【深度学习实战】一、Numpy手撸神经网络实现线性回归

目录

一、引言

二、代码实战

1、Tensor和初始化类

2、全连接层

3、模型组网

4、SGD优化器

5、均方差损失函数

6、Dataset

三、线性回归实战

四、实验结果

五、总结

---

一、引言

深度学习理论相对简单，但是深度学习框架（tensorflow/torch/paddlepaddle）源码却比较复杂，对于初学者来说，将代码和理论相结合理解是一个巨大的问题，本文的目标是使用python的numpy库从零开始实现一个简单的神经网络模型，并解决一个简单的线性回归问题。本文不会过多介绍理论，直接从代码入手，如需要了解相关理论请参考下方链接。

相关资料参考：

1、全连接层前向传播和梯度计算

2、动量梯度下降

3、ReLU

二、代码实战

1、Tensor和初始化类

Tensor：包含数据和梯度信息，神经网络层的参数已Tensor保存。

初始化类（Constant/Normal）：参数初始化方法。

# 因为层的参数需要保存值和对应的梯度，这里定义梯度，可训练的参数全部以Tensor的类别保存

import numpy as np
np.random.seed(10001)

class Tensor:
    def __init__(self, shape):
        self.data = np.zeros(shape=shape, dtype=np.float32) # 存放数据
        self.grad = np.zeros(shape=shape, dtype=np.float32) # 存放梯度

    def clear_grad(self):
        self.grad = np.zeros_like(self.grad)

    def __str__(self):
        return "Tensor shape: {}, data: {}".format(self.data.shape, self.data)


# Tensor的初始化类，目前仅提供Normal初始化和Constant初始化
class Initializer:
    """
    基类
    """
    def __init__(self, shape=None, name='initializer'):
        self.shape = shape
        self.name = name

    def __call__(self, *args, **kwargs):
        raise NotImplementedError

    def __str__(self):
        return self.name


class Constant(Initializer):
    def __init__(self, value=0., name='constant initializer', *args, **kwargs):
        super().__init__(name=name, *args, **kwargs)
        self.value = value

    def __call__(self, shape=None, *args, **kwargs):
        if shape:
            self.shape = shape
        assert shape is not None, "the shape of initializer must not be None."
        return self.value + np.zeros(shape=self.shape)


class Normal(Initializer):
    def __init__(self, mean=0., std=0.01, name='normal initializer', *args, **kwargs):
        super().__init__(name=name, *args, **kwargs)
        self.mean = mean
        self.std = std

    def __call__(self, shape=None, *args, **kwargs):
        if shape:
            self.shape = shape
        assert shape is not None, "the shape of initializer must not be None."
        return np.random.normal(self.mean, self.std, size=self.shape)

2、全连接层

Layer：层的基类，主要包括前向传播和反向传播。

Linear：全连接层，继承自Layer，具体化前向传播和反向传播的参数计算过程。

# 为了使层能够组建起来，实现前向传播和反向传播，首先定义层的基类Layer
# Layer的几个主要方法说明：
#   forward: 实现前向传播
#   backward: 实现反向传播
#   parameters: 返回该层的参数，传入优化器进行优化

class Layer:
    def __init__(self, name='layer', *args, **kwargs):
        self.name = name

    def forward(self, *args, **kwargs):
        raise NotImplementedError

    def backward(self):
        raise NotImplementedError

    def parameters(self):
        return []

    def __call__(self, *args, **kwargs):
        return self.forward(*args, **kwargs)

    def __str__(self):
        return self.name


class Linear(Layer):
    """
    input X, shape: [N, C]
    output Y, shape: [N, O]
    weight W, shape: [C, O]
    bias b, shape: [1, O]
    grad dY, shape: [N, O]
    forward formula:
        Y = X @ W + b   # @表示矩阵乘法
    backward formula:
        dW = X.T @ dY
        db = sum(dY, axis=0)
        dX = dY @ W.T
    """
    def __init__(
        self,
        in_features,
        out_features,
        name='linear',
        weight_attr=Normal(),
        bias_attr=Constant(),
        *args,
        **kwargs
        ):
        super().__init__(name=name, *args, **kwargs)
        self.weights = Tensor((in_features, out_features))
        self.weights.data = weight_attr(self.weights.data.shape)
        self.bias = Tensor((1, out_features))
        self.bias.data = bias_attr(self.bias.data.shape)
        self.input = None

    def forward(self, x):
        self.input = x
        output = np.dot(x, self.weights.data) + self.bias.data
        return output

    def backward(self, gradient):
        self.weights.grad += np.dot(self.input.T, gradient)  # dy / dw
        self.bias.grad += np.sum(gradient, axis=0, keepdims=True)  # dy / db 
        input_grad = np.dot(gradient, self.weights.data.T)  # dy / dx
        return input_grad

    def parameters(self):
        return [self.weights, self.bias]

    def __str__(self):
        string = "linear layer, weight shape: {}, bias shape: {}".format(self.weights.data.shape, self.bias.data.shape)
        return string


class ReLU(Layer):
    """
    forward formula:
        relu = x if x >= 0
             = 0 if x < 0
    backwawrd formula:
        grad = gradient * (x > 0)
    """
    def __init__(self, name='relu', *args, **kwargs):
        super().__init__(name=name, *args, **kwargs)
        self.activated = None

    def forward(self, x):
        x[x < 0] = 0             
        self.activated = x
        return self.activated

    def backward(self, gradient):
        return gradient * (self.activated > 0)  

3、模型组网

Sequential：模型组网类，将神经网络的层按照顺序叠加起来，实现顺序前向传播和反向传播。

# 模型组网的功能是将层串起来，实现数据的前向传播和梯度的反向传播
# 添加层的时候，按照顺序添加层的参数
# Sequential方法说明：
#   add: 向组网中添加层
#   forward: 按照组网构建的层顺序，依次前向传播
#   backward: 接收损失函数的梯度，按照层的逆序反向传播
class Sequential:
    def __init__(self, *args, **kwargs):
        self.graphs = []
        self._parameters = []
        for arg_layer in args:
            if isinstance(arg_layer, Layer):
                self.graphs.append(arg_layer)
                self._parameters += arg_layer.parameters()

    def add(self, layer):
        assert isinstance(layer, Layer), "The type of added layer must be Layer, but got {}.".format(type(layer))
        self.graphs.append(layer)
        self._parameters += layer.parameters()

    def forward(self, x):
        for graph in self.graphs:
            x = graph(x)
        return x

    def backward(self, grad):
        # grad backward in inverse order of graph
        for graph in self.graphs[::-1]:
            grad = graph.backward(grad)

    def __call__(self, *args, **kwargs):
        return self.forward(*args, **kwargs)

    def __str__(self):
        string = 'Sequential:\n'
        for graph in self.graphs:
            string += graph.__str__() + '\n'
        return string

    def parameters(self):
        return self._parameters

4、SGD优化器

Optimizer：优化器基类，包括参数优化方法（step）、清空梯度（clear_grad）、计算正则化（get_decay）。

SGD：随机梯度下降类，实现了动量梯度下降方法。

# 优化器主要完成根据梯度来优化参数的任务,其主要参数有学习率和正则化类型和正则化系数
# Optimizer主要方法：
#   step: 梯度反向传播后调用，该方法根据计算出的梯度，对参数进行优化
#   clear_grad: 模型调用backward后，梯度会进行累加，如果已经调用step优化过参数，需要将使用过的梯度清空
#   get_decay: 根据不同的正则化方法，计算出正则化惩罚值
class Optimizer:
    """
    optimizer base class.
    Args:
        parameters (Tensor): parameters to be optimized.
        learning_rate (float): learning rate. Default: 0.001.
        weight_decay (float): The decay weight of parameters. Defaylt: 0.0.
        decay_type (str): The type of regularizer. Default: l2.
    """
    def __init__(self, parameters, learning_rate=0.001, weight_decay=0.0, decay_type='l2'):
        assert decay_type in ['l1', 'l2'], "only support decay_type 'l1' and 'l2', but got {}.".format(decay_type)
        self.parameters = parameters
        self.learning_rate = learning_rate
        self.weight_decay = weight_decay
        self.decay_type = decay_type
        
    def step(self):
        raise NotImplementedError

    def clear_grad(self):
        for p in self.parameters:
            p.clear_grad()

    def get_decay(self, g):
        if self.decay_type == 'l1':
            return self.weight_decay
        elif self.decay_type == 'l2':
            return self.weight_decay * g

# 基本的梯度下降法为（不带正则化）：
# W = W - learn_rate * dW
# 带动量的梯度计算方法（减弱的梯度的随机性）：
# dW = (momentum * v) + (1 - momentum) * dW
class SGD(Optimizer):
    def __init__(self, momentum=0.9, *args, **kwargs):
        super().__init__(*args, **kwargs)
        self.momentum = momentum
        self.velocity = []
        for p in self.parameters:
            self.velocity.append(np.zeros_like(p.grad))

    def step(self):
        for p, v in zip(self.parameters, self.velocity):
            decay = self.get_decay(p.grad)
            v = self.momentum * v + p.grad + decay # 动量计算
            p.data = p.data - self.learning_rate * v

5、均方差损失函数

MSE：均方差损失函数。

# 损失函数的设计延续了Layer的模式，但是因为需要注意的是forward和backward部分有些不同
# MSE_loss = (predict_value - label) ^ 2
# MSE方法和Layer的区别：
#   forward：y是组网输出的值，target是目标值（这里的输入是组网的输出和目标值），前向传播的同时把dloss / dy 计算出来
#   backward: 没有参数，因为在forward的时候，计算出了dloss / dy，所以这里不需要输入参数
class MSE(Layer):
    """
    Mean Square Error:
        J = 0.5 * (y - target)^2
    gradient formula:
        dJ/dy = y - target
    """
    def __init__(self, name='mse', reduction='mean', *args, **kwargs):
        super().__init__(name=name, *args, **kwargs)
        assert reduction in ['mean', 'none', 'sum'], "reduction only support 'mean', 'none' and 'sum', but got {}.".format(reduction)
        self.reduction = reduction
        self.pred = None
        self.target = None

    def forward(self, y, target):
        assert y.shape == target.shape, "The shape of y and target is not same, y shape = {} but target shape = {}".format(y.shape, target.shape)
        self.pred = y
        self.target = target
        loss = 0.5 * np.square(y - target)
        if self.reduction is 'mean':
            return loss.mean()
        elif self.reduction is 'none':
            return loss
        else:
            return loss.sum()

    def backward(self):
        gradient = self.pred - self.target
        return gradient

6、Dataset

tensorflow/pytorch/paddlepaddle都有Dataset类，这里也简单实现一个Dataset类。

# 这里仿照PaddlePaddle，Dataset需要实现__getitem__和__len__方法
class Dataset:
    def __init__(self, *args, **kwargs):
        pass

    def __getitem__(self, idx):
        raise NotImplementedError("'{}' not implement in class {}"
                                  .format('__getitem__', self.__class__.__name__))

    def __len__(self):
        raise NotImplementedError("'{}' not implement in class {}"
                                  .format('__len__', self.__class__.__name__))


# 根据dataset和一些设置，生成每个batch在dataset中的索引
class BatchSampler:
    def __init__(self, dataset=None, shuffle=False, batch_size=1, drop_last=False):
        self.batch_size = batch_size
        self.drop_last = drop_last
        self.shuffle = shuffle

        self.num_data = len(dataset)
        if self.drop_last or (self.num_data % batch_size == 0):
            self.num_samples = self.num_data // batch_size
        else:
            self.num_samples = self.num_data // batch_size + 1
        indices = np.arange(self.num_data)
        if shuffle:
            np.random.shuffle(indices)
        if drop_last:
            indices = indices[:self.num_samples * batch_size]
        self.indices = indices

    def __len__(self):
        return self.num_samples

    def __iter__(self):
        batch_indices = []
        for i in range(self.num_samples):
            if (i + 1) * self.batch_size <= self.num_data:
                for idx in range(i * self.batch_size, (i + 1) * self.batch_size):
                    batch_indices.append(self.indices[idx])
                yield batch_indices
                batch_indices = []
            else:
                for idx in range(i * self.batch_size, self.num_data):
                    batch_indices.append(self.indices[idx])
        if not self.drop_last and len(batch_indices) > 0:
            yield batch_indices


# 根据sampler生成的索引，从dataset中取数据，并组合成一个batch
class DataLoader:
    def __init__(self, dataset, sampler=BatchSampler, shuffle=False, batch_size=1, drop_last=False):
        self.dataset = dataset
        self.sampler = sampler(dataset, shuffle, batch_size, drop_last)

    def __len__(self):
        return len(self.sampler)

    def __call__(self):
        self.__iter__()

    def __iter__(self):
        for sample_indices in self.sampler:
            data_list = []
            label_list = []
            for indice in sample_indices:
                data, label = self.dataset[indice]
                data_list.append(data)
                label_list.append(label)
            yield np.stack(data_list, axis=0), np.stack(label_list, axis=0)

三、线性回归实战

生成一组数据，利用上面完成的类，实现线性回归。

import matplotlib.pyplot as plt


class LinearDataset(Dataset):
    def __init__(self, X, Y):
        self.X = X
        self.Y = Y

    def __len__(self):
        return len(self.X)

    def __getitem__(self, idx):
        return self.X[idx], self.Y[idx]


num_data = 200  # 训练数据数量
val_number = 500    # 验证数据数量

epoches = 500
batch_size = 4
learning_rate = 0.01

X = np.linspace(-np.pi, np.pi, num_data).reshape(num_data, 1)
Y = np.sin(X) * 2 + (np.random.rand(*X.shape) - 0.5) * 0.1
y_ = np.sin(X) * 2

model = Sequential(
    Linear(1, 16, name='linear1'),
    ReLU(name='relu1'),
    Linear(16, 64, name='linear2'),
    ReLU(name='relu1'),
    Linear(64, 16, name='linear2'),
    ReLU(name='relu1'),
    Linear(16, 1, name='linear2'),
)
opt = SGD(parameters=model.parameters(), learning_rate=learning_rate, weight_decay=0.0, decay_type='l2')
loss_fn = MSE()

train_dataset = LinearDataset(X, Y)
train_dataloader = DataLoader(train_dataset, shuffle=True, batch_size=4, drop_last=True)
for epoch in range(1, epoches):
    for x, y in train_dataloader:
        pred = model(x)

        loss = loss_fn(pred, y)

        grad = loss_fn.backward()
        model.backward(grad)

        opt.step()
        opt.clear_grad()
    print("epoch: {}. loss: {}".format(epoch, loss))


X_val = np.linspace(-np.pi, np.pi, val_number).reshape(val_number, 1)
Y_val = np.sin(X_val) * 2
val_dataset = LinearDataset(X_val, Y_val)
val_dataloader = DataLoader(val_dataset, shuffle=False, batch_size=2, drop_last=False)
all_pred = []
for x, y in val_dataloader:
    pred = model(x)
    all_pred.append(pred)
all_pred = np.vstack(all_pred)

plt.scatter(X, Y, marker='x')
plt.plot(X_val, all_pred, color='red')
plt.show()

四、实验结果

实验数据如图所示：

拟合结果如图所示，可以看出本文实现的神经网络模型拟合效果。

五、总结

深度学习理论简单，而且成熟的框架很多，大部分深度学习框架的使用者都不需要关注框架底层的实现。本文使用python的numpy库实现了一个非常简单的神经网络模型，并且验证了该模型在线性回归问题上的效果，相信你一定能有所收获。

代码运行不起来？在线notebook体验链接：【深度学习实战】一、Numpy手撸神经网络实现线性回归