2,523 research outputs found
Bounding Bloat in Genetic Programming
While many optimization problems work with a fixed number of decision
variables and thus a fixed-length representation of possible solutions, genetic
programming (GP) works on variable-length representations. A naturally
occurring problem is that of bloat (unnecessary growth of solutions) slowing
down optimization. Theoretical analyses could so far not bound bloat and
required explicit assumptions on the magnitude of bloat. In this paper we
analyze bloat in mutation-based genetic programming for the two test functions
ORDER and MAJORITY. We overcome previous assumptions on the magnitude of bloat
and give matching or close-to-matching upper and lower bounds for the expected
optimization time. In particular, we show that the (1+1) GP takes (i)
iterations with bloat control on ORDER as well as
MAJORITY; and (ii) and
(and for )
iterations without bloat control on MAJORITY.Comment: An extended abstract has been published at GECCO 201
Exponential inequalities for unbounded functions of geometrically ergodic Markov chains. Applications to quantitative error bounds for regenerative Metropolis algorithms
The aim of this note is to investigate the concentration properties of
unbounded functions of geometrically ergodic Markov chains. We derive
concentration properties of centered functions with respect to the square of
the Lyapunov's function in the drift condition satisfied by the Markov chain.
We apply the new exponential inequalities to derive confidence intervals for
MCMC algorithms. Quantitative error bounds are providing for the regenerative
Metropolis algorithm of [5]
Intuitive Analyses via Drift Theory
Humans are bad with probabilities, and the analysis of randomized algorithms
offers many pitfalls for the human mind. Drift theory is an intuitive tool for
reasoning about random processes. It allows turning expected stepwise changes
into expected first-hitting times. While drift theory is used extensively by
the community studying randomized search heuristics, it has seen hardly any
applications outside of this field, in spite of many research questions which
can be formulated as first-hitting times.
We state the most useful drift theorems and demonstrate their use for various
randomized processes, including approximating vertex cover, the coupon
collector process, a random sorting algorithm, and the Moran process. Finally,
we consider processes without expected stepwise change and give a lemma based
on drift theory applicable in such scenarios without drift. We use this tool
for the analysis of the gambler's ruin process, for a coloring algorithm, for
an algorithm for 2-SAT, and for a version of the Moran process without bias
First-Hitting Times Under Additive Drift
For the last ten years, almost every theoretical result concerning the
expected run time of a randomized search heuristic used drift theory, making it
the arguably most important tool in this domain. Its success is due to its ease
of use and its powerful result: drift theory allows the user to derive bounds
on the expected first-hitting time of a random process by bounding expected
local changes of the process -- the drift. This is usually far easier than
bounding the expected first-hitting time directly.
Due to the widespread use of drift theory, it is of utmost importance to have
the best drift theorems possible. We improve the fundamental additive,
multiplicative, and variable drift theorems by stating them in a form as
general as possible and providing examples of why the restrictions we keep are
still necessary. Our additive drift theorem for upper bounds only requires the
process to be nonnegative, that is, we remove unnecessary restrictions like a
finite, discrete, or bounded search space. As corollaries, the same is true for
our upper bounds in the case of variable and multiplicative drift
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