39,237 research outputs found
Improved Pseudorandom Generators from Pseudorandom Multi-Switching Lemmas
We give the best known pseudorandom generators for two touchstone classes in
unconditional derandomization: an -PRG for the class of size-
depth- circuits with seed length , and an -PRG for the class of -sparse
polynomials with seed length . These results bring the state of the art for
unconditional derandomization of these classes into sharp alignment with the
state of the art for computational hardness for all parameter settings:
improving on the seed lengths of either PRG would require breakthrough progress
on longstanding and notorious circuit lower bounds.
The key enabling ingredient in our approach is a new \emph{pseudorandom
multi-switching lemma}. We derandomize recently-developed
\emph{multi}-switching lemmas, which are powerful generalizations of
H{\aa}stad's switching lemma that deal with \emph{families} of depth-two
circuits. Our pseudorandom multi-switching lemma---a randomness-efficient
algorithm for sampling restrictions that simultaneously simplify all circuits
in a family---achieves the parameters obtained by the (full randomness)
multi-switching lemmas of Impagliazzo, Matthews, and Paturi [IMP12] and
H{\aa}stad [H{\aa}s14]. This optimality of our derandomization translates into
the optimality (given current circuit lower bounds) of our PRGs for
and sparse polynomials
Top-Down Induction of Decision Trees: Rigorous Guarantees and Inherent Limitations
Consider the following heuristic for building a decision tree for a function
. Place the most influential variable of
at the root, and recurse on the subfunctions and on the
left and right subtrees respectively; terminate once the tree is an
-approximation of . We analyze the quality of this heuristic,
obtaining near-matching upper and lower bounds:
Upper bound: For every with decision tree size and every
, this heuristic builds a decision tree of size
at most .
Lower bound: For every and , there is an with decision tree size such that
this heuristic builds a decision tree of size .
We also obtain upper and lower bounds for monotone functions:
and
respectively. The lower bound disproves conjectures of Fiat and Pechyony (2004)
and Lee (2009).
Our upper bounds yield new algorithms for properly learning decision trees
under the uniform distribution. We show that these algorithms---which are
motivated by widely employed and empirically successful top-down decision tree
learning heuristics such as ID3, C4.5, and CART---achieve provable guarantees
that compare favorably with those of the current fastest algorithm (Ehrenfeucht
and Haussler, 1989). Our lower bounds shed new light on the limitations of
these heuristics.
Finally, we revisit the classic work of Ehrenfeucht and Haussler. We extend
it to give the first uniform-distribution proper learning algorithm that
achieves polynomial sample and memory complexity, while matching its
state-of-the-art quasipolynomial runtime
An average-case depth hierarchy theorem for Boolean circuits
We prove an average-case depth hierarchy theorem for Boolean circuits over
the standard basis of , , and gates.
Our hierarchy theorem says that for every , there is an explicit
-variable Boolean function , computed by a linear-size depth- formula,
which is such that any depth- circuit that agrees with on fraction of all inputs must have size This
answers an open question posed by H{\aa}stad in his Ph.D. thesis.
Our average-case depth hierarchy theorem implies that the polynomial
hierarchy is infinite relative to a random oracle with probability 1,
confirming a conjecture of H{\aa}stad, Cai, and Babai. We also use our result
to show that there is no "approximate converse" to the results of Linial,
Mansour, Nisan and Boppana on the total influence of small-depth circuits, thus
answering a question posed by O'Donnell, Kalai, and Hatami.
A key ingredient in our proof is a notion of \emph{random projections} which
generalize random restrictions
The Meta-Theory of Q_0 in the Calculus of Inductive Constructions, Master\u27s Thesis, May 2006
The notion of a proof is central to all of mathematics. In the language of formal logic, a proof is a finite sequence of inferences from a set of axioms, and any statement one yields from such a finitistic procedure is called a theorem. For better or for worse, this is far from the form a traditional mathematical proof takes. Mathematicians write proofs that omit routine logical steps, and details deemed tangential to the central result are often elided. These proofs are fuzzy and human-centric, and a great amount of context is assumed on the part of the reader. While traditional proofs are not overly symbolic or syntactic, and hence are easily understood, such informal proofs are susceptible to logical errors -- Fermat\u27s Last Theorem and the Four Color Theorem being prime examples. In light of this, there has been significant interest in producing formal proofs of mathematical theorems: proofs in which every intermediate logical step is supplied. Drawing on ideas from Computational Logic, Type Theory and the theory of Automated Deduction, we are able to guaranteed the correctness of these proofs. The formalization of mathematics is an endeavor that has enjoyed very encouraging progress in recent years. Major achievements include the complete formalization of the Four Color Theorem, the Prime Number Theorem, Goedel\u27s Incompleteness Theorem, the Jordan Curve Theorem (all within the past five years!). This thesis presents our work in formalizing the meta-theory of Peter Andrews\u27 classical higher-order logic in a higher-order typed lambda calculus. Our development is a completely formal one -- in addition to formalizing logical meta-theory, we have also developed and formalized the syntactic meta-theory. Our syntactic meta-theory allows for the reasoning of notions such as variable occurrences, scope and variable binding, linear replacement, etc. Our formalization is carried out in the interactive proof assistant Coq, developed as part of the LogiCal Project in INRIA. Coq is built upon the Calculus of Inductive Constructions, an extension of Coquand and Huet\u27s seminal Calculus of Construction with support for inductive data types. As far as we know, this thesis presents the first effort to formalize Andrews\u27 logical system
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