36,708 research outputs found
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
Cut-off and Escape Behaviors for Birth and Death Chains on Trees
We consider families of discrete time birth and death chains on trees, and
show that in presence of a drift towards the root of the tree, the chains
exhibit cut-off behavior along the drift and escape behavior in the opposite
direction.Comment: 22 pages, 1 figure. This new version, written after referee's
suggestions, is shorter and features a more clear presentation of the
results. The previous proofs of Section 3, based on difference equations
method, are now given in appendi
Abrupt Convergence and Escape Behavior for Birth and Death Chains
We link two phenomena concerning the asymptotical behavior of stochastic
processes: (i) abrupt convergence or cut-off phenomenon, and (ii) the escape
behavior usually associated to exit from metastability. The former is
characterized by convergence at asymptotically deterministic times, while the
convergence times for the latter are exponentially distributed. We compare and
study both phenomena for discrete-time birth-and-death chains on Z with drift
towards zero. In particular, this includes energy-driven evolutions with energy
functions in the form of a single well. Under suitable drift hypotheses, we
show that there is both an abrupt convergence towards zero and escape behavior
in the other direction. Furthermore, as the evolutions are reversible, the law
of the final escape trajectory coincides with the time reverse of the law of
cut-off paths. Thus, for evolutions defined by one-dimensional energy wells
with sufficiently steep walls, cut-off and escape behavior are related by time
inversion.Comment: 2 figure
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