s is the maximal number of possible responses from Alice and Bob in the initial game, and v' is a constant depending only on v. In this work, we simplify Raz's proof in various ways and thus shorten it significantly. Further we study the case where Alice and Bob are not restricted to local computations and can use any strategy which does not imply communication among them.
The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game G⊗k in terms of the value of the game G and the number of repetitions k. Contrary to what one might have guessed, the value of G⊗k is not val(G), but rather a more complicated expression depending on properties of G. Indeed, there are numerous works aiming to understand the value of repeated games, both for general games G and for special cases. A case of particular interest is that of projection games, where the answer of the first prover determines at most one acceptable answer for the second prover. In this work we give a simple transformation, which we call “fortification”, that can be applied to any projection game. We show that for fortified games G, the value of the k-repeated game is approximately val(G). This results in nearly a quadratic improvement in the size of projection PCP with the lowest error known today. Unlike previous proofs of the parallel repetition theorem that relied on information theory or linear algebra, our proof is purely combinatorial and quite short. We then discuss the problem of derandomizing parallel repetition, and the limitations of the fortification idea in this setting. We point out a connection between the problem of derandomizing parallel repetition and the problem of composition. This connection could shed light on the so-called Projection Games Conjecture, which asks for projection PCP with minimal error. ∗dmoshkov@csail.mit.edu. Department of Electrical Engineering and Computer Science, MIT. This material is based upon work supported by the National Science Foundation under Grant Number 1218547.