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神经网络/main.py
461
神经网络/main.py
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@ -13,142 +13,156 @@ class NeuralNetwork:
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神经网络
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"""
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HIDDEN_ACTIVATES = ["relu"]
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OUTPUT_ACTIVATES = ["linear", "softmax"]
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def __init__(
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self,
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neurons: List[int],
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structure: List[int],
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hidden_activate: Literal["relu"] = "relu",
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output_activate: Literal["linear", "softmax"] = "linear",
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epsilon: float = 1e-9,
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):
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"""
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初始化
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:param neurons: 神经元结构，例如[2, 10, 1]表示输入层为2个神经元、第一层隐含层为10个神经元、输出层为1个神经元
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:param hidden_activate: 隐含层激活函数，默认为relu
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:param output_activate: 输出层激活函数，默认为linear
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:param structure: 神经网络结构，例如[2, 10, 1]表示2层神经网络，具体为输入层2个神经元、隐含层10个神经元、输出层1个神经元
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:param hidden_activate: 隐含层的激活函数，默认为relu
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:param output_activate: 输出层的激活函数，默认为linear
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:param epsilon: 极小值，默认为1e-9
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"""
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print("正在初始化神经网络...", end="")
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# 初始化神经元结构
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self.neurons = neurons
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# 初始化神经网络结构
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self.structure = structure
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# 神经网络层数
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self.layer_counts = (
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len(structure) - 1
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) # 定义第0层为输入层，第L层为输出层（L为神经网络层数），第l层为隐含层（l=1,2,...,L-1）
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# 初始化隐含层激活函数
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if hidden_activate not in self.HIDDEN_ACTIVATES:
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raise ValueError(f"该隐含层激活函数 {hidden_activate} 暂不支持")
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self.hidden_activate = hidden_activate
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# 初始化输出层激活函数
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if output_activate not in self.OUTPUT_ACTIVATES:
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raise ValueError(f"该输出层激活函数 {output_activate} 暂不支持")
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self.output_activate = output_activate
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# 初始化神经网络结构
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self.neural_network = {}
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# 初始化神经网络所有层权重和偏置
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self._init_neural_network()
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self.paramters = {}
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# 就隐含层和输出层初始化神经网络参数
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for layer_index in range(1, self.layer_counts + 1):
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# 上一层和当前层神经元数量
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previous_layer_neuron_counts, current_layer_neuron_counts = (
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self.structure[layer_index - 1],
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self.structure[layer_index],
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)
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self.paramters[layer_index] = {
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"weight": numpy.random.randn(
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current_layer_neuron_counts, previous_layer_neuron_counts
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)
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* (
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numpy.sqrt(2 / previous_layer_neuron_counts)
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if layer_index < self.layer_counts
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else (
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numpy.sqrt(1 / previous_layer_neuron_counts)
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if self.output_activate == "linear"
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else numpy.sqrt(
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2
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/ (
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previous_layer_neuron_counts
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+ current_layer_neuron_counts
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)
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)
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)
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), # 权重，权重维度为[当前层神经元数量，上一层神经元数量]、输入维度为[上一层神经元数量，样本数]以适配加权输=权重*输入+偏移。隐含层使用He初始化权重方法、输出层激活函数若为linear则使用标准Xavier初始化权重方法否则使用改进Xavier初始化权重方法
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"bias": numpy.zeros((current_layer_neuron_counts, 1)), # 偏移
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"gamma": numpy.ones(
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(current_layer_neuron_counts, 1)
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), # 批标准化的缩放因子
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"beta": numpy.zeros(
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(current_layer_neuron_counts, 1)
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), # 批标准化的偏移因子
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"activate": (
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self.hidden_activate
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if layer_index < self.layer_counts
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else self.output_activate
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), # 激活函数
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}
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self.epsilon = epsilon
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print("已完成")
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def _init_neural_network(self):
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"""
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初始化神经网络所有层权重和偏置
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"""
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for idx in range(1, len(self.neurons)):
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# 若为隐含层则根据隐含层激活函数计算当前层权重的标准偏差，若为输出层则根据输出层激活函数计算当前层权重的标准偏差
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if idx != len(self.neurons) - 1:
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# 激活函数
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activate = self.hidden_activate
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match self.hidden_activate:
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case "relu":
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# 当前层权重的标准偏差
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standard_deviation = numpy.sqrt(
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2 / self.neurons[idx - 1]
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) # 使用He方差公式
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case _:
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raise RuntimeError(
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f"暂不支持该隐含层激活函数 {self.hidden_activate}"
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)
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else:
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# 激活函数
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activate = self.output_activate
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match self.output_activate:
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case "linear":
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# 当前层权重的标准偏差
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standard_deviation = numpy.sqrt(1 / self.neurons[idx - 1])
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case "softmax":
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# 当前层权重的标准偏差
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standard_deviation = numpy.sqrt(
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2 / (self.neurons[idx - 1] + self.neurons[idx])
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) # 使用Xavier方差公式
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case _:
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raise RuntimeError(
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f"暂不支持该输出层激活函数 {self.output_activate}"
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)
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self.neural_network[f"layer:{idx:03d}"] = {
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"weight": numpy.random.randn(self.neurons[idx - 1], self.neurons[idx])
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* standard_deviation, # 当前层权重
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"bias": numpy.zeros((1, self.neurons[idx])), # 当前层偏置
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"activate": activate, # 当前层激活函数
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"gamma": numpy.ones((1, self.neurons[idx])), # 当前层批标准化的缩放因子
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"beta": numpy.zeros((1, self.neurons[idx])), # 当前层批标准化的偏移因子
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}
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def _forward_propagate(self, x: numpy.ndarray) -> numpy.ndarray:
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def _forward_propagate(self, X: numpy.ndarray) -> numpy.ndarray:
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"""
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前向传播
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:param x: 输入层输入
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:return: 输出层预测值
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:param X: 输入层的输入，维度为[输入特征数, 样本数]
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:return: 输出层的输出预测，维度为[输出特征数, 样本数]
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"""
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activation = x # 将输入层输入作为第0层的激活值
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for layer_name, layer in self.neural_network.items():
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self.neural_network[layer_name].update(
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activation = X # 将输入层的输入作为第0层的输出
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for layer_index in range(1, self.layer_counts + 1):
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x = activation # 将上一层的输出作为当前层的输入
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self.paramters[layer_index].update(
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{
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"weighted_sum": (
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weighted_sum := numpy.dot(activation, layer["weight"])
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+ layer["bias"]
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), # 当前层加权和
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"batch_normalized_weighted_sum": (
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batch_normalized_weighted_sum := layer["gamma"]
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* (
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weighted_sum
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- numpy.mean(weighted_sum, axis=0, keepdims=True)
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"weighted_input": (
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weighted_input := numpy.dot(
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self.paramters[layer_index]["weight"], x
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)
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/ numpy.sqrt(
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numpy.var(
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weighted_sum, ddof=0, axis=0, keepdims=True
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) # 使用有偏方差公式
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+ 1e-8
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), # 加权输入
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"weighted_input_average": (
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weighted_input_average := numpy.mean(
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weighted_input, axis=1, keepdims=True
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)
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+ layer["beta"]
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), # 当前层批标准化加权和
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), # 加权输入的平均值
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"weighted_input_standard_deviation": (
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weighted_input_standard_deviation := numpy.sqrt(
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numpy.var(weighted_input, ddof=0, axis=1, keepdims=True)
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+ self.epsilon
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)
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), # 加权输入的标准差
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"batch_normalized_weighted_input": (
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batch_normalized_weighted_input := (
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weighted_input - weighted_input_average
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)
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* self.paramters[layer_index]["gamma"]
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/ weighted_input_standard_deviation
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+ self.paramters[layer_index]["beta"]
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), # 就加权输入批标准化
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"activation": (
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activation := self._activate(
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activate=layer["activate"],
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weighted_sum=batch_normalized_weighted_sum,
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activate=self.paramters[layer_index]["activate"],
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weighted_input=batch_normalized_weighted_input,
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)
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), # 当前层激活值
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), # 输出
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}
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)
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y_predict = activation # 将第L-1层（最后一层）的激活值作为输出层预测值（L为神经网络层数）
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y_predict = activation # 将第L层（输出层）的输出作为输出层的输出预测
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return y_predict
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def _activate(
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self,
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activate: Literal["relu", "linear", "softmax"],
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weighted_sum: numpy.ndarray,
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weighted_input: numpy.ndarray,
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) -> numpy.ndarray:
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"""
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激活函数
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根据激活函数计算输出
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:param activate: 激活函数
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:param weighted_sum: 加权和
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:return: 激活值
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:param weighted_input: 加权输入
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:return: 输出
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"""
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match activate:
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case "relu":
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return numpy.maximum(0, weighted_sum)
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return numpy.maximum(0, weighted_input)
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case "linear":
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return weighted_sum
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return weighted_input
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case "softmax":
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# 加权和指数项
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e_weighted_sum = numpy.exp(
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weighted_sum - numpy.max(weighted_sum, axis=1, keepdims=True)
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# 加权输入的指数项
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e_weighted_input = numpy.exp(
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weighted_input - numpy.max(weighted_input, axis=0, keepdims=True)
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)
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return e_weighted_input / numpy.sum(
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e_weighted_input, axis=0, keepdims=True
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)
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return e_weighted_sum / numpy.sum(e_weighted_sum, axis=1, keepdims=True)
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def _calculate_loss(
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self,
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@ -157,62 +171,239 @@ class NeuralNetwork:
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) -> numpy.floating:
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"""
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计算损失
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:param y_true: 输出层真实值
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:param y_predict: 输出层预测值
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:param y_true: 输出层的输出真实，维度为[输出特征数, 样本数]
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:param y_predict: 输出层的输出预测，维度为[输出特征数, 样本数]
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:return: 损失值
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"""
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# 第L-1层（最后一层）的层名
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layer_name = list(self.neural_network.keys())[-1]
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# 根据第L-1层（最后一层）的激活函数计算损失
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match activate := self.neural_network[layer_name]["activate"]:
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case "linear":
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loss = 0.5 * numpy.mean(
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numpy.square(y_true - y_predict)
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) # 使用均方误差公式
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case "softmax":
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loss = -1 * numpy.mean(
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numpy.sum(y_true * numpy.log(y_predict + 1e-8), axis=1)
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) # 使用交叉熵损失公式
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case _:
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raise RuntimeError(f"暂不支持该输出层激活函数 {activate}")
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return loss
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return (
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0.5 * numpy.mean(numpy.square(y_true - y_predict))
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if self.paramters[self.layer_counts]["activate"] == "linear"
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else -1
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* numpy.mean(
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numpy.sum(
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y_true
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* numpy.log(numpy.clip(y_predict, self.epsilon, 1 - self.epsilon)),
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axis=0,
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)
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)
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) # 若输出层的激活函数为linear则损失函数使用均方误差否则使用交叉熵
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def _backward_propagate(
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self,
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x: numpy.ndarray,
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X: numpy.ndarray,
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y_true: numpy.ndarray,
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y_predict: numpy.ndarray,
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) -> None:
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"""
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后向传播
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:param x: 输入层输入
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:param y_true: 输出层真实值
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:param y_predict: 输出层预测值
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:param X: 输入层的输入，维度为[输入特征数, 样本数]
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:param y_true: 输出层的输出真实，维度为[输出特征数, 样本数]
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:param y_predict: 输出层的输出预测，维度为[输出特征数, 样本数]
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:return: 无
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"""
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# 所有层的层名
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layer_names = list(self.neural_network.keys())
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sample_counts = X.shape[1] # 样本数
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for idx, layer_name in enumerate(reversed(layer_names)):
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# 当前层激活函数、加权和、批标准化加权和和激活值
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activate, weighted_sum, batch_normalized_weighted_sum, activation = (
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self.neural_network[layer_name]["activate"],
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self.neural_network[layer_name]["weighted_sum"],
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self.neural_network[layer_name]["batch_normalized_weighted_sum"],
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self.neural_network[layer_name]["activation"],
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# 损失函数对输出层的就加权输入批标准化的梯度
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self.paramters[self.layer_counts]["delta_batch_normalized_weighted_input"] = (
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y_predict - y_true
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) / sample_counts # 均方误差和交叉熵对输出层的输出预测的梯度是相同的
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for layer_index in range(self.layer_counts, 0, -1):
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self.paramters[layer_index].update(
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{
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"delta_gamma": numpy.sum(
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self.paramters[layer_index][
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"delta_batch_normalized_weighted_input"
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]
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* (
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self.paramters[layer_index]["weighted_input"]
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- self.paramters[layer_index]["weighted_input_average"]
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)
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/ self.paramters[layer_index][
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"weighted_input_standard_deviation"
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],
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axis=1,
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keepdims=True,
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), # 批标准化的缩放因子的梯度
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"delta_beta": numpy.sum(
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self.paramters[layer_index][
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"delta_batch_normalized_weighted_input"
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],
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axis=1,
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keepdims=True,
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), # 批标准化的偏移因子的梯度
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"delta_weighted_input": (
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delta_weighted_input := (
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sample_counts
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* self.paramters[layer_index]["gamma"]
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* self.paramters[layer_index][
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"delta_batch_normalized_weighted_input"
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]
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- numpy.sum(
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self.paramters[layer_index]["gamma"]
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* self.paramters[layer_index][
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"delta_batch_normalized_weighted_input"
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],
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axis=1,
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keepdims=True,
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)
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- (
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(
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self.paramters[layer_index]["weighted_input"]
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- self.paramters[layer_index][
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"weighted_input_average"
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]
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)
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/ self.paramters[layer_index][
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"weighted_input_standard_deviation"
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]
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)
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* numpy.sum(
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self.paramters[layer_index]["gamma"]
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* self.paramters[layer_index][
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"delta_batch_normalized_weighted_input"
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]
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* (
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(
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self.paramters[layer_index]["weighted_input"]
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- self.paramters[layer_index][
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"weighted_input_average"
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]
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)
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/ self.paramters[layer_index][
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"weighted_input_standard_deviation"
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]
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),
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axis=1,
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keepdims=True,
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)
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)
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* (1.0 / sample_counts)
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/ self.paramters[layer_index][
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"weighted_input_standard_deviation"
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]
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), # 加权输入的梯度
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"delta_weight": numpy.dot(
|
||||
delta_weighted_input,
|
||||
(
|
||||
X
|
||||
if layer_index == 1
|
||||
else self.paramters[layer_index - 1]["activation"]
|
||||
).T,
|
||||
), # 权重的梯度
|
||||
"delta_bias": numpy.sum(
|
||||
delta_weighted_input,
|
||||
axis=1,
|
||||
keepdims=True,
|
||||
), # 偏置的梯度
|
||||
}
|
||||
)
|
||||
|
||||
# 输出层的误差项
|
||||
if idx == 0:
|
||||
match activate:
|
||||
case "linear" | "softmax":
|
||||
delta = y_predict - y_true # 损失函数对第L-1层（最后一层）激活值的梯度
|
||||
case _:
|
||||
raise RuntimeError(f"暂不支持该输出层激活函数 {activate}")
|
||||
# 隐含层的误差项
|
||||
else:
|
||||
delta = numpy.dot(delta, self.neural_network[layer_names[idx - 1]]["weight"].T)
|
||||
if layer_index > 1:
|
||||
self.paramters[layer_index - 1][
|
||||
"delta_batch_normalized_weighted_input"
|
||||
] = numpy.dot(
|
||||
self.paramters[layer_index]["weight"].T,
|
||||
self.paramters[layer_index]["delta_weighted_input"],
|
||||
) * (
|
||||
self.paramters[layer_index - 1]["batch_normalized_weighted_input"]
|
||||
> 0
|
||||
).astype(
|
||||
numpy.float32
|
||||
)
|
||||
|
||||
delta = 0
|
||||
def train(
|
||||
self,
|
||||
X: numpy.ndarray,
|
||||
y_true: numpy.ndarray,
|
||||
target_loss: float = 1e-3,
|
||||
epochs: int = 200,
|
||||
learning_rate: float = 0.001,
|
||||
) -> None:
|
||||
"""
|
||||
训练神经网络
|
||||
:param X: 输入层的输入
|
||||
:param y_true: 输出层的输出真实
|
||||
:param target_loss: 目标损失
|
||||
:param epochs: 学习轮数
|
||||
:param learning_rate: 学习率
|
||||
:return: 无
|
||||
"""
|
||||
print(
|
||||
f"开始训练：目标损失为 {target_loss}，学习轮数为 {epochs}，学习率为 {learning_rate}..."
|
||||
)
|
||||
# 标准化
|
||||
X = (X - numpy.mean(X, axis=1, keepdims=True)) / (
|
||||
numpy.std(X, axis=1, keepdims=True) + self.epsilon
|
||||
)
|
||||
epoch = 1
|
||||
while True:
|
||||
# 前向传播
|
||||
y_predict = self._forward_propagate(X=X)
|
||||
|
||||
loss = self._calculate_loss(y_true=y_true, y_predict=y_predict)
|
||||
if loss < target_loss:
|
||||
print(
|
||||
f" 第 {epoch} 轮损失为 {loss}，已达到目标损失 {target_loss}，训练结束"
|
||||
)
|
||||
break
|
||||
if epoch >= epochs:
|
||||
print(
|
||||
f" 第 {epoch} 轮损失为 {loss}，已达到最大学习轮数 {epochs}，训练结束"
|
||||
)
|
||||
break
|
||||
if epoch % 50 == 0:
|
||||
print(f" 第 {epoch} 轮损失为 {loss}，继续训练...")
|
||||
|
||||
# 后向传播
|
||||
self._backward_propagate(X=X, y_true=y_true, y_predict=y_predict)
|
||||
|
||||
# 更新神经网络参数
|
||||
self._update_parameters(learning_rate=learning_rate)
|
||||
|
||||
epoch += 1
|
||||
|
||||
for idx in numpy.random.choice(X.shape[1], size=10, replace=False):
|
||||
y_true_val = y_true[0, idx]
|
||||
y_pred_val = y_predict[0, idx]
|
||||
error = abs(y_true_val - y_pred_val)
|
||||
print(f"{idx:<10} {y_true_val:<15.4f} {y_pred_val:<15.4f} {error:<15.4f}")
|
||||
|
||||
def _update_parameters(self, learning_rate: float) -> None:
|
||||
"""
|
||||
更新神经网络参数
|
||||
:param learning_rate: 学习率
|
||||
:return: 无
|
||||
"""
|
||||
for layer_index in range(1, self.layer_counts + 1):
|
||||
self.paramters[layer_index].update(
|
||||
{
|
||||
"weight": self.paramters[layer_index]["weight"]
|
||||
- self.paramters[layer_index]["delta_weight"] * learning_rate,
|
||||
"bias": self.paramters[layer_index]["bias"]
|
||||
- self.paramters[layer_index]["delta_bias"] * learning_rate,
|
||||
"gamma": self.paramters[layer_index]["gamma"]
|
||||
- self.paramters[layer_index]["delta_gamma"] * learning_rate,
|
||||
"beta": self.paramters[layer_index]["beta"]
|
||||
- self.paramters[layer_index]["delta_beta"] * learning_rate,
|
||||
}
|
||||
)
|
||||
|
||||
|
||||
# 测试代码
|
||||
if __name__ == "__main__":
|
||||
# 生成测试数据（回归任务）
|
||||
numpy.random.seed(42) # 设置随机种子保证可复现
|
||||
X = numpy.random.randn(2, 100) * 5
|
||||
# 真实函数：y = 2*x1 + 3*x2 + 1 (加噪声)
|
||||
y_true = 2 * X[0:1, :]**2 + 3 * X[1:2, :] + 1 + numpy.random.randn(1, 100) * 0.1
|
||||
|
||||
# 创建并训练神经网络
|
||||
neural_network = NeuralNetwork(
|
||||
structure=[2, 200, 100, 50, 1], # 2输入，10隐藏神经元，1输出
|
||||
)
|
||||
|
||||
# 训练
|
||||
neural_network.train(
|
||||
X=X, y_true=y_true, target_loss=0.001, epochs=10000, learning_rate=0.001
|
||||
)
|
||||
|
|
|
|||
Loading…
Reference in New Issue