14,136 research outputs found
Convergence rate and averaging of nonlinear two-time-scale stochastic approximation algorithms
The first aim of this paper is to establish the weak convergence rate of
nonlinear two-time-scale stochastic approximation algorithms. Its second aim is
to introduce the averaging principle in the context of two-time-scale
stochastic approximation algorithms. We first define the notion of asymptotic
efficiency in this framework, then introduce the averaged two-time-scale
stochastic approximation algorithm, and finally establish its weak convergence
rate. We show, in particular, that both components of the averaged
two-time-scale stochastic approximation algorithm simultaneously converge at
the optimal rate .Comment: Published at http://dx.doi.org/10.1214/105051606000000448 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Multi-level stochastic approximation algorithms
This paper studies multi-level stochastic approximation algorithms. Our aim
is to extend the scope of the multilevel Monte Carlo method recently introduced
by Giles (Giles 2008) to the framework of stochastic optimization by means of
stochastic approximation algorithm. We first introduce and study a two-level
method, also referred as statistical Romberg stochastic approximation
algorithm. Then, its extension to multi-level is proposed. We prove a central
limit theorem for both methods and describe the possible optimal choices of
step size sequence. Numerical results confirm the theoretical analysis and show
a significant reduction in the initial computational cost.Comment: 44 pages, 9 figure
Stochastic Approximation and Modern Model-Based Designs for Dose-Finding Clinical Trials
In 1951 Robbins and Monro published the seminal article on stochastic
approximation and made a specific reference to its application to the
"estimation of a quantal using response, nonresponse data." Since the 1990s,
statistical methodology for dose-finding studies has grown into an active area
of research. The dose-finding problem is at its core a percentile estimation
problem and is in line with what the Robbins--Monro method sets out to solve.
In this light, it is quite surprising that the dose-finding literature has
developed rather independently of the older stochastic approximation
literature. The fact that stochastic approximation has seldom been used in
actual clinical studies stands in stark contrast with its constant application
in engineering and finance. In this article, I explore similarities and
differences between the dose-finding and the stochastic approximation
literatures. This review also sheds light on the present and future relevance
of stochastic approximation to dose-finding clinical trials. Such connections
will in turn steer dose-finding methodology on a rigorous course and extend its
ability to handle increasingly complex clinical situations.Comment: Published in at http://dx.doi.org/10.1214/10-STS334 the Statistical
Science (http://www.imstat.org/sts/) by the Institute of Mathematical
Statistics (http://www.imstat.org
- …