410 lines
17 KiB
Python
410 lines
17 KiB
Python
# -*- coding: utf-8 -*-
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"""
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神经网络
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"""
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# 导入模块
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from typing import List, Literal
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import numpy
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class NeuralNetwork:
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"""
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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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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 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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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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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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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 _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_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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), # 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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), # 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=self.paramters[layer_index]["activate"],
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weighted_input=batch_normalized_weighted_input,
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)
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), # 输出
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}
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)
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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_input: numpy.ndarray,
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) -> numpy.ndarray:
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"""
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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_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_input)
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case "linear":
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return weighted_input
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case "softmax":
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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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def _calculate_loss(
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self,
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y_true: numpy.ndarray,
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y_predict: numpy.ndarray,
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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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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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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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sample_counts = X.shape[1] # 样本数
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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(
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delta_weighted_input,
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(
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X
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if layer_index == 1
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else self.paramters[layer_index - 1]["activation"]
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).T,
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), # 权重的梯度
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"delta_bias": numpy.sum(
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delta_weighted_input,
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axis=1,
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keepdims=True,
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), # 偏置的梯度
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}
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)
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if layer_index > 1:
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self.paramters[layer_index - 1][
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"delta_batch_normalized_weighted_input"
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] = numpy.dot(
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self.paramters[layer_index]["weight"].T,
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self.paramters[layer_index]["delta_weighted_input"],
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) * (
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self.paramters[layer_index - 1]["batch_normalized_weighted_input"]
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> 0
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).astype(
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numpy.float32
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)
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def train(
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self,
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X: numpy.ndarray,
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y_true: numpy.ndarray,
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target_loss: float = 1e-3,
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epochs: int = 200,
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learning_rate: float = 0.001,
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) -> None:
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"""
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训练神经网络
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:param X: 输入层的输入
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:param y_true: 输出层的输出真实
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:param target_loss: 目标损失
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:param epochs: 学习轮数
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:param learning_rate: 学习率
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:return: 无
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"""
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print(
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f"开始训练：目标损失为 {target_loss}，学习轮数为 {epochs}，学习率为 {learning_rate}..."
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)
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# 标准化
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X = (X - numpy.mean(X, axis=1, keepdims=True)) / (
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numpy.std(X, axis=1, keepdims=True) + self.epsilon
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)
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epoch = 1
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while True:
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# 前向传播
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y_predict = self._forward_propagate(X=X)
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loss = self._calculate_loss(y_true=y_true, y_predict=y_predict)
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if loss < target_loss:
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print(
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f" 第 {epoch} 轮损失为 {loss}，已达到目标损失 {target_loss}，训练结束"
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)
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break
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if epoch >= epochs:
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print(
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f" 第 {epoch} 轮损失为 {loss}，已达到最大学习轮数 {epochs}，训练结束"
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)
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break
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if epoch % 50 == 0:
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print(f" 第 {epoch} 轮损失为 {loss}，继续训练...")
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# 后向传播
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self._backward_propagate(X=X, y_true=y_true, y_predict=y_predict)
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# 更新神经网络参数
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self._update_parameters(learning_rate=learning_rate)
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epoch += 1
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for idx in numpy.random.choice(X.shape[1], size=10, replace=False):
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y_true_val = y_true[0, idx]
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y_pred_val = y_predict[0, idx]
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error = abs(y_true_val - y_pred_val)
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print(f"{idx:<10} {y_true_val:<15.4f} {y_pred_val:<15.4f} {error:<15.4f}")
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def _update_parameters(self, learning_rate: float) -> None:
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"""
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更新神经网络参数
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:param learning_rate: 学习率
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:return: 无
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"""
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for layer_index in range(1, self.layer_counts + 1):
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self.paramters[layer_index].update(
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{
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"weight": self.paramters[layer_index]["weight"]
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- self.paramters[layer_index]["delta_weight"] * learning_rate,
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"bias": self.paramters[layer_index]["bias"]
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- self.paramters[layer_index]["delta_bias"] * learning_rate,
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"gamma": self.paramters[layer_index]["gamma"]
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- self.paramters[layer_index]["delta_gamma"] * learning_rate,
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"beta": self.paramters[layer_index]["beta"]
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- self.paramters[layer_index]["delta_beta"] * learning_rate,
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}
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)
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# 测试代码
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if __name__ == "__main__":
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# 生成测试数据（回归任务）
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numpy.random.seed(42) # 设置随机种子保证可复现
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X = numpy.random.randn(2, 100) * 5
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# 真实函数：y = 2*x1 + 3*x2 + 1 (加噪声)
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y_true = 2 * X[0:1, :]**2 + 3 * X[1:2, :] + 1 + numpy.random.randn(1, 100) * 0.1
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# 创建并训练神经网络
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neural_network = NeuralNetwork(
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structure=[2, 200, 100, 50, 1], # 2输入，10隐藏神经元，1输出
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)
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# 训练
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neural_network.train(
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X=X, y_true=y_true, target_loss=0.001, epochs=10000, learning_rate=0.001
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)
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