708 research outputs found

### Polynomial bounds for decoupling, with applications

Let f(x) = f(x_1, ..., x_n) = \sum_{|S| <= k} a_S \prod_{i \in S} x_i be an
n-variate real multilinear polynomial of degree at most k, where S \subseteq
[n] = {1, 2, ..., n}. For its "one-block decoupled" version,
f~(y,z) = \sum_{|S| <= k} a_S \sum_{i \in S} y_i \prod_{j \in S\i} z_j,
we show tail-bound comparisons of the form
Pr[|f~(y,z)| > C_k t] t].
Our constants C_k, D_k are significantly better than those known for "full
decoupling". For example, when x, y, z are independent Gaussians we obtain C_k
= D_k = O(k); when x, y, z, Rademacher random variables we obtain C_k = O(k^2),
D_k = k^{O(k)}. By contrast, for full decoupling only C_k = D_k = k^{O(k)} is
known in these settings.
We describe consequences of these results for query complexity (related to
conjectures of Aaronson and Ambainis) and for analysis of Boolean functions
(including an optimal sharpening of the DFKO Inequality).Comment: 19 pages, including bibliograph

### Approximability and proof complexity

This work is concerned with the proof-complexity of certifying that
optimization problems do \emph{not} have good solutions. Specifically we
consider bounded-degree "Sum of Squares" (SOS) proofs, a powerful algebraic
proof system introduced in 1999 by Grigoriev and Vorobjov. Work of Shor,
Lasserre, and Parrilo shows that this proof system is automatizable using
semidefinite programming (SDP), meaning that any $n$-variable degree-$d$ proof
can be found in time $n^{O(d)}$. Furthermore, the SDP is dual to the well-known
Lasserre SDP hierarchy, meaning that the "$d/2$-round Lasserre value" of an
optimization problem is equal to the best bound provable using a degree-$d$ SOS
proof. These ideas were exploited in a recent paper by Barak et al.\ (STOC
2012) which shows that the known "hard instances" for the Unique-Games problem
are in fact solved close to optimally by a constant level of the Lasserre SDP
hierarchy.
We continue the study of the power of SOS proofs in the context of difficult
optimization problems. In particular, we show that the Balanced-Separator
integrality gap instances proposed by Devanur et al.\ can have their optimal
value certified by a degree-4 SOS proof. The key ingredient is an SOS proof of
the KKL Theorem. We also investigate the extent to which the Khot--Vishnoi
Max-Cut integrality gap instances can have their optimum value certified by an
SOS proof. We show they can be certified to within a factor .952 ($> .878$)
using a constant-degree proof. These investigations also raise an interesting
mathematical question: is there a constant-degree SOS proof of the Central
Limit Theorem?Comment: 34 page

### Coin flipping from a cosmic source: On error correction of truly random bits

We study a problem related to coin flipping, coding theory, and noise
sensitivity. Consider a source of truly random bits x \in \bits^n, and $k$
parties, who have noisy versions of the source bits y^i \in \bits^n, where
for all $i$ and $j$, it holds that \Pr[y^i_j = x_j] = 1 - \eps, independently
for all $i$ and $j$. That is, each party sees each bit correctly with
probability $1-\epsilon$, and incorrectly (flipped) with probability
$\epsilon$, independently for all bits and all parties. The parties, who cannot
communicate, wish to agree beforehand on {\em balanced} functions f_i :
\bits^n \to \bits such that $\Pr[f_1(y^1) = ... = f_k(y^k)]$ is maximized. In
other words, each party wants to toss a fair coin so that the probability that
all parties have the same coin is maximized. The functions $f_i$ may be thought
of as an error correcting procedure for the source $x$.
When $k=2,3$ no error correction is possible, as the optimal protocol is
given by $f_i(x^i) = y^i_1$. On the other hand, for large values of $k$, better
protocols exist. We study general properties of the optimal protocols and the
asymptotic behavior of the problem with respect to $k$, $n$ and \eps. Our
analysis uses tools from probability, discrete Fourier analysis, convexity and
discrete symmetrization

### Noise stability of functions with low influences: invariance and optimality

In this paper we study functions with low influences on product probability
spaces. The analysis of boolean functions with low influences has become a
central problem in discrete Fourier analysis. It is motivated by fundamental
questions arising from the construction of probabilistically checkable proofs
in theoretical computer science and from problems in the theory of social
choice in economics.
We prove an invariance principle for multilinear polynomials with low
influences and bounded degree; it shows that under mild conditions the
distribution of such polynomials is essentially invariant for all product
spaces. Ours is one of the very few known non-linear invariance principles. It
has the advantage that its proof is simple and that the error bounds are
explicit. We also show that the assumption of bounded degree can be eliminated
if the polynomials are slightly ``smoothed''; this extension is essential for
our applications to ``noise stability''-type problems.
In particular, as applications of the invariance principle we prove two
conjectures: the ``Majority Is Stablest'' conjecture from theoretical computer
science, which was the original motivation for this work, and the ``It Ain't
Over Till It's Over'' conjecture from social choice theory

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