144,431 research outputs found
Smart Sampling for Lightweight Verification of Markov Decision Processes
Markov decision processes (MDP) are useful to model optimisation problems in
concurrent systems. To verify MDPs with efficient Monte Carlo techniques
requires that their nondeterminism be resolved by a scheduler. Recent work has
introduced the elements of lightweight techniques to sample directly from
scheduler space, but finding optimal schedulers by simple sampling may be
inefficient. Here we describe "smart" sampling algorithms that can make
substantial improvements in performance.Comment: IEEE conference style, 11 pages, 5 algorithms, 11 figures, 1 tabl
Efficient generation of random derangements with the expected distribution of cycle lengths
We show how to generate random derangements efficiently by two different
techniques: random restricted transpositions and sequential importance
sampling. The algorithm employing restricted transpositions can also be used to
generate random fixed-point-free involutions only, a.k.a. random perfect
matchings on the complete graph. Our data indicate that the algorithms generate
random samples with the expected distribution of cycle lengths, which we
derive, and for relatively small samples, which can actually be very large in
absolute numbers, we argue that they generate samples indistinguishable from
the uniform distribution. Both algorithms are simple to understand and
implement and possess a performance comparable to or better than those of
currently known methods. Simulations suggest that the mixing time of the
algorithm based on random restricted transpositions (in the total variance
distance with respect to the distribution of cycle lengths) is
with and the length of the
derangement. We prove that the sequential importance sampling algorithm
generates random derangements in time with probability of
failing.Comment: This version corrected and updated; 14 pages, 2 algorithms, 2 tables,
4 figure
Generalizing Informed Sampling for Asymptotically Optimal Sampling-based Kinodynamic Planning via Markov Chain Monte Carlo
Asymptotically-optimal motion planners such as RRT* have been shown to
incrementally approximate the shortest path between start and goal states. Once
an initial solution is found, their performance can be dramatically improved by
restricting subsequent samples to regions of the state space that can
potentially improve the current solution. When the motion planning problem lies
in a Euclidean space, this region , called the informed set, can be
sampled directly. However, when planning with differential constraints in
non-Euclidean state spaces, no analytic solutions exists to sampling
directly.
State-of-the-art approaches to sampling in such domains such as
Hierarchical Rejection Sampling (HRS) may still be slow in high-dimensional
state space. This may cause the planning algorithm to spend most of its time
trying to produces samples in rather than explore it. In this paper,
we suggest an alternative approach to produce samples in the informed set
for a wide range of settings. Our main insight is to recast this
problem as one of sampling uniformly within the sub-level-set of an implicit
non-convex function. This recasting enables us to apply Monte Carlo sampling
methods, used very effectively in the Machine Learning and Optimization
communities, to solve our problem. We show for a wide range of scenarios that
using our sampler can accelerate the convergence rate to high-quality solutions
in high-dimensional problems
Higher order scrambled digital nets achieve the optimal rate of the root mean square error for smooth integrands
We study a random sampling technique to approximate integrals
by averaging the function
at some sampling points. We focus on cases where the integrand is smooth, which
is a problem which occurs in statistics. The convergence rate of the
approximation error depends on the smoothness of the function and the
sampling technique. For instance, Monte Carlo (MC) sampling yields a
convergence of the root mean square error (RMSE) of order (where
is the number of samples) for functions with finite variance. Randomized
QMC (RQMC), a combination of MC and quasi-Monte Carlo (QMC), achieves a RMSE of
order under the stronger assumption that the integrand
has bounded variation. A combination of RQMC with local antithetic sampling
achieves a convergence of the RMSE of order (where
is the dimension) for functions with mixed partial derivatives up to
order two. Additional smoothness of the integrand does not improve the rate of
convergence of these algorithms in general. On the other hand, it is known that
without additional smoothness of the integrand it is not possible to improve
the convergence rate. This paper introduces a new RQMC algorithm, for which we
prove that it achieves a convergence of the root mean square error (RMSE) of
order provided the integrand satisfies the strong
assumption that it has square integrable partial mixed derivatives up to order
in each variable. Known lower bounds on the RMSE show that this rate
of convergence cannot be improved in general for integrands with this
smoothness. We provide numerical examples for which the RMSE converges
approximately with order and , in accordance with the
theoretical upper bound.Comment: Published in at http://dx.doi.org/10.1214/11-AOS880 the Annals of
Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical
Statistics (http://www.imstat.org
The Devil is in the Tails: Fine-grained Classification in the Wild
The world is long-tailed. What does this mean for computer vision and visual
recognition? The main two implications are (1) the number of categories we need
to consider in applications can be very large, and (2) the number of training
examples for most categories can be very small. Current visual recognition
algorithms have achieved excellent classification accuracy. However, they
require many training examples to reach peak performance, which suggests that
long-tailed distributions will not be dealt with well. We analyze this question
in the context of eBird, a large fine-grained classification dataset, and a
state-of-the-art deep network classification algorithm. We find that (a) peak
classification performance on well-represented categories is excellent, (b)
given enough data, classification performance suffers only minimally from an
increase in the number of classes, (c) classification performance decays
precipitously as the number of training examples decreases, (d) surprisingly,
transfer learning is virtually absent in current methods. Our findings suggest
that our community should come to grips with the question of long tails
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