11,478 research outputs found
Sharp lower bounds for the asymptotic entropy of symmetric random walks
The entropy, the spectral radius and the drift are important numerical
quantities associated to random walks on countable groups. We prove sharp
inequalities relating those quantities for walks with a finite second moment,
improving upon previous results of Avez, Varopoulos, Carne, Ledrappier. We also
deduce inequalities between these quantities and the volume growth of the
group. Finally, we show that the equality case in our inequality is rather
rigid.Comment: v2: minor corrections v3: reorganization, stronger rigidity
statement
Statistics and compression of scl
We obtain sharp estimates on the growth rate of stable commutator length on
random (geodesic) words, and on random walks, in hyperbolic groups and groups
acting nondegenerately on hyperbolic spaces. In either case, we show that with
high probability stable commutator length of an element of length is of
order .
This establishes quantitative refinements of qualitative results of
Bestvina-Fujiwara and others on the infinite dimensionality of 2-dimensional
bounded cohomology in groups acting suitably on hyperbolic spaces, in the sense
that we can control the geometry of the unit balls in these normed vector
spaces (or rather, in random subspaces of their normed duals).
As a corollary of our methods, we show that an element obtained by random
walk of length in a mapping class group cannot be written as a product of
fewer than reducible elements, with probability going to 1 as
goes to infinity. We also show that the translation length on the complex
of free factors of a random walk of length on the outer automorphism group
of a free group grows linearly in .Comment: Minor edits arising from referee's comments; 45 page
The Right Mutation Strength for Multi-Valued Decision Variables
The most common representation in evolutionary computation are bit strings.
This is ideal to model binary decision variables, but less useful for variables
taking more values. With very little theoretical work existing on how to use
evolutionary algorithms for such optimization problems, we study the run time
of simple evolutionary algorithms on some OneMax-like functions defined over
. More precisely, we regard a variety of
problem classes requesting the component-wise minimization of the distance to
an unknown target vector . For such problems we see a crucial
difference in how we extend the standard-bit mutation operator to these
multi-valued domains. While it is natural to select each position of the
solution vector to be changed independently with probability , there are
various ways to then change such a position. If we change each selected
position to a random value different from the original one, we obtain an
expected run time of . If we change each selected position
by either or (random choice), the optimization time reduces to
. If we use a random mutation strength with probability inversely proportional to and change
the selected position by either or (random choice), then the
optimization time becomes , bringing down
the dependence on from linear to polylogarithmic. One of our results
depends on a new variant of the lower bounding multiplicative drift theorem.Comment: an extended abstract of this work is to appear at GECCO 201
Upper Bounds on the Runtime of the Univariate Marginal Distribution Algorithm on OneMax
A runtime analysis of the Univariate Marginal Distribution Algorithm (UMDA)
is presented on the OneMax function for wide ranges of its parameters and
. If for some constant and
, a general bound on the expected runtime
is obtained. This bound crucially assumes that all marginal probabilities of
the algorithm are confined to the interval . If for a constant and , the
behavior of the algorithm changes and the bound on the expected runtime becomes
, which typically even holds if the borders on the marginal
probabilities are omitted.
The results supplement the recently derived lower bound
by Krejca and Witt (FOGA 2017) and turn out as
tight for the two very different values and . They also improve the previously best known upper bound by Dang and Lehre (GECCO 2015).Comment: Version 4: added illustrations and experiments; improved presentation
in Section 2.2; to appear in Algorithmica; the final publication is available
at Springer via http://dx.doi.org/10.1007/s00453-018-0463-
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