以向量为中心的神经网络Python实现

Question

你好，我尝试使用一个以向量为中心的神经网络，一个输入节点，一个输出节点和两个隐藏层，每个隐藏层有3个节点，以适应一个非常简单的x**2函数。-只是为了验证它的功能。因此，我使用下面的代码。结果我得到了橙色的线，蓝色的线是真实的线

正如你所看到的，有些东西不起作用。我试图改变迭代次数和学习率的值，但没有成功。如果我绘制迭代过程中的损失，我会得到100次迭代的下图：

我还没有添加偏差，但我认为这个简单的函数应该适合没有额外偏差节点。此外，我假设代码中的失败最有可能出现在代码的“计算相对于权重的梯度”部分

所以原则上我有两个问题：

1. 我的代码中是否有任何基本故障导致代码无法工作
2. 如果不是，为什么我的模型不能适应easy数据

提前谢谢你的帮助

下面是代码-它已准备好播放：

class Neural_Net:
    """
    """
    def __init__(self, activation_function, learning_rate, runs):
        self.activation_function = activation_function
        self.X_train = np.linspace(0,1,1000)
        self.y_train = self.X_train**2
        plt.plot(self.X_train, self.y_train)
        self.y_pred = None
        self.W_input = np.random.randn(1, 3)
        self.Partials_W_input = np.random.randn(1, 3)
        self.W_hidden = np.random.randn(3,3)
        self.Partials_W_hidden = np.random.randn(3,3)
        self.W_output = np.random.randn(3,1)
        self.Partials_W_output = np.random.randn(3,1)
        self.Activations = np.ones((3,2))
        self.Partials = np.ones((3,2))
        self.Output_Gradient = None
        self.Loss = 0
        self.learning_rate = learning_rate
        self.runs = runs
        self.Losses = []
        self.i = 0

    def apply_activation_function(self, activation_vector):
            return 1/(1+np.exp(-activation_vector))

    def forward_pass(self, training_instance):
        for layer in range(len(self.Activations[0])):
            # For the first layer between X and the first hidden layer
            pre_activation_first = self.W_input.T @ training_instance.reshape(1,1)
                # print('pre activation: ', pre_activation)
                # Apply the activation function
                self.Activations[:,0] = self.apply_activation_function(pre_activation_first).ravel()
            else:
                pre_activation_hidden = self.W_hidden.T @ self.Activations[:, layer-1]
                self.Activations[:, layer] = self.apply_activation_function(pre_activation_hidden)
                # print('Activations: ', self.Activations)
        output = self.W_output.T @ self.Activations[:, -1]
        # print('output: ', output)
        return output

    def backpropagation(self, y_true, training_instance):
        if self.activation_function == 'linear':
            # Calculate the ouput gradient
            self.Output_Gradient = -(y_true-self.y_pred)
            # print('Output Gradient: ', self.Output_Gradient)

            # Calculate the partial gradients of the Error with respect to the pre acitvation values in the nodes
            self.Partials[:, 1] = self.Activations[:, 1]*(1-self.Activations[:, 1])*(self.W_output @ self.Output_Gradient)
            self.Partials[:, 0] = self.Activations[:, 0]*(1-self.Activations[:, 0])*(self.W_hidden @ self.Partials[:, 1])
            # print('Partials: ', self.Partials)

            # Calculate the Gradients with respect to the weights
            self.Partials_W_output = self.Output_Gradient * self.Activations[:, -1]
            # print('Partials_W_output: ', self.Partials_W_output)
            self.Partials_W_hidden = self.Partials[:, -1].reshape(3,1) * self.Activations[:, 0].reshape(1,3)
            # print('Partials_W_hidden: ',self.Partials_W_hidden)
            self.Partials_W_input = (self.Partials[:, 0].reshape(3,1) * training_instance.T).T
            # print('Partials_W_input: ', self.Partials_W_input)

    def weight_update(self, training_instance, learning_rate):

        # Output Layer weights
        w_output_old = self.W_output.copy()
        self.W_output = w_output_old - learning_rate*self.Output_Gradient

        # Hidden Layer weights
        w_hidden_old = self.W_hidden.copy()
        self.W_hidden = w_hidden_old - learning_rate * self.W_hidden
        # print('W_hidden new: ', self.W_hidden)

        # Input Layer weights
        w_input_old = self.W_input.copy()
        self.W_input = w_input_old - learning_rate * self.W_input
        # print('W_input new: ', self.W_input)


    def train_model(self):
        for _ in range(self.runs):
            for instance in range(len(self.X_train)):
                # forward pass
                self.y_pred = self.forward_pass(self.X_train[instance])

                # Calculate loss
                self.Loss = self.calc_loss(self.y_pred, self.y_train[instance])
                # print('Loss: ', self.Loss)

                # Calculate backpropagation
                self.backpropagation(self.y_train[instance], self.X_train[instance])

                # Update weights
                self.weight_update(self.X_train[instance], self.learning_rate)

        # print(self.Losses)
        # plt.plot(range(len(self.Losses)), self.Losses)
        # plt.show()

        # Make predictions on training data to check if the model is basically able to fit the training data
        predictions = []
        for i in np.linspace(0,1,1000):
            predictions.append(self.make_prediction(i))
        plt.plot(np.linspace(0,1,1000), predictions)


    def make_prediction(self, X_new):
        return self.forward_pass(X_new)


    def calc_loss(self, y_pred, y_true):
        loss = (1/2)*(y_true-y_pred)**2
        self.Losses.append(loss[0])
        return (1/2)*(y_true-y_pred)**2

    def accuracy(self):
        pass


Neural_Net('linear', 0.0001, 10).train_model()

Answer 1

我已经解决了这个问题： 首先，我混淆了一些我已经纠正过的维度。尽管如此，真正的问题是重量更新，我尝试用例如

self.W_hidden = w_hidden_old - learning_rate * self.W_hidden

但这是错误的，因为学习率必须乘以关于权重的部分误差，而不是权重矩阵本身。因此，正确的方法是：

self.W_hidden = w_hidden_old - learning_rate * self.Partials_W_hidden

之后，我收到以下结果和损失曲线：

最后的代码是：

class Neural_Net:
    """
    """
    def __init__(self, activation_function, learning_rate, runs):
        self.activation_function = activation_function

        self.Data = pd.read_csv(r"U:\19_035_Machine_Learning_Workshop\2_Workshopinhalt\Weitere\Neural Networks\AirQualityUCI\AirQualityUCI.csv", sep=';', decimal=b',').iloc[:, :-2].dropna()
        self.X_train = np.linspace(0,5,1000)
        self.y_train = np.sin(self.X_train)
        plt.plot(self.X_train, self.y_train)
        self.y_pred = None
        self.W_input = np.random.randn(1, 3)
        self.Partials_W_input = np.random.randn(1, 3)
        self.W_hidden = np.random.randn(3,3)
        self.Partials_W_hidden = np.random.randn(3,3)
        self.W_output = np.random.randn(3,1)
        self.Partials_W_output = np.random.randn(3,1)
        self.Activations = np.zeros((3,2))
        self.Partials = np.zeros((3,2))
        self.Output_Gradient = None
        self.Loss = 0
        self.learning_rate = learning_rate
        self.runs = runs
        self.Losses = []
        self.i = 0

    def apply_activation_function(self, activation_vector):
        # print('activation: ', 1/(1+np.exp(-activation_vector)))
        return 1/(1+np.exp(-activation_vector))

    def forward_pass(self, training_instance):
        for layer in range(len(self.Activations[0])):
            # For the first layer between X and the first hidden layer
            if layer == 0:
                pre_activation_first = self.W_input.T @ training_instance.reshape(1,1)
                # print('pre activation: ', pre_activation)
                # Apply the activation function
                self.Activations[:,0] = self.apply_activation_function(pre_activation_first).ravel()
            else:
                pre_activation_hidden = self.W_hidden.T @ self.Activations[:, layer-1]
                self.Activations[:, layer] = self.apply_activation_function(pre_activation_hidden)
                # print('Activations: ', self.Activations)
        output = self.W_output.T @ self.Activations[:, -1].reshape(-1,1)
        # print('output: ', output)
        return output

    def backpropagation(self, y_true, training_instance):
        if self.activation_function == 'sigmoid':
            pass
        if self.activation_function == 'linear':
            # Calculate the ouput gradient
            self.Output_Gradient = -(y_true-self.y_pred)
            # print('Output Gradient: ', self.Output_Gradient)

            # Calculate the partial gradients of the Error with respect to the pre acitvation values in the nodes
            self.Partials[:, 1] = ((self.Activations[:, 1]*(1-self.Activations[:, 1])).reshape(-1,1)*(self.W_output @ self.Output_Gradient)).ravel()
            self.Partials[:, 0] = self.Activations[:, 0]*(1-self.Activations[:, 0])*(self.W_hidden @ self.Partials[:, 1])
            # print('Partials: ', self.Partials)

            # Calculate the Gradients with respect to the weights
            self.Partials_W_output = self.Output_Gradient * self.Activations[:, -1]
            # print('Partials_W_output: ', self.Partials_W_output)
            self.Partials_W_hidden = self.Partials[:, -1].reshape(3,1) * self.Activations[:, 0].reshape(1,3)
            # print('Partials_W_hidden: ',self.Partials_W_hidden)
            self.Partials_W_input = (self.Partials[:, 0].reshape(3,1) * training_instance.T).T
            # print('Partials_W_input: ', self.Partials_W_input)

    def weight_update(self, training_instance, learning_rate):

        # Output Layer weights
        w_output_old = self.W_output.copy()
        self.W_output = w_output_old - learning_rate*self.Partials_W_output.reshape(-1,1)

        # Hidden Layer weights
        w_hidden_old = self.W_hidden.copy()
        self.W_hidden = w_hidden_old - learning_rate * self.Partials_W_hidden
        # print('W_hidden new: ', self.W_hidden)

        # Input Layer weights
        w_input_old = self.W_input.copy()
        self.W_input = w_input_old - learning_rate * self.Partials_W_input
        # print('W_input new: ', self.W_input)


    def train_model(self):
        # print('Initially predicted Value: ', self.make_prediction(self.X_test[0]))
        # print('True value: ', self.y_test[0])
        for _ in range(self.runs):
            for instance in range(len(self.X_train)):
                # forward pass
                self.y_pred = self.forward_pass(self.X_train[instance])

                # Calculate loss
                self.Loss = self.calc_loss(self.y_pred, self.y_train[instance])
                # print('Loss: ', self.Loss)

                # Calculate backpropagation
                self.backpropagation(self.y_train[instance], self.X_train[instance])

                # Update weights
                self.weight_update(self.X_train[instance], self.learning_rate)

        # print(self.Losses)
        # plt.plot(range(len(self.Losses)), self.Losses)
        # plt.show()

        # Make predictions
        predictions = []
        for i in np.linspace(0,5,1000):
            predictions.append(self.make_prediction(i)[0])
        plt.plot(np.linspace(0,5,1000), predictions)


    def make_prediction(self, X_new):
        return self.forward_pass(X_new)

    def calc_loss(self, y_pred, y_true):
        loss = (1/2)*(y_true-y_pred)**2
        self.Losses.append(loss[0])
        return (1/2)*(y_true-y_pred)**2

    def accuracy(self):
        pass

Neural_Net('linear', 0.1, 1500).train_model()

为了优化代码，我们必须在隐藏层的输入中添加偏差

Answer 2

只要你的激活函数是线性的，整个ANN将提供一个简单的加权和，因此是线性输出。实际上你现在正在做线性回归。这有助于验证学习在某种程度上是否有效（尝试教授一些线性函数），但仅此而已，对于真正需要的非线性函数

参见维基百科上的https://en.wikipedia.org/wiki/Activation_function#Comparison_of_activation_functions了解想法。事实上，函数的比较首先是对理想特性的回顾，第一个是非线性的：

Comparison of activation functions

Some desirable properties in an activation function include:

• Nonlinear – When the activation function is non-linear, then a two-layer neural network can be proven to be a universal function approximator.[6] The identity activation function does not satisfy this property. When multiple layers use the identity activation function, the entire network is equivalent to a single-layer model.