How to Fool People to Work on Circuit Lower Bounds Ran Raz Weizmann Institute & Microsoft Research.
-
Upload
myles-wride -
Category
Documents
-
view
221 -
download
3
Transcript of How to Fool People to Work on Circuit Lower Bounds Ran Raz Weizmann Institute & Microsoft Research.
How to Fool People to Work
on Circuit Lower Bounds
Ran RazWeizmann Institute &
Microsoft Research
The Only Barrier for Proving Super-Poly Lower Bounds…
Why Super-Poly Lower Bounds Were Still not Proved ?
Maybe because not enough people are working on it…
The Secret Plan:
Fooling people to work on circuit lower bounds…
Coming up with innocent lookingclean and simple problems thatare seemingly unrelated to
provingcircuit lower bounds, and whose solution would imply strong
circuitlower bounds
Arithmetic Circuits:Field: FVariables: X1,...,Xn
Gates:
Every gate in the circuit computes
a polynomial in F[X1,...,Xn]
Example: (X1 ¢ X1) ¢ (X2 + 1)
The Holy Grail:
Super-polynomial lower bounds
for circuit or formula size
I will present two innocent looking
problems that imply such bounds
Elusive Functions and Lower
Bounds for Arithmetic
Circuits
Polynomial Mappings:
f = (f1,...,fm) : Cn ! Cm is a polynomial mapping of degree d iff1,...,fm are polynomials of (total)degree d
f is explicit if given a monomial M and index i, the coefficient of M infi can be computed in poly time [Val]
The Moments Curve:
f: C ! Cm
f(x) = (x,x2,x3,...,xm)
Fact: 8 affine subspace A ( Cm
8 :Cm-1 ! Cm of (total) degree 1,
The Exercise that Was Never Given:
Give an explicit f: C ! Cm s.t.:8 : Cm-1 ! Cm of degree 2,
We require: f of degree ·
[R08]: Any explicit f ) super-polynomial lower bounds for the permanent
Elusive Functions:
f: Cn ! Cm is (s,r)-elusive if8 : Cs ! Cm of degree r,
[R08]: explicit constructions ofelusive functions imply lower
boundsfor the size of arithmetic
circuits
Proof Idea:Consider : Cs ! Cm of degree r, that mapsa circuit to the polynomial computed by it = polynomials that can be computed by small circuits.Proving lower bounds ,Finding points outside Since f hits a hard functionAdd input variables of f as additionalinput variables
Lower Bounds for Depth-d Circuits:
[SS91], [R08]:
Lower bounds of n1+(1/d)
(using elusive functions)
Tensor-Rank and Lower
Bounds for Arithmetic
Formulas
Tensor-Rank:A: [n]r ! F is of rank 1 if9 a1,…,ar : [n] ! F s.t.A = a1 a2 … ar , that isA(i1,…,ir) = a1(i1) ¢¢¢ ar(ir)
Rank(A) = Min k s.t. A=A1+…+Ak
where A1,…,Ak are of rank 1
8 A: [n]r ! F Rank(A) · nr-1
(generalization of matrix rank)
Tensors and Polynomials:
Given A: [n]r ! F and n¢r variables
x1,1,…,xr,n define
Tensor-Rank and Arithmetic Circuits:
[Str73]: explicit A:[n]3!F of rank m ) explicit lower bound of (m) for arithmetic circuits (for fA)(may give lower bounds of up to (n2))(best known bound: (n))
[R09]: 8 r · logn/loglognexplicit A:[n]r!F of rank nr(1-o(1)) ) explicit super-poly lower bound for arithmetic formulas (for fA)
Depth-3 vs. General Formulas:
Tensor-rank corresponds to depth-3
set-multilinear formulas (for fA)Corollary: strong enough lower
boundsfor depth-3 formulas ) super-polylower bounds for general formulasFolklore: strong enough bounds for depth-4circuits ) exp bounds for general circuits[AV08]: any exp bound for depth-4circuits ) exp bound for general circuits
The Tensor-Product Approach [Str]:Given A1:[n1]r!F, A2:[n2]r!F
Define A = A1 A2 : [n1¢n2]r ! F by
A((i1,j1),…,(ir,jr)) = A1(i1,…,ir) ¢ A2(j1,…jr)
For r=2, Rank(A) = Rank(A1)¢Rank(A2)
Is Rank(A) > Rank(A1)¢Rank(A2)/no(1) (8r) ?
YES ) super-poly lower bounds for arithmetic formulas
The Tensor-Product Approach [Str]:Given A1:[n1]r!F, A2:[n2]r!F
Define A = A1 A2 : [n1¢n2]r ! F by
A((i1,j1),…,(ir,jr)) = A1(i1,…,ir) ¢ A2(j1,…jr)
For r=2, Rank(A) = Rank(A1)¢Rank(A2)
Is Rank(A) > Rank(A1)¢Rank(A2)/no(1) (8r) ?
YES ) super-poly lower bounds for arithmetic formulas
Proof: Let m=n1/r Take A1,…,Ar:[m]r!F of high rank
Let A = A1 A2 … Ar : [n]r ! F
How do we find A1,…,Ar of high rank ?
We fix their r¢n entries as inputs !
Main Steps of the Proof:1) New homogenization and multilinearization techniques2) Defining syntactic-rank of a formula (bounds the tensor-
rank)3) 8s we find the formula of size s with the largest syntactic-rank4) Compute the largest syntactic-
rank of a poly-size formula
Conclusions (of Step 1):For r · logn/loglogn1) super-poly lower bounds for homogenous formulas ) super-
polylower bounds for general
formulas 2) super-poly lower bounds for set-mult formulas ) super-polylower bounds for general
formulas
Homogenization:Given a formula C of size s for ahomogenous polynomial f of deg r give a homogenous formula D for
f[Str73]: D of size sO(log r)
(optimality conjectured in [NW95])
[R09]: D of size
(where d = product depth of C) If s=poly(n), and r · logn/loglognSize(D)=poly(n)
Thanks!