Monte Carlo Tuning#

Where can I check the acceptance ratio?#

For exchange and pamc, the acceptance ratio can be checked as follows:

Standard output during execution:

pamc displays the acceptance ratio in the standard output at each temperature step. exchange does not print it to the standard output; check fx.txt below instead.

# beta  mean[f]  Err[f]  nreplica  log(Z/Z0)  acceptance_ratio

Output file ``fx.txt``:

Column 6 of output/fx.txt contains the acceptance ratio.

# $1: beta (= 1/T)
# $2: mean of f(x)
# $3: std err of f(x)
# $4: number of replicas
# $5: log(Z/Z0)
# $6: acceptance ratio

What is an appropriate acceptance ratio?#

General guidelines:

  • Too high (> 0.8): Step size is too small. Exploration of parameter space is slow.

  • Too low (< 0.1): Step size is too large. Most proposals are rejected and the search stagnates.

  • Appropriate range: Around 0.2-0.5 is generally efficient.

The acceptance ratio varies with temperature (inverse temperature \(\beta\)). It is normal for it to be higher at high temperatures (low \(\beta\)) and lower at low temperatures (high \(\beta\)).

What should I do if the acceptance ratio is too low?#

Consider the following remedies:

  1. Reduce the step size

    Adjust the step size in the [algorithm.param] section using step_list.

    [algorithm.param]
step_list = [0.1, 0.1, 0.1]

    Smaller values increase the acceptance ratio. Set an appropriate scale relative to the search range of each parameter.

  2. Increase the number of temperature points

    For exchange, if the temperature difference between replicas is too large, exchanges become unlikely. Increase the number of temperature points to reduce the temperature interval.

    For pamc, increasing Tnum (number of temperature points) similarly makes the temperature change at each step more gradual.

  3. Increase the number of MCMC steps

    If equilibration at each temperature is insufficient, increase the number of MCMC steps per temperature.

What should I do if the acceptance ratio is too high?#

Increase the step_list values to make larger moves per step. This commonly occurs when the step size is too small relative to the search range (max_list - min_list).

Monte Carlo results differ each run#

Monte Carlo methods are stochastic, so results differ with different random seeds. To reproduce results, fix the seed in the [algorithm] section.

[algorithm]
seed = 12345

With sufficient steps, statistically equivalent results will be obtained.

If the optimization does not converge or the results are not what you expect, see Troubleshooting.