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Given the following text description, write Python code to implement the functionality described below step by step
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Description:
 The structure of GPflowOpt
Joachim van der Herten
In this document, the structure of the GPflowOpt library is explained, including some small examples. First the Domain and Optimizer interfaces are shortly illustrated, followed by a description of the BayesianOptimizer. At the end, a step-by-step walkthrough of the BayesianOptimizer is given.
Optimization
The underlying design principles of GPflowOpt were chosen to address the following task
Step1:  Domain
Then, we represent $\mathcal{X}$ by composing parameters. This is how a simple continuous square domain is defined
Step2:  Optimize
Based on the domain and a valid objective function, we can now easily apply one of the included optimizers to optimize objective functions. GPflowOpt defines an intuitive Optimizer interface which can be used to specify the domain, the initial point(s), constraints (to be implemented) etc. Some popular optimization approaches are provided.
Here is how our function is optimized using one of the available methods of SciPy's minimize
Step3:  And here is how we optimize it Monte-Carlo. We can pass the same function as the gradients are automatically discarded.
Step4:  Bayesian Optimization
In Bayesian Optimization (BO), the typical assumption is that $f$ is expensive to evaluate and no gradients are available. The typical approach is to sequentially select a limited set of decisions $\boldsymbol{x}0, \boldsymbol{x}_1, ... \boldsymbol{x}{n-1}$ using a sampling policy. Hence each decision $\boldsymbol{x}_i \in \mathcal{X}$ itself is the result of an optimization problem
$$\boldsymbol{x}_i = \underset{\boldsymbol{x}}{\operatorname{argmax}} \alpha_i(\boldsymbol{x})$$
Each iteration, a function $\alpha_i$ which is cheap-to-evaluate acts as a surrogate for the expensive function. It is typically a mapping of the predictive distribution of a (Bayesian) model built on all decisions and their corresponding (noisy) evaluations. The mapping introduces an order in $\mathcal{X}$ to obtain a certain goal. The typical goal within the context of BO is the search for optimality or feasibility while keeping the amount of required evaluations ($n$) a small number. As we can have several functions $f$ representing objectives and constraints, BO may invoke several models and mappings $\alpha$. These mappings are typically referred to as acquisition functions (or infill criteria). GPflowOpt defines an Acquisition interface to implement these mappings and provides implementations of some default choices. In combination with a special Optimizer implementation for BO, following steps are required for a typical workflow
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Python Code:
import numpy as np
def fx(X):
    X = np.atleast_2d(X)
    # Return objective & gradient
    return np.sum(np.square(X), axis=1)[:,None], 2*X
Explanation: The structure of GPflowOpt
Joachim van der Herten
In this document, the structure of the GPflowOpt library is explained, including some small examples. First the Domain and Optimizer interfaces are shortly illustrated, followed by a description of the BayesianOptimizer. At the end, a step-by-step walkthrough of the BayesianOptimizer is given.
Optimization
The underlying design principles of GPflowOpt were chosen to address the following task: optimizing problems of the form
$$\underset{\boldsymbol{x} \in \mathcal{X}}{\operatorname{argmin}} f(\boldsymbol{x}).$$ The objective function $f: \mathcal{X} \rightarrow \mathbb{R}^p$ maps a candidate optimum to a score (or multiple). Here $\mathcal{X}$ represents the input domain. This domain encloses all candidate solutions to the optimization problem and can be entirely continuous (i.e., a $d$-dimensional hypercube) but may also consist of discrete and categorical parameters. 
In GPflowOpt, the Domain and Optimizer interfaces and corresponding subclasses are used to explicitly represent the optimization problem.
Objective function
The objective function itself is provided and must be implemented as any python callable (function or object with implemented call operator), accepting a two dimensional numpy array with shape $(n, d)$ as an input, with $n$ the batch size. It returns a tuple of numpy arrays: the first element of the tuple has shape $(n, p)$ and returns the objective scores for each point to evaluate. The second element is the gradient in every point, shaped either $(n, d)$ if we have a single-objective optimization problem, or $(n, d, p)$ in case of a multi-objective function. If the objective function is passed on to a gradient-free optimization method, the gradient is automatically discarded. GPflowOpt provides decorators which handle batch application of a function along the n points of the input matrix, or dealing with functions which accept each feature as function argument.
Here, we define a simple quadratic objective function:
End of explanation
from gpflowopt.domain import ContinuousParameter
domain = ContinuousParameter('x1', -2, 2) + ContinuousParameter('x2', -1, 2)
domain
Explanation: Domain
Then, we represent $\mathcal{X}$ by composing parameters. This is how a simple continuous square domain is defined:
End of explanation
from gpflowopt.optim import SciPyOptimizer
optimizer = SciPyOptimizer(domain, method='SLSQP')
optimizer.set_initial([-1,-1])
optimizer.optimize(fx)
Explanation: Optimize
Based on the domain and a valid objective function, we can now easily apply one of the included optimizers to optimize objective functions. GPflowOpt defines an intuitive Optimizer interface which can be used to specify the domain, the initial point(s), constraints (to be implemented) etc. Some popular optimization approaches are provided.
Here is how our function is optimized using one of the available methods of SciPy's minimize:
End of explanation
from gpflowopt.optim import MCOptimizer
optimizer = MCOptimizer(domain, 200)
optimizer.optimize(fx)
Explanation: And here is how we optimize it Monte-Carlo. We can pass the same function as the gradients are automatically discarded.
End of explanation
from gpflowopt.bo import BayesianOptimizer
from gpflowopt.design import FactorialDesign
from gpflowopt.acquisition import ExpectedImprovement
import gpflow
# The Bayesian Optimizer does not expect gradients to be returned
def fx(X):
    X = np.atleast_2d(X)
    # Return objective & gradient
    return np.sum(np.square(X), axis=1)[:,None]
    
X = FactorialDesign(2, domain).generate()
Y = fx(X)
# initializing a standard BO model, Gaussian Process Regression with
# Matern52 ARD Kernel
model = gpflow.gpr.GPR(X, Y, gpflow.kernels.Matern52(2, ARD=True))
alpha = ExpectedImprovement(model)
# Now we must specify an optimization algorithm to optimize the acquisition 
# function, each iteration. 
acqopt = SciPyOptimizer(domain)
# Now create the Bayesian Optimizer
optimizer = BayesianOptimizer(domain, alpha, optimizer=acqopt)
with optimizer.silent():
    r = optimizer.optimize(fx, n_iter=15)
print(r)
Explanation: Bayesian Optimization
In Bayesian Optimization (BO), the typical assumption is that $f$ is expensive to evaluate and no gradients are available. The typical approach is to sequentially select a limited set of decisions $\boldsymbol{x}0, \boldsymbol{x}_1, ... \boldsymbol{x}{n-1}$ using a sampling policy. Hence each decision $\boldsymbol{x}_i \in \mathcal{X}$ itself is the result of an optimization problem
$$\boldsymbol{x}_i = \underset{\boldsymbol{x}}{\operatorname{argmax}} \alpha_i(\boldsymbol{x})$$
Each iteration, a function $\alpha_i$ which is cheap-to-evaluate acts as a surrogate for the expensive function. It is typically a mapping of the predictive distribution of a (Bayesian) model built on all decisions and their corresponding (noisy) evaluations. The mapping introduces an order in $\mathcal{X}$ to obtain a certain goal. The typical goal within the context of BO is the search for optimality or feasibility while keeping the amount of required evaluations ($n$) a small number. As we can have several functions $f$ representing objectives and constraints, BO may invoke several models and mappings $\alpha$. These mappings are typically referred to as acquisition functions (or infill criteria). GPflowOpt defines an Acquisition interface to implement these mappings and provides implementations of some default choices. In combination with a special Optimizer implementation for BO, following steps are required for a typical workflow: 
1) Define the problem domain. Its dimensionality matches the input to the objective and constraint functions. (like normal optimization)
2) Specify the (GP) models for the constraints and objectives. This involves choice of kernels, priors, fixes, transforms... this step follows the standard way of setting up GPflow models. GPflowOpt does not further wrap models hence it is possible to implement custom models in GPflow and use them directly in GPflowOpt
3) Set up the acquisition function(s) using the available built-in implementations in GPflowOpt, or design your own by implementing the Acquisition interface.
4) Set up an optimizer for the acquisition function.
5) Run the high-level BayesianOptimizer which implements a typical BO flow. BayesianOptimizer in itself is compliant with the Optimizer interface. Exceptionally, the BayesianOptimizer requires that the objective function returns no gradients.
Alternatively, advanced users requiring finer control can easily implement their own flow based on the low-level interfaces of GPflowOpt, as the coupling between these objects was intentionally kept loose.
As illustration of the described flow, the previous example is optimized using Bayesian optimization instead, with the well-known Expected Improvement acquisition function:
End of explanation
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