4,980 research outputs found
Algorithms and lower bounds for de Morgan formulas of low-communication leaf gates
The class consists of Boolean functions
computable by size- de Morgan formulas whose leaves are any Boolean
functions from a class . We give lower bounds and (SAT, Learning,
and PRG) algorithms for , for classes
of functions with low communication complexity. Let
be the maximum -party NOF randomized communication
complexity of . We show:
(1) The Generalized Inner Product function cannot be computed in
on more than fraction of inputs
for As a corollary, we get an average-case lower bound for
against .
(2) There is a PRG of seed length that -fools . For
, we get the better seed length . This gives the first
non-trivial PRG (with seed length ) for intersections of half-spaces
in the regime where .
(3) There is a randomized -time SAT algorithm for , where In particular, this implies a nontrivial
#SAT algorithm for .
(4) The Minimum Circuit Size Problem is not in .
On the algorithmic side, we show that can be
PAC-learned in time
Consistency of Spectral Hypergraph Partitioning under Planted Partition Model
Hypergraph partitioning lies at the heart of a number of problems in machine
learning and network sciences. Many algorithms for hypergraph partitioning have
been proposed that extend standard approaches for graph partitioning to the
case of hypergraphs. However, theoretical aspects of such methods have seldom
received attention in the literature as compared to the extensive studies on
the guarantees of graph partitioning. For instance, consistency results of
spectral graph partitioning under the stochastic block model are well known. In
this paper, we present a planted partition model for sparse random non-uniform
hypergraphs that generalizes the stochastic block model. We derive an error
bound for a spectral hypergraph partitioning algorithm under this model using
matrix concentration inequalities. To the best of our knowledge, this is the
first consistency result related to partitioning non-uniform hypergraphs.Comment: 35 pages, 2 figures, 1 tabl
Conspiracies between learning algorithms, circuit lower bounds, and pseudorandomness
We prove several results giving new and stronger connections between learning theory, circuit
complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)]
denote n-variable C-circuits of size ≤ s(n). We show:
Learning Speedups. If C[poly(n)] admits a randomized weak learning algorithm under the
uniform distribution with membership queries that runs in time 2n/nω(1), then for every k ≥ 1
and ε > 0 the class C[n
k
] can be learned to high accuracy in time O(2n
ε
). There is ε > 0 such that
C[2n
ε
] can be learned in time 2n/nω(1) if and only if C[poly(n)] can be learned in time 2(log n)
O(1)
.
Equivalences between Learning Models. We use learning speedups to obtain equivalences
between various randomized learning and compression models, including sub-exponential
time learning with membership queries, sub-exponential time learning with membership and
equivalence queries, probabilistic function compression and probabilistic average-case function
compression.
A Dichotomy between Learnability and Pseudorandomness. In the non-uniform setting,
there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure
pseudorandom functions computable in C[poly(n)].
Lower Bounds from Nontrivial Learning. If for each k ≥ 1, (depth-d)-C[n
k
] admits a
randomized weak learning algorithm with membership queries under the uniform distribution
that runs in time 2n/nω(1), then for each k ≥ 1, BPE * (depth-d)-C[n
k
]. If for some ε > 0 there
are P-natural proofs useful against C[2n
ε
], then ZPEXP * C[poly(n)].
Karp-Lipton Theorems for Probabilistic Classes. If there is a k > 0 such that BPE ⊆
i.o.Circuit[n
k
], then BPEXP ⊆ i.o.EXP/O(log n). If ZPEXP ⊆ i.o.Circuit[2n/3
], then ZPEXP ⊆
i.o.ESUBEXP.
Hardness Results for MCSP. All functions in non-uniform NC1
reduce to the Minimum
Circuit Size Problem via truth-table reductions computable by TC0
circuits. In particular, if
MCSP ∈ TC0
then NC1 = TC0
Conspiracies Between Learning Algorithms, Circuit Lower Bounds, and Pseudorandomness
We prove several results giving new and stronger connections between learning theory, circuit complexity and pseudorandomness. Let C be any typical class of Boolean circuits, and C[s(n)] denote n-variable C-circuits of size <= s(n). We show:
Learning Speedups: If C[s(n)] admits a randomized weak learning algorithm under the uniform distribution with membership queries that runs in time 2^n/n^{omega(1)}, then for every k >= 1 and epsilon > 0 the class C[n^k] can be learned to high accuracy in time O(2^{n^epsilon}). There is epsilon > 0 such that C[2^{n^{epsilon}}] can be learned in time 2^n/n^{omega(1)} if and only if C[poly(n)] can be learned in time 2^{(log(n))^{O(1)}}.
Equivalences between Learning Models: We use learning speedups to obtain equivalences between various randomized learning and compression models, including sub-exponential time learning with membership queries, sub-exponential time learning with membership and equivalence queries, probabilistic function compression and probabilistic average-case function compression.
A Dichotomy between Learnability and Pseudorandomness: In the non-uniform setting, there is non-trivial learning for C[poly(n)] if and only if there are no exponentially secure pseudorandom functions computable in C[poly(n)].
Lower Bounds from Nontrivial Learning: If for each k >= 1, (depth-d)-C[n^k] admits a randomized weak learning algorithm with membership queries under the uniform distribution that runs in time 2^n/n^{omega(1)}, then for each k >= 1, BPE is not contained in (depth-d)-C[n^k]. If for some epsilon > 0 there are P-natural proofs useful against C[2^{n^{epsilon}}], then ZPEXP is not contained in C[poly(n)].
Karp-Lipton Theorems for Probabilistic Classes: If there is a k > 0 such that BPE is contained in i.o.Circuit[n^k], then BPEXP is contained in i.o.EXP/O(log(n)). If ZPEXP is contained in i.o.Circuit[2^{n/3}], then ZPEXP is contained in i.o.ESUBEXP.
Hardness Results for MCSP: All functions in non-uniform NC^1 reduce to the Minimum Circuit Size Problem via truth-table reductions computable by TC^0 circuits. In particular, if MCSP is in TC^0 then NC^1 = TC^0
Circuit complexity, proof complexity, and polynomial identity testing
We introduce a new algebraic proof system, which has tight connections to
(algebraic) circuit complexity. In particular, we show that any
super-polynomial lower bound on any Boolean tautology in our proof system
implies that the permanent does not have polynomial-size algebraic circuits
(VNP is not equal to VP). As a corollary to the proof, we also show that
super-polynomial lower bounds on the number of lines in Polynomial Calculus
proofs (as opposed to the usual measure of number of monomials) imply the
Permanent versus Determinant Conjecture. Note that, prior to our work, there
was no proof system for which lower bounds on an arbitrary tautology implied
any computational lower bound.
Our proof system helps clarify the relationships between previous algebraic
proof systems, and begins to shed light on why proof complexity lower bounds
for various proof systems have been so much harder than lower bounds on the
corresponding circuit classes. In doing so, we highlight the importance of
polynomial identity testing (PIT) for understanding proof complexity.
More specifically, we introduce certain propositional axioms satisfied by any
Boolean circuit computing PIT. We use these PIT axioms to shed light on
AC^0[p]-Frege lower bounds, which have been open for nearly 30 years, with no
satisfactory explanation as to their apparent difficulty. We show that either:
a) Proving super-polynomial lower bounds on AC^0[p]-Frege implies VNP does not
have polynomial-size circuits of depth d - a notoriously open question for d at
least 4 - thus explaining the difficulty of lower bounds on AC^0[p]-Frege, or
b) AC^0[p]-Frege cannot efficiently prove the depth d PIT axioms, and hence we
have a lower bound on AC^0[p]-Frege.
Using the algebraic structure of our proof system, we propose a novel way to
extend techniques from algebraic circuit complexity to prove lower bounds in
proof complexity
- …