190 Beam-based Correction and Optimization for Accelerators
Some traditional optimization algorithms employ simple modeling of the
sample data. For example, the derivatives or parabolic fitting can be considered local models around one point. But these algorithms do not attempt to
build models in an extended area of the parameters space. There are algorithms that take all or a significant subset of previously evaluated data points
into account in order to build a model that represents the objective function,
which is in turn used to guide the search for the optimum. These algorithms
are considered machine learning methods as they actually learn about the
objective function through probing the parameter space.
Gaussian Process (GP) optimizer: Gaussian Process optimization is
a type of Bayesian optimization [72, 129, 87, 69, 98]. Bayesian optimization
combines prior assumptions and the observed data to establish a posterior
model of the objective function, based on Bayes’ theorem of conditional probabilities. Specifically, given the prior probability distribution of the objective,
P (f ), and the likelihood of measuring the sample data under function f , the
posterior function can be obtained with
P (f |D t ) = P (D t |f )P (f ),
(7.24)
where P (a|b) denotes the conditional probability of a under condition b, D t =
{(x i , f (x i )|i ∈ (1, 2, · · · t)}, and t is the number of data points. P (f |D t ) is a
surrogate function that approximates the actual objective, which can be used
to help find the optimum, for example, by predicting the next trial solution
that has the best chance to minimize the objective.
In a GP optimizer the prior and posterior distributions of the objective
function are both Gaussian processes. A Gaussian process is a multi-variate
normal distribution of variables distributed over space or time. The function
values over all points in the parameter space is a Gaussian process, which is
characterized by the mean value function, m(x), and the covariance function
between any pair of points, k(x, x
). The prior mean function is usually assumed m(x) = 0. The covariance, known as the kernel function, is commonly
chosen to be the squared exponential function,
k(x, x
) = Σ
2
f exp
−
1
2
(x − x
)
T Θ
−2 (x − x
)
,
(7.25)
where Σ
2
f is the prior variance of f over the parameter space, Θ =
diag(θ 1 , θ 2 , · · · , θ n ), and θ i is a parameter that characterizes the correlation
length of the objective function over parameter x i . A large θ i indicates that
the function values at two points separated with a large distance in the x i
parameter are still highly correlated.
After the data sample D t is obtained, we would like to know the posterior
distribution of function value f t+1 = f (x t+1 ) at an arbitrary trial solution
x t+1 . This can be derived from the joint distribution of f t+1 and the function
values on the previous data points, f t+1 = (f t , f t+1 ), with f t = (f 1 , f 2 , · · · , f t ).
Some traditional optimization algorithms employ simple modeling of the
sample data. For example, the derivatives or parabolic fitting can be considered local models around one point. But these algorithms do not attempt to
build models in an extended area of the parameters space. There are algorithms that take all or a significant subset of previously evaluated data points
into account in order to build a model that represents the objective function,
which is in turn used to guide the search for the optimum. These algorithms
are considered machine learning methods as they actually learn about the
objective function through probing the parameter space.
Gaussian Process (GP) optimizer: Gaussian Process optimization is
a type of Bayesian optimization [72, 129, 87, 69, 98]. Bayesian optimization
combines prior assumptions and the observed data to establish a posterior
model of the objective function, based on Bayes’ theorem of conditional probabilities. Specifically, given the prior probability distribution of the objective,
P (f ), and the likelihood of measuring the sample data under function f , the
posterior function can be obtained with
P (f |D t ) = P (D t |f )P (f ),
(7.24)
where P (a|b) denotes the conditional probability of a under condition b, D t =
{(x i , f (x i )|i ∈ (1, 2, · · · t)}, and t is the number of data points. P (f |D t ) is a
surrogate function that approximates the actual objective, which can be used
to help find the optimum, for example, by predicting the next trial solution
that has the best chance to minimize the objective.
In a GP optimizer the prior and posterior distributions of the objective
function are both Gaussian processes. A Gaussian process is a multi-variate
normal distribution of variables distributed over space or time. The function
values over all points in the parameter space is a Gaussian process, which is
characterized by the mean value function, m(x), and the covariance function
between any pair of points, k(x, x
). The prior mean function is usually assumed m(x) = 0. The covariance, known as the kernel function, is commonly
chosen to be the squared exponential function,
k(x, x
) = Σ
2
f exp
−
1
2
(x − x
)
T Θ
−2 (x − x
)
,
(7.25)
where Σ
2
f is the prior variance of f over the parameter space, Θ =
diag(θ 1 , θ 2 , · · · , θ n ), and θ i is a parameter that characterizes the correlation
length of the objective function over parameter x i . A large θ i indicates that
the function values at two points separated with a large distance in the x i
parameter are still highly correlated.
After the data sample D t is obtained, we would like to know the posterior
distribution of function value f t+1 = f (x t+1 ) at an arbitrary trial solution
x t+1 . This can be derived from the joint distribution of f t+1 and the function
values on the previous data points, f t+1 = (f t , f t+1 ), with f t = (f 1 , f 2 , · · · , f t ).