391 lines
15 KiB
Python
391 lines
15 KiB
Python
# -*- coding: utf-8 -*-
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"""
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神经网络（多层感知机）
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"""
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# 导入模块
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from typing import List, Literal, Optional, Dict
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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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seed: int = 52,
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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 seed: 随机种子，默认为52
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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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if not (
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all(x >= 1 if isinstance(x, int) else False for x in structure)
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if isinstance(structure, list) and len(structure) >= 3
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else False
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):
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raise RuntimeError(
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"神经网络结构应为长度大于等于3的列表且列表元素应为大于等于1的整数"
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)
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# 初始化神经网络结构
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self.structure = structure
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if hidden_activate not in self.HIDDEN_ACTIVATES:
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raise RuntimeError(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 RuntimeError(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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# 神经网络层数（定义，第0层为输入层，第l层为隐含层（l=1,2,...,L-1），第L层为输出层（L为神经网络层数），深度为L+1）
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self.layer_counts = len(structure) - 1
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numpy.random.seed(seed) # 设置随机种子
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self.parameters = {}
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# 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
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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.parameters[layer_index] = {
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"activate": (
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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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)
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), # 激活函数
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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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* self._calculate_init_weight_scale(
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activate=activate,
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previous_layer_neuron_counts=previous_layer_neuron_counts,
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current_layer_neuron_counts=current_layer_neuron_counts,
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), # 权重，维度为[当前层神经元数，上一层神经元数]，适配加权输入=权重*输入+平移
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"bias": numpy.zeros((current_layer_neuron_counts, 1)), # 平移
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}
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self.epsilon = epsilon
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print("已完成")
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def _calculate_init_weight_scale(
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self,
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activate: Literal["relu", "linear", "softmax"],
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previous_layer_neuron_counts: int,
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current_layer_neuron_counts: int,
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) -> numpy.floating:
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"""
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计算初始化权重时缩放因子
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:param activate: 激活函数
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:param previous_layer_neuron_counts: 上一层神经元数
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:param current_layer_neuron_counts: 当前层神经元数
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:return: 初始化权重时缩放因子
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"""
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match activate:
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case "relu":
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return numpy.sqrt(
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2 / previous_layer_neuron_counts
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) # 使用He初始化权重方法
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case "linear":
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return numpy.sqrt(
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2 / previous_layer_neuron_counts
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) # 使用He初始化权重方法
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case "softmax":
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return numpy.sqrt(
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2 / (previous_layer_neuron_counts + current_layer_neuron_counts)
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) # 使用Xavier初始化权重方法
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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，维度为[输入神经元数, 样本数]
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:param y_true: 真实输出，维度为[输出神经元数, 样本数]
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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"开始训练神经网络：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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if not (
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X.shape[1] == y_true.shape[1]
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and X.shape[0] == self.structure[0]
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and y_true.shape[0] == self.structure[-1]
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if isinstance(X, numpy.ndarray) and isinstance(y_true, numpy.ndarray)
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else False
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):
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raise RuntimeError(
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"输入和真实输出应为数组，其中输入维度应为[输入神经元数, 样本数]，真实输出维度应为[输出神经元数, 样本数]，样本数应需相同"
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)
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# 归一化输入并将其作为输入层的输出
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self.parameters[0] = {"activation": self._normalize(input=X)}
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epoch = 1
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while True:
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# 前向传播
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self._forward_propagate()
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# 计算损失
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loss = self._calculate_loss(y_true=y_true)
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if loss <= target_loss:
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print(
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f"第 {epoch:6d} 轮损失已达到目标损失 {target_loss:9.3f}，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(f"已达到最大学习轮数，损失为 {loss:9.3f}，训练结束")
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break
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# 后向传播
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self._backward_propagate(y_true=y_true)
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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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if epoch % 100 == 0:
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print(f"第 {epoch:6d} 轮损失为 {loss:9.3f}，继续训练...")
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epoch += 1
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def _normalize(
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self,
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input: numpy.ndarray,
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) -> numpy.ndarray:
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"""
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归一化
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:param input: 输入
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:return: 归一化后的输入，维度与输入相同
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"""
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return (input - numpy.mean(input, axis=1, keepdims=True)) / numpy.sqrt(
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numpy.var(input, ddof=0, axis=1, keepdims=True) + self.epsilon
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)
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def _forward_propagate(self) -> None:
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"""
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前向传播
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:return: 输出层的输出预测，维度为[输出神经元数, 样本数]
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"""
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for layer_index in range(1, self.layer_counts + 1):
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self.parameters[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.parameters[layer_index]["weight"],
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self.parameters[layer_index - 1][
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"activation"
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], # 将上一层的输出作为当前层的输入
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)
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+ self.parameters[layer_index]["bias"]
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), # 加权输入，维度为[当前层神经元数，样本数]，将上一层的输出作为当前层的输入
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"activation": self._activate(
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activate=self.parameters[layer_index]["activate"],
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input=weighted_input,
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), # 输出
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}
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)
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def _activate(
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self,
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activate: Literal["relu", "linear", "softmax"],
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input: numpy.ndarray,
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) -> numpy.ndarray:
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"""
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激活
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:param activate: 激活函数
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:param input: 输入
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:return: 经过激活函数计算后的输入，维度与输入相同
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"""
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match activate:
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case "relu":
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return numpy.maximum(0, input)
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case "linear":
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return 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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input - numpy.max(input, axis=0, keepdims=True)
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) # 减去各样本所有神经元最大值以避免指数溢出
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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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) -> numpy.floating:
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"""
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计算损失
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:param y_true: 真实输出，维度为[输出神经元数, 样本数]
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:return: 损失
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"""
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match self.parameters[self.layer_counts]["activate"]:
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case "linear":
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return 0.5 * numpy.mean(
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numpy.square(
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y_true - self.parameters[self.layer_counts]["activation"]
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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0.5*均方误差
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case "softmax":
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return numpy.mean(
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(
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numpy.max(
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self.parameters[self.layer_counts]["weighted_input"],
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axis=0,
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keepdims=True,
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)
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+ numpy.log(
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numpy.sum(
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numpy.exp(
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self.parameters[self.layer_counts]["weighted_input"]
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- numpy.max(
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self.parameters[self.layer_counts][
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"weighted_input"
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],
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axis=0,
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keepdims=True,
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)
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),
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axis=0,
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keepdims=True,
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)
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)
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).squeeze()
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- numpy.sum(
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y_true * self.parameters[self.layer_counts]["weighted_input"],
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axis=0,
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)
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) # 若输出层的激活函数为softmax则损失函数使用交叉熵
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case _:
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raise RuntimeError(
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f"该激活函数 {self.parameters[self.layer_counts]["activate"]} 暂不支持"
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)
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def _backward_propagate(
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self,
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y_true: numpy.ndarray,
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) -> None:
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"""
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后向传播
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:param y_true: 真实输输出，维度为[输出神经元数, 样本数]
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:return: 无
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"""
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sample_counts = y_true.shape[1] # 样本数
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for layer_index in range(self.layer_counts, 0, -1):
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self.parameters[layer_index].update(
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{
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"delta_activation": (
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delta_activation := (
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(self.parameters[self.layer_counts]["activation"] - y_true)
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/ sample_counts
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if layer_index == self.layer_counts
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else numpy.dot(
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self.parameters[layer_index + 1]["weight"].T,
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self.parameters[layer_index + 1][
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"delta_weighted_input"
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],
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)
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)
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), # 若为输出层则直接计算输出的梯度，否则基于下一层的权重转置和加权输入的梯度计算当前层的输出梯度
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"delta_weighted_input": (
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delta_weighted_input := delta_activation
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* self._activate_derivative(
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activate=self.parameters[layer_index]["activate"],
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input=self.parameters[layer_index]["weighted_input"],
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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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(self.parameters[layer_index - 1]["activation"]).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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def _activate_derivative(
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self,
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activate: Literal["relu", "linear", "softmax"],
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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入的导数
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:param activate: 激活函数
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:param input: 输入
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:return: 经过激活函数计算后的输入，维度与输入相同
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"""
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match activate:
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case "relu":
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return numpy.where(input > 0, 1, 0)
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case "linear":
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return numpy.ones_like(input)
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case "softmax":
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activation = self._activate(
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activate=activate,
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input=input,
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)
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return activation * (1 - activation)
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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.parameters[layer_index].update(
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{
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"weight": self.parameters[layer_index]["weight"]
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- self.parameters[layer_index]["delta_weight"]
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* learning_rate, # 权重
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"bias": self.parameters[layer_index]["bias"]
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- self.parameters[layer_index]["delta_bias"]
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* learning_rate, # 平移
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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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X = numpy.random.randn(2, 100)
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# 真实函数：y = 2*x1 + 3*x2 + 1
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y_true = 2 * X[0:1, :] ** 2 + 3 * X[1:2, :] + 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, 256, 128, 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=0.05, epochs=1_000, learning_rate=0.05
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)
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