22 research outputs found
Parallel Repetition From Fortification
The Parallel Repetition Theorem upper-bounds the value of a repeated (tensored) two prover game in terms of the value of the base game and the number of repetitions. In this work we give a simple transformation on games – “fortification” – and show that for fortified games, the value of the repeated game decreases perfectly exponentially with the number of repetitions, up to an arbitrarily small additive error. Our proof is combinatorial and short. As corollaries, we obtain: (1) Starting from a PCP Theorem with soundness error bounded away from 1, we get a PCP with arbitrarily small constant soundness error. In particular, starting with the combinatorial PCP of Dinur, we get a combinatorial PCP with low error. The latter can be used for hardness of approximation as in the work of Hastad. (2) Starting from the work of the author and Raz, we get a projection PCP theorem with the smallest soundness error known today. The theorem yields nearly a quadratic improvement in the size compared to previous work. 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.National Science Foundation (U.S.) (Grant 1218547
On Fortification of Projection Games
A recent result of Moshkovitz \cite{Moshkovitz14} presented an ingenious
method to provide a completely elementary proof of the Parallel Repetition
Theorem for certain projection games via a construction called fortification.
However, the construction used in \cite{Moshkovitz14} to fortify arbitrary
label cover instances using an arbitrary extractor is insufficient to prove
parallel repetition. In this paper, we provide a fix by using a stronger graph
that we call fortifiers. Fortifiers are graphs that have both and
guarantees on induced distributions from large subsets. We then show
that an expander with sufficient spectral gap, or a bi-regular extractor with
stronger parameters (the latter is also the construction used in an independent
update \cite{Moshkovitz15} of \cite{Moshkovitz14} with an alternate argument),
is a good fortifier. We also show that using a fortifier (in particular
guarantees) is necessary for obtaining the robustness required for
fortification.Comment: 19 page
A parallel repetition theorem for all entangled games
The behavior of games repeated in parallel, when played with quantumly
entangled players, has received much attention in recent years. Quantum
analogues of Raz's classical parallel repetition theorem have been proved for
many special classes of games. However, for general entangled games no parallel
repetition theorem was known. We prove that the entangled value of a two-player
game repeated times in parallel is at most for a
constant depending on , provided that the entangled value of is
less than 1. In particular, this gives the first proof that the entangled value
of a parallel repeated game must converge to 0 for all games whose entangled
value is less than 1. Central to our proof is a combination of both classical
and quantum correlated sampling.Comment: To appear in the 43rd International Colloquium on Automata,
Languages, and Programming (ICALP
Strong Parallel Repetition for Unique Games on Small Set Expanders
Strong Parallel Repetition for Unique Games on Small Set Expanders
The strong parallel repetition problem for unique games is to efficiently
reduce the 1-delta vs. 1-C*delta gap problem of Boolean unique games (where C>1
is a sufficiently large constant) to the 1-epsilon vs. epsilon gap problem of
unique games over large alphabet. Due to its importance to the Unique Games
Conjecture, this problem garnered a great deal of interest from the research
community. There are positive results for certain easy unique games (e.g.,
unique games on expanders), and an impossibility result for hard unique games.
In this paper we show how to bypass the impossibility result by enlarging the
alphabet sufficiently before repetition. We consider the case of unique games
on small set expanders for two setups: (i) Strong small set expanders that
yield easy unique games. (ii) Weaker small set expanders underlying possibly
hard unique games as long as the game is mildly fortified. We show how to
fortify unique games in both cases, i.e., how to transform the game so
sufficiently large induced sub-games have bounded value. We then prove strong
parallel repetition for the fortified games. Prior to this work fortification
was known for projection games but seemed hopeless for unique games
Direct Sum Theorems From Fortification
We revisit the direct sum questions in communication complexity which asks
whether the resource needed to solve communication problems together is
(approximately) the sum of resources needed to solve these problems separately.
Our work starts with the observation that Dinur and Meir's fortification lemma
can be generalized to a general fortification lemma for a sub-additive measure
over set. By applying this lemma to the case of cover number, we obtain a dual
form of cover number, called "-fooling set" which is a generalized
fooling set. Any rectangle which contains enough number of elements from a
-fooling set can not be monochromatic.
With this fact, we are able to reprove the classic direct sum theorem of
cover number with a simple double counting argument. Formally, let and be two
communication problems, where
denotes the cover number. One issue of current deterministic
direct sum theorems about communication complexity is that they provide no
information when is small, especially when . In this work, we prove a
new direct sum theorem about protocol size which imply a better direct sum
theorem for two functions in terms of protocol size. Formally, let
denotes complexity of the protocol size of a communication problem, given a
communication problem , . All
our results are obtained in a similar way using the -fooling set to
construct a hardcore for the direct sum problem.Comment: fix some typos; remove a section of agree problem with a gap in the
proof pointed by an anonymous reviewer, the author has tried to fix it but
could not fix it for no
A No-Go Theorem for Derandomized Parallel Repetition: Beyond Feige-Kilian
In this work we show a barrier towards proving a randomness-efficient
parallel repetition, a promising avenue for achieving many tight
inapproximability results. Feige and Kilian (STOC'95) proved an impossibility
result for randomness-efficient parallel repetition for two prover games with
small degree, i.e., when each prover has only few possibilities for the
question of the other prover. In recent years, there have been indications that
randomness-efficient parallel repetition (also called derandomized parallel
repetition) might be possible for games with large degree, circumventing the
impossibility result of Feige and Kilian. In particular, Dinur and Meir
(CCC'11) construct games with large degree whose repetition can be derandomized
using a theorem of Impagliazzo, Kabanets and Wigderson (SICOMP'12). However,
obtaining derandomized parallel repetition theorems that would yield optimal
inapproximability results has remained elusive.
This paper presents an explanation for the current impasse in progress, by
proving a limitation on derandomized parallel repetition. We formalize two
properties which we call "fortification-friendliness" and "yields robust
embeddings." We show that any proof of derandomized parallel repetition
achieving almost-linear blow-up cannot both (a) be fortification-friendly and
(b) yield robust embeddings. Unlike Feige and Kilian, we do not require the
small degree assumption.
Given that virtually all existing proofs of parallel repetition, including
the derandomized parallel repetition result of Dinur and Meir, share these two
properties, our no-go theorem highlights a major barrier to achieving
almost-linear derandomized parallel repetition
Multiplayer Parallel Repetition for Expanding Games
We investigate the value of parallel repetition of one-round games with any number of players k>=2. It has been an open question whether an analogue of Raz\u27s Parallel Repetition Theorem holds for games with more than two players, i.e., whether the value of the repeated game decays exponentially with the number of repetitions. Verbitsky has shown, via a reduction to the density Hales-Jewett theorem, that the value of the repeated game must approach zero, as the number of repetitions increases. However, the rate of decay obtained in this way is extremely slow, and it is an open question whether the true rate is exponential as is the case for all two-player games.
Exponential decay bounds are known for several special cases of multi-player games, e.g., free games and anchored games. In this work, we identify a certain expansion property of the base game and show all games with this property satisfy an exponential decay parallel repetition bound. Free games and anchored games satisfy this expansion property, and thus our parallel repetition theorem reproduces all earlier exponential-decay bounds for multiplayer games. More generally, our parallel repetition bound applies to all multiplayer games that are *connected* in a certain sense.
We also describe a very simple game, called the GHZ game, that does not satisfy this connectivity property, and for which we do not know an exponential decay bound. We suspect that progress on bounding the value of this the parallel repetition of the GHZ game will lead to further progress on the general question
Quantum-Proof Extractors: Optimal up to Constant Factors
We give the first construction of a family of quantum-proof extractors that has optimal seed
length dependence O(log(n/ǫ)) on the input length n and error ǫ. Our extractors support any
min-entropy k = Ω(log n + log1+α
(1/ǫ)) and extract m = (1 − α)k bits that are ǫ-close to uniform,
for any desired constant α > 0. Previous constructions had a quadratically worse seed length or
were restricted to very large input min-entropy or very few output bits.
Our result is based on a generic reduction showing that any strong classical condenser is automatically
quantum-proof, with comparable parameters. The existence of such a reduction for
extractors is a long-standing open question; here we give an affirmative answer for condensers.
Once this reduction is established, to obtain our quantum-proof extractors one only needs to consider
high entropy sources. We construct quantum-proof extractors with the desired parameters
for such sources by extending a classical approach to extractor construction, based on the use of
block-sources and sampling, to the quantum setting.
Our extractors can be used to obtain improved protocols for device-independent randomness
expansion and for privacy amplification