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- Randomness and Computation
- Introduction to randomized algorithms
- Introduction to randomized algorithms
This chapter describes routines for multidimensional Monte Carlo integration. Each algorithm computes an estimate of a multidimensional definite integral of the form,. The routines also provide a statistical estimate of the error on the result.
In computing , a Las Vegas algorithm is a randomized algorithm that always gives correct results; that is, it always produces the correct result or it informs about the failure. However, the runtime of a Las Vegas algorithm differs depending on the input. The usual definition of a Las Vegas algorithm includes the restriction that the expected runtime be finite, where the expectation is carried out over the space of random information, or entropy, used in the algorithm. An alternative definition requires that a Las Vegas algorithm always terminate is effective , but may output a symbol not part of the solution space to indicate failure in finding a solution. Las Vegas algorithms are prominent in the field of artificial intelligence, and in other areas of computer science and operations research.
Randomness and Computation
Class vegas. Integrator gives Monte Carlo estimates of arbitrary multidimensional integrals using the vegas algorithm G. Lepage, J. The algorithm has two components. First an automatic transformation is applied to to the integration variables in an attempt to flatten the integrand.
Then a Monte Carlo estimate of the integral is made using the transformed variables. Flattening the integrand makes the integral easier and improves the estimate. The transformation applied to the integration variables is optimized over several iterations of the algorithm: information about the integrand that is collected during one iteration is used to improve the transformation used in the next iteration.
This makes Monte Carlo integration unusually robust. It also makes it well suited for adaptive integration. Adaptive strategies are essential for multidimensional integration, especially in high dimensions, because multidimensional space is large, with lots of corners, making it easy to lose important features in the integrand.
Monte Carlo integration also provides efficient and reliable methods for estimating the accuracy of its results. In particular, each Monte Carlo estimate of an integral is a random number from a distribution whose mean is the correct value of the integral. This distribution is Gaussian or normal provided the number of integrand samples is sufficiently large.
In practice we generate multiple estimates of the integral in order to verify that the distribution is indeed Gaussian. Error analysis is straightforward if the integral estimates are Gaussian. The algorithm used here is significantly improved over the original implementation, and that used in most other implementations. It uses two adaptive strategies: importance sampling, as in the original implementation, and adaptive stratified sampling, which is new.
The new algorithm is described in G. This module is written in Cython, so it is almost as fast as compiled Fortran or C, particularly when the integrand is also coded in Cython or some other compiled language , as discussed below. The following sections describe how to use vegas. Almost every example shown is a complete code, which can be copied into a file and run with Python.
It is worthwhile playing with the parameters to see how things change. About Printing: The examples in this tutorial use the print function as it is used in Python 3. Drop the outermost parenthesis in each print statement if using Python 2, or add. Here we illustrate the use of vegas by estimating the integral. The following code shows how this can be done:. First we define the integrand f x where x[d] specifies a point in the 4-dimensional space.
We then create an integrator, integ , which is an integration operator that can be applied to any 4-dimensional function. It is where we specify the integration volume. Each iteration produces an independent estimate of the integral. The final estimate is the weighted average of the results from all 10 iterations, and is returned by integ f The call result. Adaptation: Integration estimates are shown for each of the 10 iterations, giving both the estimate from just that iteration, and the weighted average of results from all iterations up to that point.
It uses information from the samples in one iteration, however, to remap the integration variables for subsequent iterations, concentrating samples where the function is largest and reducing errors.
As a result, the per-iteration error is reduced to 4. Eventually the per-iteration error stops decreasing because vegas has found the optimal remapping, at which point it has fully adapted to the integrand.
Weighted Average: The final result, 1. The individual estimates are statistical: each is a random number drawn from a distribution whose mean equals the correct value of the integral, and the errors quoted are estimates of the standard deviations of those distributions. The distributions are Gaussian provided the number of integrand evaluations per iteration neval is sufficiently large, in which case the standard deviation is a reliable estimate of the error. The weighted average minimizes.
If the are Gaussian, should be of order the number of degrees of freedom plus or minus the square root of double that number ; here the number of degrees of freedom is the number of iterations minus 1. The distributions are likely non-Gaussian, and error estimates unreliable, if is much larger than the number of iterations. This criterion is quantified by the Q or p-value of the , which is the probability that a larger could result from random Gaussian fluctuations.
A very small Q less than 0. This means that neval is not sufficiently large to guarantee Gaussian behavior, and must be increased if the error estimates are to be trusted.
RAvg , that has the following attributes:. In this example the final Q is 0. Precision: The precision of vegas estimates is determined by nitn , the number of iterations of the vegas algorithm, and by neval , the maximum number of integrand evaluations made per iteration. The computing cost is typically proportional to the product of nitn and neval. The number of integrand evaluations per iteration varies from iteration to iteration, here between and Typically vegas needs more integration points in early iterations, before it has fully adapted to the integrand.
We can increase precision by increasing either nitn or neval , but it is generally far better to increase neval. For example, adding the following lines to the code above. Typically you want to use no more than 10 or 20 iterations beyond the point where vegas has fully adapted.
You want some number of iterations so that you can verify Gaussian behavior by checking the and Q , but not too many. It is also generally useful to compare two or more results from values of neval that differ by a significant factor 4—10, say. These should agree within errors. If they do not, it could be due to non-Gaussian artifacts caused by a small neval. One is the statistical error, which is what is quoted by vegas.
The other is a systematic error due to residual non-Gaussian effects. The systematic error can bias the Monte Carlo estimate, however, if neval is insufficiently large. This usually results in a large and small Q , but a more reliable check is to compare results that use signficantly different values of neval.
The systematic errors due to non-Gaussian behavior are likely negligible if the different estimates agree to within the statistical errors. The possibility of systematic biases is another reason for increasing neval rather than nitn to obtain more precision.
Making nitn larger and larger, on the other hand, is guaranteed eventually to give the wrong answer. Early Iterations: Integral estimates from early iterations, before vegas has adapted, can be quite crude. With very peaky integrands, these are often far from the correct answer with highly unreliable error estimates.
For example, the integral above becomes more difficult if we double the length of each side of the integration volume by redefining integ as:. It is common practice in using vegas to discard estimates from the first several iterations, before the algorithm has adapted, in order to avoid ruining the final result in this way.
This is done by replacing the single call to integ f The integrator is trained in the first step, as it adapts to the integrand, and so is more or less fully adapted from the start in the second step, which yields:. Other Integrands: Once integ has been trained on f x , it can be usefully applied to other functions with similar structure. For example, adding the following at the end of the original code,. Again the grid is almost optimal for g x from the start, because g x peaks in the same region as f x.
The exact value for this integral is very close to 0. Note that vegas. The is useful for costly integrations that might need to be reanalyzed later since the integrator remembers the variable transformations made to minimize errors, and so need not be readapted to the integrand when used later. Non-Rectangular Volumes: vegas can integrate over volumes of non-rectangular shape.
For example, we can replace integrand f x above by the same Gaussian, but restricted to a 4-sphere of radius 0. The normalization is adjusted to again make the exact integral equal 1. Integrating as before gives:. Note, finally, that integration to infinity is also possible: map the relevant variable into a different variable of finite range.
For example, an integral over from 0 to infinity is easily reexpressed as an integral over from 0 to 1, where the transformation emphasizes the region in of order free parameter. Parameter alpha controls the speed with which vegas adapts, with smaller alpha s giving slower adaptation. Here we reduce alpha to 0. Notice how the errors fluctuate less from iteration to iteration with the smaller alpha in this case.
Persistent, large fluctuations in the size of the per-iteration errors is often a signal that alpha should be reduced. With larger alpha s, vegas can over-react to random fluctuations it encounters as it samples the integrand. In general, we want alpha to be large enough so that vegas adapts quickly to the integrand, but not so large that it has difficulty holding on to the optimal tuning once it has found it.
The best value depends upon the integrand. There are three reasons one might do this. The first is if vegas is exhibiting the kind of instability discussed in the previous section — one might use the following code, instead of that presented there:. The second reason is that vegas runs slightly faster when it is no longer adapting to the integrand.
The difference is not signficant for complicated integrands, but is noticable in simpler cases. Unweighted averages are not biased.
They have no systematic error of the sort discussed above, and so give correct results even for very large numbers of iterations, nitn. The lack of systematic biases is not a strong reason for turning off adaptation, however, since the biases are usually negligible see above. The most important reason is the first: stability.
Introduction to randomized algorithms
Randomized Algorithms Set 1 Introduction and Analysis. Las Vegas: These algorithms always produce correct or optimum result. Time complexity of these algorithms is based on a random value and time complexity is evaluated as expected value. Monte Carlo: Produce correct or optimum result with some probability. These algorithms have deterministic running time and it is generally easier to find out worst case time complexity.
Introduction to randomized algorithms
One of the most remarkable developments in Computer Science over the past 30 years has been the realization that the ability of computers to toss coins can lead to algorithms that are more efficient, conceptually simpler and more elegant that their best known deterministic counterparts. Randomization has by now become a ubiquitous tool in computation. This course will survey several of the most widely used techniques in this context, illustrating them with examples taken from algorithms, random structures and combinatorics. Our goal is to provide a solid background in the key ideas used in the design and analysis of randomized algorithms and probabilistic processes.
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