结合sklearn的例子理解神经网络的基本概念

先看代码（sklearn的示例代码）：

from sklearn.neural_network import MLPClassifier
    X = [[0., 0.], [1., 1.]]
    y = [0, 1]

    clf = MLPClassifier(solver='lbfgs', alpha=1e-5,
                    hidden_layer_sizes=(5, 2), random_state=1)

    clf.fit(X, y)
    print 'predict\t',clf.predict([[2., 2.], [-1., -2.]])
    print 'predict\t',clf.predict_proba([[2., 2.], [1., 2.]])
    print 'clf.coefs_ contains the weight matrices that constitute the model parameters:\t',[coef.shape for coef in clf.coefs_]
    print clf
    c=0
    for i in clf.coefs_:
        c+=1
        print c,len(i),i

说明：

   MLPclassifier，MLP 多层感知器的的缩写（Multi-layer Perceptron）

   fit(X,y) 与正常特征的输入输出相同

solver='lbfgs',  MLP的求解方法：L-BFGS 在小数据上表现较好，Adam 较为鲁棒，SGD在参数调整较优时会有最佳表现（分类效果与迭代次数）；
         SGD标识随机梯度下降。疑问：SGD与反向传播算法的关系
alpha:L2的参数：MLP是可以支持正则化的，默认为L2，具体参数需要调整
hidden_layer_sizes=(5, 2) hidden层2层,第一层5个神经元，第二层2个神经元)
      
计算的时间复杂度（非常高。。。。）：
Suppose there are n training samples, m features, k hidden layers, each containing h neurons - for simplicity, and o output neurons. The time complexity of backpropagation is O(n\cdot m \cdot h^k \cdot o \cdot i), where i is the number of iterations. Since backpropagation has a high time complexity, it is advisable to start with smaller number of hidden neurons and few hidden layers for training.
 涉及到的设置：隐藏层数量k，每层神经元数量h，迭代次数i。


整体计算流程：
输入：the input layer, consists of a set of neurons \{x_i | x_1, x_2, ..., x_m\} representing the input features 
各个层间的计算： Each neuron in the hidden layer transforms the values from the previous layer with a weighted linear summation w_1x_1 + w_2x_2 + ... + w_mx_m, followed by a non-linear activation function g(\cdot):R \rightarrow R - like the hyperbolic tan function.   （疑问： 如果a non-linear activation function 是logit function 那么每个节点就是逻辑回归？，那么整个神经网络是变为多层的逻辑回归了么？）
输出：The output layer receives the values from the last hidden layer and transforms them into output values