1,281 research outputs found
Imitation Games and Computation
TAn imitation game is a finite two person normal form game in which the two players have the same set of pure strategies and the goal of the second player is to choose the same pure strategy as the first player. Gale et al. (1950) gave a way of passing from a given two person game to a symmetric game whose symmetric Nash equilibria are in oneto-one correspondence with the Nash equilibria of the given game. We give a way of passing from a given symmetric two person game to an imitation game whose Nash equilibria are in one-to-one correspondence with the symmetric Nash equilibria of the given symmetric game. Lemke (1965) portrayed the Lemke-Howson algorithm as a special case of the Lemke paths algorithm. Using imitation games, we show how Lemke paths may be obtained by projecting Lemke-Howson paths.
Superpolynomial lower bounds for general homogeneous depth 4 arithmetic circuits
In this paper, we prove superpolynomial lower bounds for the class of
homogeneous depth 4 arithmetic circuits. We give an explicit polynomial in VNP
of degree in variables such that any homogeneous depth 4 arithmetic
circuit computing it must have size .
Our results extend the works of Nisan-Wigderson [NW95] (which showed
superpolynomial lower bounds for homogeneous depth 3 circuits),
Gupta-Kamath-Kayal-Saptharishi and Kayal-Saha-Saptharishi [GKKS13, KSS13]
(which showed superpolynomial lower bounds for homogeneous depth 4 circuits
with bounded bottom fan-in), Kumar-Saraf [KS13a] (which showed superpolynomial
lower bounds for homogeneous depth 4 circuits with bounded top fan-in) and
Raz-Yehudayoff and Fournier-Limaye-Malod-Srinivasan [RY08, FLMS13] (which
showed superpolynomial lower bounds for multilinear depth 4 circuits). Several
of these results in fact showed exponential lower bounds.
The main ingredient in our proof is a new complexity measure of {\it bounded
support} shifted partial derivatives. This measure allows us to prove
exponential lower bounds for homogeneous depth 4 circuits where all the
monomials computed at the bottom layer have {\it bounded support} (but possibly
unbounded degree/fan-in), strengthening the results of Gupta et al and Kayal et
al [GKKS13, KSS13]. This new lower bound combined with a careful "random
restriction" procedure (that transforms general depth 4 homogeneous circuits to
depth 4 circuits with bounded support) gives us our final result
A Nearly Optimal Lower Bound on the Approximate Degree of AC
The approximate degree of a Boolean function is the least degree of a real polynomial that
approximates pointwise to error at most . We introduce a generic
method for increasing the approximate degree of a given function, while
preserving its computability by constant-depth circuits.
Specifically, we show how to transform any Boolean function with
approximate degree into a function on variables with approximate degree at least . In particular, if , then
is polynomially larger than . Moreover, if is computed by a
polynomial-size Boolean circuit of constant depth, then so is .
By recursively applying our transformation, for any constant we
exhibit an AC function of approximate degree . This
improves over the best previous lower bound of due to
Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of
that holds for any function. Our lower bounds also apply to
(quasipolynomial-size) DNFs of polylogarithmic width.
We describe several applications of these results. We give:
* For any constant , an lower bound on the
quantum communication complexity of a function in AC.
* A Boolean function with approximate degree at least ,
where is the certificate complexity of . This separation is optimal
up to the term in the exponent.
* Improved secret sharing schemes with reconstruction procedures in AC.Comment: 40 pages, 1 figur
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