How vegas
Works¶
vegas
uses two adaptive strategies: importance sampling, and
adaptive stratified sampling. Here we discuss the ideas behind each,
in turn.
Importance Sampling¶
The most important adaptive strategy vegas
uses is
its remapping of the integration variables in each
direction, before it makes Monte Carlo estimates of the integral.
This is equivalent to a standard Monte Carlo optimization
called “importance sampling.”
vegas
chooses transformations
for each integration variable
that minimize the statistical errors in
Monte Carlo estimates whose integrand
samples are uniformly distributed
in the new variables.
The idea in one-dimension, for
example, is to replace the original integral over ,
by an equivalent integral over a new variable ,
where is the Jacobian of the transformation.
A simple Monte Carlo estimate of the transformed
integral is given by
where the sum is over random points
uniformly distributed between 0 and 1.
The estimate is a itself a random number from a distribution
whose mean is the exact integral and whose variance is:
The standard deviation is an estimate of the possible
error in the Monte Carlo estimate.
A straightforward variational calculation, constrained by
shows that is minimized if
Such transformations greatly reduce the standard deviation when the integrand has high peaks. Since
the regions in space where
is large are
stretched out in
space. Consequently, a uniform Monte Carlo in
space
places more samples in the peak regions than it would
if were we integrating in
space — its samples are concentrated
in the most important regions, which is why this is called “importance
sampling.” The product
has no peaks when
the transformation is optimal.
The distribution of the Monte Carlo estimates becomes
Gaussian in the limit of large
. Non-Gaussian corrections
vanish like
. For example, it is easy to show that
This moment would equal , which falls like
,
if the distribution was Gaussian. The corrections to the Gaussian result
fall as
and so become negligible at large
.
These results assume
that
is integrable for all
,
which need not be the case
if
has (integrable) singularities.
The vegas
Map¶
vegas
implements the transformation of an integration variable
into a new variable
using a grid in
space:
The grid specifies the transformation function at the points
for
:
Linear interpolation is used between those points. The Jacobian for this transformation function is piecewise constant:
for .
The variance for a Monte Carlo estimate using this transformation becomes
Treating the as independent variables, with the
constraint
it is trivial to show that the standard deviation is minimized when
for all .
vegas
adjusts the grid until this last condition is
satisfied. As a result grid increments are
small in regions where
is large.
vegas
typically has no knowledge of the integrand initially, and
so starts with a uniform grid. As it samples the integrand
it also estimates the integrals
and use this information to refine
its choice of s, bringing them closer to their optimal
values, for use
in subsequent iterations. The grid usually converges,
after several iterations,
to the optimal grid.
This analysis generalizes easily to multi-dimensional integrals.
vegas
applies a similar transformation in each direction, and
the grid increments along an axis
are made smaller in regions where the
projection of the integral onto that axis is larger. For example,
the optimal grid for the four-dimensional Gaussian integral
in the section on Basic Integrals looks like:


These grids transform into uniformly-spaced grids in space.
Consequently a uniform,
-space Monte Carlo places the same
number of integrand evaluations, on average, in every rectangle
of these pictures. (The average number is typically much less one
in higher dimensions.) Integrand evaluations are concentrated
in regions where the
-space rectangles are small
(and therefore numerous) —
here in the vicinity of
x = [0.5, 0.5, 0.5, 0.5]
, where the
peak is.
These plots were obtained by including the line
integ.map.show_grid(30)
in the integration code after the integration is finished.
It causes matplotlib
(if it is installed) to create
images showing the locations of 30 nodes
of
the grid in each direction. (The grid uses 99 nodes in all
on each axis, but that is too many to display at low resolution.)
Adaptive Stratified Sampling¶
A limitation of vegas
’s remapping strategy becomes obvious if we look
at the grid for the following integral, which has two Gaussians
arranged along the diagonal of the hypercube:
import vegas
import math
def f2(x):
dx2 = 0
for d in range(4):
dx2 += (x[d] - 1/3.) ** 2
ans = math.exp(-dx2 * 100.) * 1013.2167575422921535
dx2 = 0
for d in range(4):
dx2 += (x[d] - 2/3.) ** 2
ans += math.exp(-dx2 * 100.) * 1013.2167575422921535
return ans / 2.
integ = vegas.Integrator(4 * [[0, 1]])
integ(f2, nitn=10, neval=4e4)
result = integ(f2, nitn=30, neval=4e4)
print('result = %s Q = %.2f' % (result, result.Q))
integ.map.show_grid(70)
This code gives the following grid, now showing 70 nodes in each direction:

The grid shows that vegas
is concentrating on the regions
around x=[0.33, 0.33, 0.33, 0.33]
and
x=[0.67, 0.67, 0.67, 0.67]
, where the peaks are.
Unfortunately it is also concentrating on regions around
points like x=[0.67, 0.33, 0.33, 0.33]
where the integrand
is very close to zero. There are 14 such phantom peaks
that vegas
’s new integration variables emphasize,
in addition to the 2 regions
where the integrand actually is large. This grid gives
much better results
than using a uniform grid, but it obviously
wastes integration resources.
The waste occurs because
vegas
remaps the integration variables in
each direction separately. Projected on the x[0]
axis, for example,
this integrand appears to have two peaks and so vegas
will
focus on both regions of x[0]
, independently of what it does
along the x[1]
axis.
vegas
uses axis-oriented remappings because other
alternatives are much more complicated and expensive; and vegas
’s
principal adaptive strategy has proven very effective in
many realistic applications.
An axis-oriented
strategy will always have difficulty adapting to structures that
lie along diagonals of the integration volume. To address such problems,
the new version of vegas
introduces a second adaptive strategy,
based upon another standard Monte Carlo technique called “stratified
sampling.” vegas
divides the -dimensional
-space volume into
hypercubes using
a uniform
-space grid with
or
stratifications on each
axis. It estimates
the integral by doing a separate Monte Carlo integration in each of
the hypercubes, and adding the results together to provide an estimate
for the integral over the entire integration region.
Typically
this
-space grid is much coarser than the
-space grid used to
remap the integration variables. This is because
vegas
needs
at least two integrand evaluations in each -space hypercube, and
so must keep the number of hypercubes
smaller than
neval/2
.
This can restrict severely when
is large.
Older versions of vegas
also divide -space into hypercubes and
do Monte Carlo estimates in the separate hypercubes. These versions, however,
use the same number of integrand evaluations in each hypercube.
In the new version,
vegas
adjusts the number of evaluations used
in a hypercube in proportion to the standard deviation of
the integrand estimates (in space) from that hypercube.
It uses information about the hypercube’s standard deviation in one
iteration to set the number of evaluations for that hypercube
in the next iteration. In this way it concentrates
integrand evaluations where the potential statistical errors are
largest.
In the two-Gaussian example above, for example,
the new vegas
shifts
integration evaluations away from the phantom peaks, into
the regions occupied by the real peaks since this is where all
the error comes from. This improves vegas
’s ability to estimate
the contributions from the real peaks and
reduces statistical errors,
provided neval
is large enough to permit a large number (more
than 2 or 3) of
stratifications on each axis. With
neval=4e4
,
statistical errors for the two-Gaussian
integral are reduced by more than a factor of 3 relative to what older
versions of vegas
give. This is a relatively easy integral;
the difference can be much larger for more difficult (and realistic)
integrals.