196 Beam-based Correction and Optimization for Accelerators
0
50
100
150
200
250
300
evaluations
10
-4
10
-2
10
0
y = f(x)
Powell, N=300
=0.001
=0.01
=0.1
0
20
40
60
80
100
cases
10
-4
10
-2
10
0
f
min
Powell, N=300
=0.001
=0.01
=0.1
Figure 7.7 Testing of Powell’s method with function f1(x) in Eq. (7.31) and three
random noise levels. The golden section method is used as the line optimizer. Left:
typical convergence histories (for cases whose final minimum ranks the 30th); right:
minimum function values within 300 evaluations for 100 runs.
0
50
100
150
200
250
300
evaluations
10
-4
10
-2
10
0
y = f(x)
GP, N=300
=0.001
=0.01
=0.1
0
20
40
60
80
100
cases
10
-4
10
-2
10
0
f
min
GP, N=300
=0.001
=0.01
=0.1
Figure 7.8 Testing of the Gaussian Process optimizer with function f1(x) and three
random noise levels. Left: typical convergence histories; right: minimum function
values within 300 evaluations.
and 0.01), the GP optimizer can successfully find the minimum. However, with
a high noise level (σ = 0.1), it fails to converge to the minimum.
It is worth noting that the GP optimizer becomes slow as the number of
data points grows. For example, it takes 9 hrs to complete 100 runs for the
test while on the same computer it takes only a few seconds for all other
algorithms. It would be even slower if the number of function evaluations is
larger.
Summary of test results:
From the test results shown in Figures 7.5 – 7.8, it is clear that noise can
significantly impact the behaviors of the optimization algorithms. At low noise
levels, the algorithms may have similar performances as for smooth functions.
However, noise at high levels usually will disrupt the actions of the algorithms
and cause failures to converge to the minimum. The level of noise that leads
to failures depends on the working principles of the algorithms as well as
the nature of the objective functions. For example, for a function with a large,
monotonic drop between the starting point and the minimum, it would require
a high noise level to cause the algorithms to fail. If the terrain between the
0
50
100
150
200
250
300
evaluations
10
-4
10
-2
10
0
y = f(x)
Powell, N=300
=0.001
=0.01
=0.1
0
20
40
60
80
100
cases
10
-4
10
-2
10
0
f
min
Powell, N=300
=0.001
=0.01
=0.1
Figure 7.7 Testing of Powell’s method with function f1(x) in Eq. (7.31) and three
random noise levels. The golden section method is used as the line optimizer. Left:
typical convergence histories (for cases whose final minimum ranks the 30th); right:
minimum function values within 300 evaluations for 100 runs.
0
50
100
150
200
250
300
evaluations
10
-4
10
-2
10
0
y = f(x)
GP, N=300
=0.001
=0.01
=0.1
0
20
40
60
80
100
cases
10
-4
10
-2
10
0
f
min
GP, N=300
=0.001
=0.01
=0.1
Figure 7.8 Testing of the Gaussian Process optimizer with function f1(x) and three
random noise levels. Left: typical convergence histories; right: minimum function
values within 300 evaluations.
and 0.01), the GP optimizer can successfully find the minimum. However, with
a high noise level (σ = 0.1), it fails to converge to the minimum.
It is worth noting that the GP optimizer becomes slow as the number of
data points grows. For example, it takes 9 hrs to complete 100 runs for the
test while on the same computer it takes only a few seconds for all other
algorithms. It would be even slower if the number of function evaluations is
larger.
Summary of test results:
From the test results shown in Figures 7.5 – 7.8, it is clear that noise can
significantly impact the behaviors of the optimization algorithms. At low noise
levels, the algorithms may have similar performances as for smooth functions.
However, noise at high levels usually will disrupt the actions of the algorithms
and cause failures to converge to the minimum. The level of noise that leads
to failures depends on the working principles of the algorithms as well as
the nature of the objective functions. For example, for a function with a large,
monotonic drop between the starting point and the minimum, it would require
a high noise level to cause the algorithms to fail. If the terrain between the