20,377 research outputs found
Concentration Bounds for Stochastic Approximations
We obtain non asymptotic concentration bounds for two kinds of stochastic
approximations. We first consider the deviations between the expectation of a
given function of the Euler scheme of some diffusion process at a fixed
deterministic time and its empirical mean obtained by the Monte-Carlo
procedure. We then give some estimates concerning the deviation between the
value at a given time-step of a stochastic approximation algorithm and its
target. Under suitable assumptions both concentration bounds turn out to be
Gaussian. The key tool consists in exploiting accurately the concentration
properties of the increments of the schemes. For the first case, as opposed to
the previous work of Lemaire and Menozzi (EJP, 2010), we do not have any
systematic bias in our estimates. Also, no specific non-degeneracy conditions
are assumed.Comment: 14 page
A Rigorous Computational Approach to Linear Response
We present a general setting in which the formula describing the linear
response of the physical measure of a perturbed system can be obtained. In this
general setting we obtain an algorithm to rigorously compute the linear
response. We apply our results to expanding circle maps. In particular, we
present examples where we compute, up to a pre-specified error in the
-norm, the response of expanding circle maps under stochastic and
deterministic perturbations. Moreover, we present an example where we compute,
up to a pre-specified error in the -norm, the response of the intermittent
family at the boundary; i.e., when the unperturbed system is the doubling map.Comment: Revised version following reports. A new example which contains the
computation of the linear response at the boundary of the intermittent family
has been adde
Different Approaches on Stochastic Reachability as an Optimal Stopping Problem
Reachability analysis is the core of model checking of time systems. For
stochastic hybrid systems, this safety verification method is very little supported mainly
because of complexity and difficulty of the associated mathematical problems. In this
paper, we develop two main directions of studying stochastic reachability as an optimal
stopping problem. The first approach studies the hypotheses for the dynamic programming
corresponding with the optimal stopping problem for stochastic hybrid systems.
In the second approach, we investigate the reachability problem considering approximations
of stochastic hybrid systems. The main difficulty arises when we have to prove the
convergence of the value functions of the approximating processes to the value function
of the initial process. An original proof is provided
Fluid passage-time calculation in large Markov models
Recent developments in the analysis of large Markov models facilitate the fast approximation of transient characteristics of the underlying stochastic process. So-called fluid analysis makes it possible to consider previously intractable models whose underlying discrete state space grows exponentially as model components are added. In this work, we show how fluid approximation techniques may be used to extract passage-time measures from performance models. We focus on two types of passage measure: passage-times involving individual components; as well as passage-times which capture the time taken for a population of components to evolve. Specifically, we show that for models of sufficient scale, passage-time distributions can be well approximated by a deterministic fluid-derived passage-time measure. Where models are not of sufficient scale, we are able to generate approximate bounds for the entire cumulative distribution function of these passage-time random variables, using moment-based techniques. Finally, we show that for some passage-time measures involving individual components the cumulative distribution function can be directly approximated by fluid techniques
Oblivious Bounds on the Probability of Boolean Functions
This paper develops upper and lower bounds for the probability of Boolean
functions by treating multiple occurrences of variables as independent and
assigning them new individual probabilities. We call this approach dissociation
and give an exact characterization of optimal oblivious bounds, i.e. when the
new probabilities are chosen independent of the probabilities of all other
variables. Our motivation comes from the weighted model counting problem (or,
equivalently, the problem of computing the probability of a Boolean function),
which is #P-hard in general. By performing several dissociations, one can
transform a Boolean formula whose probability is difficult to compute, into one
whose probability is easy to compute, and which is guaranteed to provide an
upper or lower bound on the probability of the original formula by choosing
appropriate probabilities for the dissociated variables. Our new bounds shed
light on the connection between previous relaxation-based and model-based
approximations and unify them as concrete choices in a larger design space. We
also show how our theory allows a standard relational database management
system (DBMS) to both upper and lower bound hard probabilistic queries in
guaranteed polynomial time.Comment: 34 pages, 14 figures, supersedes: http://arxiv.org/abs/1105.281
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