poly-logarithmic independence fools circuits, a survey
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Poly-logarithmic independence fools circuits, a survey. Mirmojtaba (Mojtaba) Gharibi December 2010. Overview. In 1990, Linial and Nisan conjectured that no circuit can distinguish the uniform distribution from any poly-logarithmic independent distribution - PowerPoint PPT PresentationTRANSCRIPT
POLY-LOGARITHMIC
INDEPENDENCE FOOLS
CIRCUITS, A SURVEY
Mirmojtaba (Mojtaba) Gharibi
December 2010
OVERVIEW
In 1990, Linial and Nisan conjectured that no
circuit can distinguish the uniform
distribution from any poly-logarithmic
independent distribution
In a recent breakthrough, after about two
decades of no progress, the conjecture was
settled by Mark Braverman
OVERVIEW : CIRCUITS
A circuit consisting of polynomial number of ,
and gates.
gates only appear at the input nodes
Depth of the circuit is constant and is the
number of gates from the input to the output
OVERVIEW:R-INDEPENDENT DISTRIBUTION
Consider a set of random variables distributed
according to an r-independent distribution over .
Looking at any subset of size at most , the
probability that any bit in that subset be is
independent of any other outcomes in that subset
AN EXAMPLE
Consider the following probability distribution over :
where are set uniformly and independently at random.
This is an example of a -independent distribution, since choosing any subset of size at most , each variable is totally independent from any outcome within that set:
AN EXAMPLE
Subsets of size at most of are:
OVERVIEW:R-INDEPENDENT DISTRIBUTION
So, -independent distributions are in some
sense locally random
Whereas on the other hand, the uniform
distribution is globally random
LINIAL AND NISAN 1990’S CONJECTURE
circuit cannot distinguish local randomness
from global randomness.
i.e. any function computable by an circuit
aiming to distinguish the Uniform distribution
from an -independent distribution output the
same thing on inputs drawn from both
distributions except with a negligible bias.
A DEFINITION
A distribution is said to the Boolean function
if:
Or equivalently
LINIAL AND NISAN 1990’S CONJECTURE
We are interested to know how large needs
to be in order for to any circuit of size
operating on
LN conjecture (with relaxed parameters):
SMALL BIAS
So for a polynomially small , a polylogarithmic is good enough.
We cannot take for granted that a polynomially small , can be boosted to a constant probability by taking majority. Why? circuits are not capable of taking majority.
Polynomially small bias is taken as negligible for circuits.
MOTIVATION AND APPLICATIONS
It says that if an circuit accepts truly random bits,
then it also accepts pseudorandom bits. So if one
wants to distinguish random bits from
pseudorandom bits, he needs a more powerful
circuit, possibly with exponentially more gates or
more powerful gates like XOR and MAJORITY or with
greater depth (e.g. logarithmic).
So it gives a better understanding of limitations of an
important complexity class. The result may be later
used as a tool for proving lower bounds.
MOTIVATION AND APPLICATIONS
Since 1980’s we have known many serious
limitations of circuits like many specific
pseudorandom distributions that fool
circuits. The conjecture actually says that
any r-independent distribution will fool
them. So this gives a large class of
distributions that look random to circuits.
For instance, linear codes with poly-logarithmic
seed length can be PRGs for .
HISTORY
In 2007, in the first noticeable development,
Bazzi settled the conjecture for circuits. His proof
was about 50 pages.
In 2008, Razborov simplified Bazzi’s proof to a 3-
page proof.
Finally, in 2009, Mark Braverman settled the
conjecture with a short proof.
BAZZI’S RESULTS
Bazzi’s theorem: - independence depth 2
circuits where
BAZZI’S RESULTS
Bazzi’s proof was based on harmonic and poset
analysis techniques. He also used Linial,
Mansour and Nisan celebrated result of 1993 of
low-degree real polynomial approximation.
Razborov’s proof does not use Fourier analysis
techniques except of making connection to
Linial, Mansour, and Nisan’s theorem.
REST OF THIS PRESENTATION
Razborov-Smolensky’s approximation
technique by low-degree polynomials over
finite fields
Linial, Mansour and Nisan’s approximation
technique by low-degree real polynomials
Mark Braverman’s proof of the conjecture
RAZBOROV-SMOLENSKY’S APPROXIMATION
Recall : the technique was used in the class to prove
It involves approximating a Boolean function using a low-degree polynomial over finite fields.
Then, knowing the properties of the low-degree polynomials, we can talk about the properties of F.
RAZBOROV-SMOLENSKY’S APPROXIMATION
For example, for PARITY:
Any function can be well approximated by low-
degree polynomials
can be represented with a high-degree
polynomial
A low-degree polynomial cannot approximate a
high-degree polynomial
Hence
RAZBOROV-SMOLENSKY’S APPROXIMATION
Denote the approximation of Boolean
function with a low-degree polynomial .
In Rozborov-Smolenskey’s technique, the
criteria for a good approximation is that for
a large fraction of inputs. However,
when , they may largely disagree.
RAZBOROV-SMOLENSKY’S APPROXIMATION
0
0
1
1
𝑭
𝒇
LINIAL, MANSOUR AND NISAN’S APPROXIMATION
Denote the approximator of the Boolean
function with a low-degree real polynomial
LMN says that approximation of is possible
via low-degree real polynomials. But there
is no guarantee that on any inputs.
Most likely for any inputs,
LINIAL, MANSOUR AND NISAN’S APPROXIMATION
[LMN93]: Every Boolean function
computable by an circuit of size and depth ,
can be approximated by a real low-degree
polynomial of degree :
LINIAL, MANSOUR AND NISAN’S APPROXIMATION
0
1𝑭
𝒇
THE CONJECTURE
We wish to prove : For any -independent distribution where :
THE BASIC IDEA
If we can find a “good” low-degree polynomial
approximation of F we are done.
Because:
Low-degree polynomials are composed of low-degree
terms.
The expectation of any polynomial is the sum of the
expectation of its terms.
Each term’s expectation is exactly the same under
and distribution. So the polynomial’s expectation is
also the same.
THE FIRST STEP IN THE PROOF
We will construct a distribution on the
polynomial over a proper finite field such
that with high probability agrees with on
any given input. So for any given measure ,
with high probability we have an
approximator having a small error, which
implies that there exists a specific
approximator having a small error with
respect to .
THE FIRST STEP IN THE PROOF
If the polynomial is a good approximator (i.e.
for some small ), one can transform to a
good approximator and show that the
conjecture holds.
However, most likely it is not the case! Why?
0
0
1
1
𝑭
𝒇
Small fraction
THE FIRST STEP IN THE PROOF
We want a good approximation too, but is
behaving wildly in its bad region.
Let’s first construct , then we will deal with
this problem in the second step of the proof!
APPROXIMATOR’S CONSTRUCTION
NOT gates will all appear at input nodes
which are easy to approximate.
Anywhere else, we have AND/OR gates.
As usual we use induction on the depth of
the circuit.
We describe the construction of AND
approximator. OR’s construction follows from
the symmetry between 1 & 0 and AND & OR.
AND’S APPROXIMATOR
Consider set of indexes . For a parameter , we prepare a collection of of its subsets in the following way:
For each of we prepare at least random subsets of
We include in each subset each of the indexes with independent probability . We also include . Denote these subsets with
⋀𝐺1 𝐺𝑘…
𝐹
For convenience let us assume is a power of 2, e.g. .
AND’S APPROXIMATOR
Construct:
is the approximator of .
Let us for now focus on the case that all have approximated correctly. We later bound all the errors by the union bound.
AND’S APPROXIMATOR
approximates correctly when .
However, we may err when .
AND’S APPROXIMATOR
When , let us say of the ’s have had been zero.( is
the number of zeros). approximates correctly if at
least one of hits exactly one zero.
The probability of a wrong approximation can be
shown to be at most . Since it is true for any value
of , we can actually find a collection of that yields
that error bound.
By the union bound
The degree of the polynomial is
INSIGHT
Our goal was to make our behave nicely in its bad region.
Here is the idea:
Given our choices for , there exist another
Boolean formula computable by an circuit of
slightly more depth and size which can
determine if has erred or not. Denote it with .
0
0
1
1
𝑭
𝒇
0
1 ℰ𝝂
INSIGHT
How can we use this to make have a better behaviour in its bad region?
Set Compute .
0
0
1
1
𝑭
𝒇
0
1 ℰ𝝂
0
1𝑭 ′
INSIGHT
One can show that is a good approximator of
with respect to both measures and .
and are small:
Also is a good approximator of . But it is
Boolean, not a polynomial. We wish was
exactly behaving like !!!
THE SECOND STEP IN THE PROOF
Here is our new strategy:
Since we failed to find a good approximator of
directly, we try to find a good low-degree
approximator of which we denote by .
Since is a good approximator of , is also a good
approximator of .
By a good approximator we mean an approximator
which by known techniques can be transformed into
an approximator.
THE SECOND STEP IN THE PROOF
We approximate with a low-degree real polynomial of degree based on Linial, Nisan and Mansour technique. Denote the approximation by .
We use this approximation to form .
We choose t large enough to have close to 0 when is close to 1.
0
1
𝟏− ℰ̂𝝂
01
𝑭 ′
0
1
0
1 𝒇
0
𝑭1
THE SECOND STEP IN THE PROOF
Since we have used LMN technique, we can
just say that is a good approximation of with
respect to uniform distribution. (though it is
also good with respect to , since we believe
in the conjecture that a Boolean formula like
outputs the same things most of the times
with respect to measure or )
THE SECOND STEP IN THE PROOF
However, this is enough for us, since we can
transform into by
And then using and playing with inequalities
-which is skipped for our present purpose-
will lead us to the proof.
FINALLY
By setting the parameter properly (i.e. ) one
can settle the conjecture:
Any independent distribution circuits of size
and depth :
REFERENCES [1] Linial, N., and Nisan, N. "Approximate inclusion exclusion." Combinatorica,
1990: 349- 365.
[2] Braverman, M. "Poly-logarithmic independence fools AC0 circuits." IEEE conference on Computational Complexity, 2009: 3-8.
[3] Bazzi, L. M. J. "Polylogarithmic independence can fool DNF formulas." Proceedings of the 48th annual IEEE symposium on Foundations of Computer Science, 2007: 63-73.
[4] Bazzi, L. M. J. "Polylogarithmic independence can fool DNF formulas." SIAM Journal on Computing (SICOMP), 2009.
[5] Razborov, A. A. "A simple proof of Bazzi's theorem." Electronic Colloquium on Computational Complexity, 2008: Report No. 81.
[6] Razborov, A. A. "Lower bounds for the size of circuits of bounded depth with basis {&,⊕}." Mathematicheskie Zametki, 1987: 598–607. English translation in Math. Notes. Acad.Sci. USSR, 1987: 333–338.
[7] Smolensky, R. "Algebraic methods in the theory of lower bounds for Boolean circuit complexity." Proceedings of the nineteenth annual ACM symposium on Theory of computing, 1987: 77-82.
[8] Linial, N., Mansour, Y. and Nisan, N. "Constant depth circuits, Fourier transform, and learnability." Journal of the ACM, 1993: 607-620.
[9] www.scottaaronson.com/blog/ retrieved on Nov 25th, 2010