1,239 research outputs found
Mining Circuit Lower Bound Proofs for Meta-algorithms
We show that circuit lower bound proofs based on the method of random restrictions yield non-trivial compression algorithms for “easy ” Boolean functions from the corresponding circuit classes. The compression problem is defined as follows: given the truth table of an n-variate Boolean function f computable by some unknown small circuit from a known class of circuits, find in deterministic time poly(2n) a circuit C (no restriction on the type of C) computing f so that the size of C is less than the trivial circuit size 2n/n. We get non-trivial compression for functions computable by AC0 circuits, (de Morgan) formulas, and (read-once) branching programs of the size for which the lower bounds for the corresponding circuit class are known. These compression algorithms rely on the structural characterizations of “easy ” functions, which are useful both for proving circuit lower bounds and for designing “meta-algorithms” (such as Circuit-SAT). For (de Morgan) formulas, such structural characterization is provided by the “shrinkage under random restrictions ” results [Sub61, H̊as98], strengthened to the “high-probability ” version by [San10, IMZ12, KR13]. We give a new, simple proof of the “high-probability ” version of the shrinkage result for (de Morgan) formulas, with improved parameters. We use this shrinkage result to get both compression and #SAT algorithms for (de Morgan) formulas of size about n2. We also use this shrinkage result to get an alternative proof of the recent result by Komargodski and Raz [KR13] of the average-case lower bound against small (de Morgan) formulas. Finally, we show that the existence of any non-trivial compression algorithm for a circuit class C ⊆ P/poly would imply the circuit lower bound NEXP 6 ⊆ C; a similar implication is independently proved also by Williams [Wil13]. This complements Williams’s result [Wil10] that any non-trivial Circuit-SAT algorithm for a circuit class C would imply a superpolynomial lower bound against C for a language in NEXP
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
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
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