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Algorithms and lower bounds for de-Morgan formulas of low-communication leaf gatesSajin Koroth (Simon Fraser University)
ValentineKabanets
Joint with
ZhenjianLu
Dimitrios Myrisiotis
Igor Carboni Oliveira
Outline
ā¢ Background
ā¢ Circuit model :
ā¢ Prior work
ā¢ Results
ā¢ Lower bounds
ā¢ PRGās
ā¢ SAT algorithmās
ā¢ Learning algorithms
ā¢ Overview of the lower bound technique
Formula[s] ā š¢
Parallel vs Sequential computation
ā¢ Most of linear algebra can be done in parallel
ā¢ Gaussian elimination is an outlier
ā¢ Intuitively its an inherently sequential procedure
ā¢ There are theoretical reasons to believe so
ā¢ There is an efficient sequential algorithm
P vs NC1
Are there problems with efficient sequential algorithms which do not have efficient parallel algorithms ?
Class P of poly-time solvable problems
Modeled as circuits
Circuit complexity
ā¢ Complexity parameters :
ā¢ Size : # of gates
ā¢ Depth : length of the longest path from root to leaf
ā¢ Fan in : 2, Fan out
ā¢ Formulas :
ā¢ Underlying DAG is a tree
ā¢ No reuse of computation
ā¢ Depth = log ( Size )
Internal gates
Leaf gates
Circuit complexityClass = Poly-Size FormulasNC1
ā¢ Efficient parallel computation (formally CREW PRAM):
ā¢ Polynomially many processors
ā¢ Logarithmic computation time
size(F) = nO(1)
x1 x2 xn x5
F
depth(F) = O(log n)
In formula, depth(F) = O(log size(F))
Circuit complexityP vs rephrasedNC1
ā¢ A Boolean function (candidates: Perfect matching, Gaussian elimination etc)
ā¢ That can be computed in poly-time ( )
ā¢ Any de-Morgan formula computing it has super-poly size ( )
f
f ā P
f ā NC1
P vs NC1State of the art
ā¢ Andreevā87 : for a function in called the Andreev function
ā¢ Also, Andreevā87 : , where is the shrinkage exponent
ā¢ Paterson and Zwickā93 :
ā¢ Hastadā98 (breakthrough) :
ā¢ Talā14 :
ā¢ Best l.b. for Andreevās function (Talā14) :
ā¢ Best l.b. for a function in (Talā16) :
Ī©(n2.5āo(1)) P
Ī©(n1+Īāo(1)) Ī
Ī ā„ 1.63
Ī ā„ 2 ā o(1)
Ī = 2
Ī© ( n3
log2 n log log n )P Ī© ( n3
log n(log log n)2 )
Cubic formula lower boundsAndreevās function
f (
)
x1 x2 x3 xnāÆāÆ ,Truth Table of a bit function ( )log n h 2log n = n
y1 y2 y3 ynāÆāÆ =
y1 y2 y3 y nlog nāÆāÆ
ā
z1
y nlog n
y 2nlog n
āÆāÆ
ā
z2
ynā nlog n
ynāÆāÆ
ā
zlog n
h
nlog n
Cubic formula lower boundsHastadās result
ā¢ (Talā14) :
ā¢ Doesnāt work if there are parity gates at bottom
Ī© ( n3
log2 n log log n )
Our ModelAugmenting de-Morgan formulas
ā¢ de-Morgan formulas : leaf gates, input literals
ā¢ Our model : leaf gates, low communication functions
Leaf gatesLeaf gates of low cc
Our modelReformulation
ā¢
ā¢ Size s de-Morgan formula
ā¢ : A family of Boolean functions
ā¢ Leaf gates are functions
ā¢ Our model :
ā¢ - low communication complexity Boolean functions
ā¢
Formula[s] ā š¢
š¢
g ā š¢
š¢
s = O(n2)
Communication complexity
ā¢ Yaoās 2-party model
ā¢ Input divided into 2 parts
ā¢ Goal : compute with minimal communication
x, y
f(x, y) x yf(x, y)
m1m2
mk
Our modelComplexity of Andreevās function
f (
)
x1 x2 x3 xnāÆāÆ ,Truth Table of a bit function ( )log n h 2log n = n
y1 y2 y3 ynāÆāÆ =
y1 y2 y3 y nlog nāÆāÆ
ā
z1
y nlog n
y 2nlog n
āÆāÆ
ā
z2
ynā nlog n
ynāÆāÆ
ā
zlog n
h
nlog n
de-Morgan formula of size n
log n
Leaf gates
Communication complexityOf Parity = 2 bits
Our modelPrior work - Bipartite Formulas
ā¢ Input is divided into two parts,
ā¢ Every leaf can gate can access any Boolean function of either or but not both
ā¢ Models a well known measure - graph complexity
ā¢ Talā16: Bipartite formula complexity of is
ā¢ Earlier methods could not do super linear
x, y
x y
IPn Ī©(n2)
g1 g2 g3 gs
F
x1 x2 x3 xnāÆāÆ y1 y2 y3 ynāÆāÆ
Communication complexityOf a bipartite function = 1 bit
Our modelConnection to Hardness Magnification
ā¢ : Given the truth table of a function on bits ( )
ā¢ Yes : if has a circuit of size at most
ā¢ No : otherwise
ā¢ Meta computational problem with connections to Crypto, learning theory, circuit complexity etc
ā¢ OPSā19:
ā¢ If there exists an such that is not in
ā¢ then,
MCSPN[k] f n N = 2n
f k
Ļµ MCSPN[2o(n)] Formula[N1+Ļµ] ā XOR
NP ā NC1
Our modelConnection to PRG for polytopes
ā¢ Polytope : AND of LTFās
ā¢ LTF :
ā¢
ā¢ Ex :
ā¢ Nisanā94 : Randomized communication complexity
ā¢ PRGās for polytopes : Approximate volume computation
sign(w1x1 + ā¦ + wnxn ā Īø)
w1, ā¦, wn, Īø ā ā
3x1 + 4x2 + 5x7 ā„ 12
O(log n)
Our modelInteresting low communication bottom gates
ā¢ Bipartite functions
ā¢ Parities
ā¢ LTFās (Linear threshold functions)
ā¢ PTFās (Polynomial threshold functions)
Our resultsTarget function - Generalized inner product
ā¢ Generalization of binary inner product
ā¢
ā¢
IPn(x, y) = āiā[n]
xiyi
GIPkn(x1, x2, ā¦, xk) = ā
iā[n/k]ājā[k]
xji
x21 x2
2 x23 x2
nk
āÆāÆ
x11 x1
2 x13 x1
nk
āÆāÆ
xk1 xk
2 xk3 xk
nk
āÆāÆ
ā§
ā§
ā§z1
ā§
ā§
ā§z2
ā§
ā§
ā§zkā0/1
Our resultsLower bound
ā¢ Let be computed on average by ,
ā¢ That is,
ā¢ Then,
ā¢ : Randomized communication of with error in the number on forehead communication complexity model
GIPkn F ā Formula[s] ā š¢
Prx
[F(x) = GIPkn(x)] ā„ 1/2 + Ļµ
s = Ī© ( n2
k2 ā 16k ā RkĻµ/2n2(š¢) ā log2(1/Ļµ) )
RkĻµ/2n2(š¢) š¢ Ļµ/2n2
Our resultsMCSP lower bounds
ā¢ If is computed , then
ā¢ Contrast : OPSā19:
ā¢ If there exists an such that is not in
ā¢ then,
ā¢ Our techniques cannot handle
MCSPN[2cn] Formula[s] ā XOR s = O(n2)
Ļµ MCSPN[2o(n)]Formula[N1+Ļµ] ā XOR
NP ā NC1
MCSPN[2o(n)]
Our resultsPRG
ā¢ A pseudo random generator is said to fool a function class if
ā¢
ā¢ is any function from
ā¢
ā¢ is the seed,
ā¢ Smaller the seed length compared to the better
G Ļµ ā±
Przā{0,1}l(n) [f(G(z)) = 1] ā Pr
xā{0,1}n [f(x) = 1] ā¤ Ļµ
f ā±
G : {0,1}l(n) ā {0,1}n
z l(n) ā n
n
Our resultsPRG
ā¢ Parities at the bottom can make things harder.
ā¢ best known PRG seed length
ā¢ best known only
AC0 poly(log n)
AC0 ā XOR (1 ā o(1))n
Our resultsPRG
ā¢ There is a PRG that -fools
ā¢ Seed length :
ā¢ Seed length is optimal, unless lower bound can be improved
Ļµ Formula[s] ā XOR
O( s ā log s ā log(1/Ļµ) + log n)
Our resultsPRG
ā¢ Natural generalization to
ā¢ There is a PRG that -fools
ā¢ Seed length :
ā¢ Number in hand
Formula[s] ā š¢
Ļµ Formula[s] ā š¢
n/k + O( s ā (RkāNIHĻµ/6s (š¢) + log s) ā log(1/Ļµ) + log k) ā log k
Our resultsPRG - Corollaries
ā¢ (Ours + Vio15) : There is a PRG
ā¢ Seed length :
ā¢ -fools intersection of halfspaces over
ā¢ Our results beats earlier results when and
O(n1/2 ā m1/4 ā log n ā log(n/Ļµ))
Ļµ m {0,1}n
m = O(n) Ļµ ā¤ 1/n
Our resultsPRG - Corollaries
ā¢ There is a PRG
ā¢ Seed length :
ā¢ -fools
ā¢ First of its kind
ā¢ Blackbox counting algorithm (Whitebox due to CW19)
O(n1/2 ā s1/4 ā log n ā log(n/Ļµ))
Ļµ Formula[s] ā SYM
Our resultsSAT Algorithm
ā¢ Given circuit class
ā¢ Circuit SAT : Given , is there an ,
ā¢ #Circuit SAT : Given , how many ,
š
C ā š x C(x) = 1
C ā š x C(x) = 1
Our resultsSAT Algorithm
ā¢ Randomized #SAT algorithm for
ā¢ Running time
ā¢
ā¢ First of its kind #SAT for unbounded depth Boolean circuits with PTFās at the bottom
Formula[s] ā š¢
2nāt
t = Ī©n
s ā log2 s ā R21/3(š¢)
1/2
for LTFslog n
Our resultsLearning algorithm
ā¢ There is PAC-learning algorithm
ā¢ Learns
ā¢ Accuracy : , Confidence :
ā¢ Time complexity :
ā¢ can be learned in [Rei11]
ā¢ Crypto connection:
ā¢ is assumed to compute PRFs (BIP+18)
ā¢ If true, canāt be learned in time
Formula[n2āĪ³] ā XOR
Ļµ Ī“
poly(2n/log n,1/Ļµ, log(1/Ī“))
Formula[n2āĪ³] 2o(n)
MOD3 ā XOR
Formula[n2.8] ā XOR 2o(n)
Lower bound techniqueOutline
ā¢ cannot even be weakly approximated by low communication complexity functions
ā¢ Weakness of : Size formula can be āapproximatedā by degree polynomial
ā¢ is weakly approximated by a collection of leaf gates
GIPkn
Formula[s] ā š¢ ss
GIPkn
Lower bound techniquePart I
ā¢ cannot even be weakly approximated by low communication complexity functions
ā¢ In the number on forehead model
ā¢ Protocol computes with error (uniform distribution)
ā¢ Then commn.comp >
GIPkn
GIPkn Ļµ
n/4k ā log(1/(1 ā 2Ļµ))
Lower bound techniquePart II
ā¢ Weakness of : Size formula can be āapproximatedā by degree polynomial
ā¢ Reichardtā11 : Approximation of Boolean formulas by Polynomials
ā¢ be a formula of size
ā¢ There is a real polynomial of degree
ā¢ For every
ā¢ Fact : For any ,
ā¢ Corollary : For any formula of size ,
Formula[s] ā š¢ s s
F(y1, ā¦, ym) s
p(y1, ā¦, ym) O( s)
y ā {0,1}m, |F(a) ā p(a) | ā¤ 1/10
0 < Ļµ < 1 degĻµ( f ) ā¤ deg( f ) ā log(1/Ļµ)
F s degĻµ(F) ā¤ s ā log(1/Ļµ)
Lower bound - proof sketch
g1 g2 g3 gs
F
size(F) = sā d ā¤ s
pS gi1
gid
ā¤ s sā
p(x)
ā¢ correlates well ( ) with
ā¢ correlates well ( ) with a monomial ( )
F Ļµ p
F1
s spS ā
jā[S],|S|ā¤ s
gij
ā¢ Since each has low communication complexity, so does
ā¢ correlated well with the target function , thus it correlates well with the monomial ( a low communication function) !!!!!!!
gi
ājā[S],|S|ā¤ s
gij
F f
Reichardt ā2011
āx ā {0,1}n, |F(x) ā p(x) | ā¤ Ļµdeg(p) ā¤ s
Limitations of our approach
ā¢ To get better lower bounds, find a smaller degree approximating polynomial
ā¢ Approximate degree bound of Reichardt ( ) cannot be improved
ā¢ function can be computed by a size de-Morgan formula
ā¢ Approximate degree of is
s
ANDn n
ANDn Īø( n)
Future directions
ā¢ Extend lower bounds to when
ā¢ Design a PRG of seed length and error for intersection of half spaces
ā¢ Learn in time
Formula[s] ā š¢ s = Ļ(n2)
no(1) Ļµ ā¤ 1/n n
Formula[s] ā XOR 2O( s)
Questions?
Thank you