基于高斯过程的贝叶斯优化python代码

以下是基于高斯过程的贝叶斯优化的Python代码示例：

import numpy as np
from scipy.stats import norm
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import ConstantKernel, Matern

def acquisition_function(x, X_sample, Y_sample, gpr, xi=0.01):
    mu, sigma = gpr.predict(x.reshape(1, -1), return_std=True)
    mu_sample = Y_sample.max()
    with np.errstate(divide='warn'):
        imp = mu - mu_sample - xi
        Z = imp / sigma
        ei = imp * norm.cdf(Z) + sigma * norm.pdf(Z)
        ei[sigma == 0.0] = 0.0
    return -ei[0]

def bayesian_optimization(n_iters, sample_loss, bounds):
    X = []
    Y = []
    for i in range(n_iters):
        if i == 0:
            X_sample = np.array([[np.random.uniform(x[0], x[1])] for x in bounds])
        else:
            X_sample = np.array(X)
        Y_sample = np.array(Y)
        kernel = ConstantKernel(1.0) * Matern(length_scale=1.0)
        gpr = GaussianProcessRegressor(kernel=kernel, alpha=0.01, n_restarts_optimizer=25, normalize_y=True)
        gpr.fit(X_sample, Y_sample)
        x_next = None
        y_next = None
        max_acq = -1
        n_points = 10000
        X_points = np.zeros((n_points, len(bounds)))
        for j, (a, b) in enumerate(bounds):
            X_points[:, j] = np.random.uniform(a, b, n_points)
        for x in X_points:
            acq = acquisition_function(x, X_sample, Y_sample, gpr)
            if acq > max_acq:
                x_next = x
                max_acq = acq
        y_next = sample_loss(x_next)
        X.append(x_next)
        Y.append(y_next)
    return X, Y

其中，`acquisition_function`函数计算采样点的期望改进值（expected improvement），用于优化过程中选择下一个采样点。`bayesian_optimization`函数实现了贝叶斯优化的主要逻辑，包括随机初始化、高斯过程回归、期望改进值计算和下一个采样点的选择。最终返回优化得到的采样点和相应的函数值。需要注意的是，这里的损失函数应该是一个黑盒函数，无法直接求导，因此使用高斯过程回归来近似估计函数的值和方差，进而计算期望改进值。