890,574 research outputs found
Variance Reduction Techniques in Monte Carlo Methods
Monte Carlo methods are simulation algorithms to estimate a numerical quantity in a statistical model of a real system. These algorithms are executed by computer programs. Variance reduction techniques (VRT) are needed, even though computer speed has been increasing dramatically, ever since the introduction of computers. This increased computer power has stimulated simulation analysts to develop ever more realistic models, so that the net result has not been faster execution of simulation experiments; e.g., some modern simulation models need hours or days for a single ’run’ (one replication of one scenario or combination of simulation input values). Moreover there are some simulation models that represent rare events which have extremely small probabilities of occurrence), so even modern computer would take ’for ever’ (centuries) to execute a single run - were it not that special VRT can reduce theses excessively long runtimes to practical magnitudes.common random numbers;antithetic random numbers;importance sampling;control variates;conditioning;stratied sampling;splitting;quasi Monte Carlo
Stochastic Variance Reduction Methods for Saddle-Point Problems
We consider convex-concave saddle-point problems where the objective
functions may be split in many components, and extend recent stochastic
variance reduction methods (such as SVRG or SAGA) to provide the first
large-scale linearly convergent algorithms for this class of problems which is
common in machine learning. While the algorithmic extension is straightforward,
it comes with challenges and opportunities: (a) the convex minimization
analysis does not apply and we use the notion of monotone operators to prove
convergence, showing in particular that the same algorithm applies to a larger
class of problems, such as variational inequalities, (b) there are two notions
of splits, in terms of functions, or in terms of partial derivatives, (c) the
split does need to be done with convex-concave terms, (d) non-uniform sampling
is key to an efficient algorithm, both in theory and practice, and (e) these
incremental algorithms can be easily accelerated using a simple extension of
the "catalyst" framework, leading to an algorithm which is always superior to
accelerated batch algorithms.Comment: Neural Information Processing Systems (NIPS), 2016, Barcelona, Spai
Some variance reduction methods for numerical stochastic homogenization
We overview a series of recent works devoted to variance reduction techniques
for numerical stochastic homogenization. Numerical homogenization requires
solving a set of problems at the micro scale, the so-called corrector problems.
In a random environment, these problems are stochastic and therefore need to be
repeatedly solved, for several configurations of the medium considered. An
empirical average over all configurations is then performed using the
Monte-Carlo approach, so as to approximate the effective coefficients necessary
to determine the macroscopic behavior. Variance severely affects the accuracy
and the cost of such computations. Variance reduction approaches, borrowed from
other contexts of the engineering sciences, can be useful. Some of these
variance reduction techniques are presented, studied and tested here
American Options Based on Malliavin Calculus and Nonparametric Variance Reduction Methods
This paper is devoted to pricing American options using Monte Carlo and the
Malliavin calculus. Unlike the majority of articles related to this topic, in
this work we will not use localization fonctions to reduce the variance. Our
method is based on expressing the conditional expectation E[f(St)/Ss] using the
Malliavin calculus without localization. Then the variance of the estimator of
E[f(St)/Ss] is reduced using closed formulas, techniques based on a
conditioning and a judicious choice of the number of simulated paths. Finally,
we perform the stopping times version of the dynamic programming algorithm to
decrease the bias. On the one hand, we will develop the Malliavin calculus
tools for exponential multi-dimensional diffusions that have deterministic and
no constant coefficients. On the other hand, we will detail various
nonparametric technics to reduce the variance. Moreover, we will test the
numerical efficiency of our method on a heterogeneous CPU/GPU multi-core
machine
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