Search by population annealing#

This tutorial describes how to solve the optimization problem of the Himmelblau function by using the population annealing Monte Carlo method (PAMC).

Sample files#

Sample files are available from sample/analytical/pamc. This directory includes the following files:

input.toml

The input file of odatse
plot_result_2d.py, plot_result_2dmap.py

Programs to visualize the results
do.sh

Script for running this tutorial

Input files#

This subsection describes the input file. For details, see the “Input file” chapter and Population Annealing Monte Carlo pamc.

[base]
dimension = 2
output_dir = "output"

[solver]
name = "analytical"
function_name = "himmelblau"

[runner]
[runner.log]
interval = 20

[algorithm]
name = "pamc"
seed = 12345

[algorithm.param]
max_list = [6.0, 6.0]
min_list = [-6.0, -6.0]
step_list = [0.1, 0.1]

[algorithm.pamc]
Tmin = 1.0
Tmax = 100.0
Tnum = 21
Tlogspace = true
numsteps_annealing = 100
nreplica_per_proc = 100

In the following, we will briefly describe this input file. For details, see the manual of Population Annealing Monte Carlo pamc.

The contents of [base], [solver], and [runner] sections are the same as those for the search by the Nelder-Mead method (minsearch).

[algorithm] section specifies the algorithm to use and its settings.

name is the name of the algorithm you want to use. In this tutorial we will use the population annealing Monte Carlo (PAMC) algorithm, so specify pamc.
seed is the seed that a pseudo-random number generator uses.

[algorithm.param] section sets the parameter space to be explored.

min_list is a lower bound and max_list is an upper bound.
step_list is the step length of one Monte Carlo update (standard deviation of the Gaussian distribution).

[algorithm.pamc] section sets the parameters for PAMC.

numsteps_annealing is the number of Monte Carlo steps between temperature decrements.
Tmin, Tmax are the minimum and the maximum of temperature, respectively. (Inverse temperature can be set instead with bmin/bmax.)
Tnum is the number of temperature points.
When Tlogspace is true, the temperature points are distributed uniformly in the logarithmic space.
nreplica_per_proc is the number of replicas (MC walkers) in one MPI process.

Calculation#

First, move to the folder where the sample file is located. (Hereinafter, it is assumed that you are in the root directory of ODAT-SE.)

$ cd sample/analytical/pamc

Then, run the main program. It will take a few seconds on a normal PC.

$ mpiexec -np 4 odatse input.toml | tee log.txt

Here, the calculation is performed using MPI parallel with 4 processes. If you are using Open MPI and you request more processes than the number of cores, add the --oversubscribe option to the mpiexec command.

When executed, a folder for each MPI rank will be created under the directory output, and trial_TXXX.txt files containing the parameters evaluated in each Monte Carlo step and the value of the objective function at each temperature (XXX is the index of the temperature points), and result_TXXX.txt files containing the parameters actually adopted will be created. These files are concatenated into result.txt and trial.txt.

These files have the same format: the first two columns are time (step) and the index of walker in the process, the third is the (inverse) temperature, the fourth column is the value of the objective function, and the fifth and subsequent columns are the parameters. The final two columns are the weight of the walker (Neal-Jarzynski weight) and the index of the grand ancestor (the replica index at the beginning of the calculation).

# step walker T fx x1 x2 weight ancestor
0 100.0 187.94429125133564 5.155393113805774 -2.203493345018569 1.0 0
1 100.0 3.179380982615041 -3.7929742598748666 -3.5452766573635235 1.0 1
2 100.0 108.25464277273859 0.8127003489802398 1.1465364357510186 1.0 2
3 100.0 483.84183395038843 5.57417423682746 1.8381251624588506 1.0 3
4 100.0 0.43633134370869153 2.9868796504069426 1.8428384502208246 1.0 4
5 100.0 719.7992581349758 2.972577711255287 5.535680832873856 1.0 5
6 100.0 452.4691017123836 -5.899340424701358 -4.722667479627368 1.0 6
7 100.0 45.5355817998709 -2.4155554347674215 1.8769341969872393 1.0 7
8 100.0 330.7972369561986 3.717750630491217 4.466110964691396 1.0 8
9 100.0 552.0479484091458 5.575771168463163 2.684224163039442 1.0 9
10 100.0 32.20027165958588 1.7097039347500953 2.609443449748964 1.0 10
...

output/best_result.txt is filled with information about the parameter with the optimal objective function, the rank from which it was obtained, and the Monte Carlo step.

nprocs = 4
rank = 3
step = 1806
walker = 74
fx = 4.748689609377718e-06
x1 = -2.805353724219707
x2 = 3.131045687296453

Finally, output/fx.txt stores the statistics at each temperature point:

# $1: 1/T
# $2: mean of f(x)
# $3: standard error of f(x)
# $4: number of replicas
# $5: log(Z/Z0)
# $6: acceptance ratio
0.01 130.39908953806298 6.023477428315198 400 0.0 0.9378
0.01258925411794167 83.6274790817115 3.83620542622489 400 -0.2971072297035158 0.930325
0.015848931924611134 60.25390522675298 2.73578884504734 400 -0.5426399088244793 0.940375
0.01995262314968879 47.20146188151557 2.3479083531465976 400 -0.7680892360649545 0.93715
0.025118864315095794 41.118822390166315 1.8214854089575818 400 -0.9862114670289625 0.9153
...

The first column is the inverse temperature, and the second/third ones are the mean and standard error of \(f(x)\), respectively. The fourth column is the number of replicas and the fifth one is the logarithm of the ratio of the partition functions, \(\log(Z_n/Z_0)\), where \(Z_0\) is the partition function at the first temperature. The sixth column is the acceptance ratio of MC updates.

Visualization#

By visualizing result_T%.txt, you can estimate regions where the function values become small. Running plot_result_2dmap.py reads samples from all temperature points and generates result_T.png (colored by temperature T), result_fx.png (colored by the objective function value f(x)), and result.png (which places both plots side by side).

$ python3 plot_result_2dmap.py

plot_result_2d.py instead creates one plot for the result_T%.txt file given as an argument, with samples colored by f(x). Its default output file is res.png.

It is seen from the figures that the samples are concentrated near the minima of f(x) where the objective function has a small value.