1,547 research outputs found
No Free Lunch versus Occam's Razor in Supervised Learning
The No Free Lunch theorems are often used to argue that domain specific
knowledge is required to design successful algorithms. We use algorithmic
information theory to argue the case for a universal bias allowing an algorithm
to succeed in all interesting problem domains. Additionally, we give a new
algorithm for off-line classification, inspired by Solomonoff induction, with
good performance on all structured problems under reasonable assumptions. This
includes a proof of the efficacy of the well-known heuristic of randomly
selecting training data in the hope of reducing misclassification rates.Comment: 16 LaTeX pages, 1 figur
Transition asymptotics for reaction-diffusion in random media
We describe a universal transition mechanism characterizing the passage to an
annealed behavior and to a regime where the fluctuations about this behavior
are Gaussian, for the long time asymptotics of the empirical average of the
expected value of the number of random walks which branch and annihilate on
, with stationary random rates. The random walks are
independent, continuous time rate , simple, symmetric, with . A random walk at , binary branches at rate ,
and annihilates at rate . The random environment has coordinates
which are i.i.d. We identify a natural way to describe
the annealed-Gaussian transition mechanism under mild conditions on the rates.
Indeed, we introduce the exponents
, and assume
that for
small enough, where and
denotes the average of the expected value of the number of particles
at time and an environment of rates , given that initially there was
only one particle at 0. Then the empirical average of over a box of
side has different behaviors: if for some and large enough , a law of large
numbers is satisfied; if for some
and large enough , a CLT is satisfied. These statements are
violated if the reversed inequalities are satisfied for some negative
. Applications to potentials with Weibull, Frechet and double
exponential tails are given.Comment: To appear in: Probability and Mathematical Physics: A Volume in Honor
of Stanislav Molchanov, Editors - AMS | CRM, (2007
- …