【深度学习实战】一、Numpy手撸神经网络实现线性回归

一、简介

在学习深度学习时，在理论学习完成后，我们常常会直接使用框架（paddle/torch/tensorflow）来搭建我们的模型，常常忽略了各种层结构的底层实现。学习完成深度学习理论的你，能不能手撸一个简单的模型呢？本文旨在从基础开始，一步一步实现深度学习的参数优化，模型搭建过程，巩固基础知识，从理论到实践，一步一步探索深度学习的奥秘。

本文不会过多介绍深度学习的理论，直接从代码层面来实现全连接层、激活函数和SGD优化器，搭建一个简单的全连接模型，并且以一个线性回归示例，验证模型的效果。

二、目标

本文以学习为目的，以f(x) = sin(x)为目标函数，建立神经网络模型来拟合目标曲线。

数据如下图所示：

拟合结果如下图所示：

本文涉及的参考资料：

1、全连接层前向传播和梯度计算
2、动量梯度下降
3、ReLU

三、实现思路

在深度学习框架中，数据都是以tensor的形式进行计算，这里为了简单，数据的输入和输入都是以numpy.ndarray的格式传输。
本小节内容包含了相关类的实现。

1、tensor和初始化

tensor包含data和grad，保存data和对应的梯度数据。

# 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、Layer

这里实现了全连接层Linear和ReLU激活函数。
1、全连接层前向传播和梯度计算
2、ReLU

# 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

将层串联起来，实现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、优化器

实现了SGD优化器（带动量）
1、动量梯度下降

# 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、损失函数

实现了MSE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

# 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)

四、线性回归示例

本小节的目标是使用上面完成的类，搭建一个简单的模型，并且实现线性拟合的过程。

1、提取数据

# 提取训练数据（这里是一个预先生成的f(x) = sin(x) + noise的数据）
!unzip -oq ~/data/data119921/sin_data.zip

2、查看数据分布

# 绘制原始数据图像
import matplotlib.pyplot as plt
%matplotlib inline

x_path = "x.npy"
y_path = "y.npy"

X = np.load(x_path)
Y = np.load(y_path)

plt.scatter(X, Y)

<matplotlib.collections.PathCollection at 0x7f0b843ee410>

3、搭建模型，设置超参数

epoches = 1000
batch_size = 4
learning_rate = 0.01
weight_decay = 0.0
train_number = 100 # 选择的训练数据数量，总共200，这里仅挑选一部分训练，否则数据太多过拟合看不出来

# 继承之前定义的Dataset，定义一个简单的Dataset
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]

# 搭建一个简单的模型
model = Sequential(
    Linear(1, 16, name='linear1'),
    ReLU(name='relu1'),
    Linear(16, 64, name='linear2'),
    ReLU(name='relu2'),
    Linear(64, 16, name='linear2'),
    ReLU(name='relu3'),
    Linear(16, 1, name='linear2'),
)
opt = SGD(parameters=model.parameters(), learning_rate=learning_rate, weight_decay=weight_decay, decay_type='l2')
loss_fn = MSE()

print(model)

Sequential:
linear layer, weight shape: (1, 16), bias shape: (1, 16)
relu1
linear layer, weight shape: (16, 64), bias shape: (1, 64)
relu2
linear layer, weight shape: (64, 16), bias shape: (1, 16)
relu3
linear layer, weight shape: (16, 1), bias shape: (1, 1)

4、训练

# 挑选部分数据训练，数据分布图绘制
indexes = np.arange(X.shape[0])
train_indexes = np.random.choice(indexes, train_number)
X = X[train_indexes]
Y = Y[train_indexes]
plt.scatter(X, Y)

<matplotlib.collections.PathCollection at 0x7f0b843ee3d0>

# 构建dataset和dataloader，开始训练
train_dataset = LinearDataset(X, Y)
train_dataloader = DataLoader(train_dataset, shuffle=True, batch_size=batch_size, drop_last=True)

for epoch in range(1, epoches):
    losses = []
    for x, y in train_dataloader:
        pred = model(x)
        loss = loss_fn(pred, y)
        losses.append(loss)

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

        opt.step()
        opt.clear_grad()
    print("epoch: {}. loss: {}".format(epoch, np.array(losses).mean()))

5、验证效果

# 训练结束，生成比较密集的点，绘制曲线查看模型效果
val_number = 500  # 验证点的个数

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.plot(X_val, Y_val, color='green', label='true')
plt.plot(X_val, all_pred, color='red', label='predict')
plt.legend()
plt.show()

# 打印模型权重
for g in model.graphs:
    try:
        print(g.name, "  weights: ", g.weights.data)
        print(g.name, "  bias: ", g.bias.data)
    except:
        # relu 没有参数
        pass

linear1   weights:  [[-3.39505853e-01  1.82815127e-01  3.41670755e-04  4.51586227e-01
   1.53022752e-01  4.51654343e-01 -3.72304150e-01  2.76332489e-01
  -1.38630030e-01 -9.45745032e-02 -2.80274033e-02  3.21501804e-01
   5.63259058e-04  3.02464553e-01  4.12779030e-01 -5.02756806e-01]]
linear1   bias:  [[-0.27559667  0.25060406 -0.00106264  0.25735576  0.15667835 -0.29261948
  -0.22068097  0.34773508 -0.06852324 -0.06383495 -0.00121021 -0.20815822
  -0.00207523  0.41023867 -0.14955467 -0.27659916]]
linear2   weights:  [[ 0.00802045 -0.01371165 -0.02685921 ...  0.02362987 -0.00621883
  -0.02786108]
 [-0.00452856 -0.00503155  0.04844489 ... -0.00561967  0.0025664
   0.00678349]
 [-0.00615242 -0.00192324  0.00115901 ... -0.00903875  0.00314179
  -0.01176954]
 ...
 [-0.00625044 -0.00103386  0.12367338 ... -0.0048607  -0.01353281
  -0.00611369]
 [ 0.00415564 -0.01963549  0.12541482 ...  0.01609308 -0.00733272
  -0.01286687]
 [ 0.03625054 -0.03395289  0.00589992 ...  0.02610544  0.00226727
  -0.01638553]]
linear2   bias:  [[-5.80917490e-02  5.01950195e-02 -2.29461260e-01  8.53813886e-01
   0.00000000e+00  5.57391247e-03  0.00000000e+00 -1.83248948e-01
   2.48837634e-01 -1.11183245e-01 -3.48240873e-01 -4.50779643e-02
  -1.28934035e-02  1.12025269e-01  3.79346683e-01  1.35687659e-01
   1.21481402e-01 -8.63197975e-02  1.85562909e-03 -2.77419326e-01
   7.55994579e-01  0.00000000e+00  0.00000000e+00 -1.42549552e-01
   2.88624148e-01 -1.72867527e-01  1.70860914e-01  2.40404679e-01
  -8.84156448e-02 -8.03972453e-02 -2.88965818e-01  9.83171145e-02
   0.00000000e+00  3.17059611e-01 -1.04739710e-01 -1.16109983e-03
   4.49676180e-01  5.43205541e-01  0.00000000e+00  2.16567560e-01
   2.66316055e-01  6.52556933e-02  4.21085572e-01 -1.75897451e-01
   1.70725040e-01  4.57763929e-01  5.90660615e-02  0.00000000e+00
   2.24770074e-01  4.92650106e-01  3.92872747e-01 -1.09088249e-03
   3.87059634e-01  1.32970903e-01 -8.24098597e-04  6.95018101e-01
  -2.67006851e-01 -3.10753157e-03  0.00000000e+00 -4.12923279e-02
  -1.75980184e-02 -4.42488935e-02  0.00000000e+00  4.16345087e-02]]
linear2   weights:  [[-0.00030464 -0.01316401 -0.00232969 ... -0.00735164  0.02166657
   0.00125336]
 [ 0.01472182 -0.01163708  0.00238465 ... -0.01319246  0.02852089
   0.00038934]
 [-0.01692646 -0.00244135 -0.00771588 ...  0.05227914 -0.14514223
   0.01070569]
 ...
 [ 0.00063122  0.00322329 -0.00613279 ... -0.00629652  0.02223584
  -0.00575858]
 [ 0.00141447  0.00212887 -0.01180259 ... -0.00223564  0.00415438
   0.00539367]
 [-0.00545804 -0.01563078  0.00269196 ... -0.01580513  0.0176982
  -0.00294621]]
linear2   bias:  [[ 2.24308947e-02  4.52850508e-02 -6.85257176e-04  4.71155020e-01
  -2.25538467e-02  4.94627319e-01 -3.26158083e-04  8.32043208e-01
   1.27510619e+00 -1.93002987e-02  8.65411471e-01 -1.87107957e-02
  -2.39317258e-02  3.22988423e-02  8.12310457e-01  1.43143661e-02]]
linear2   weights:  [[-0.02986098]
 [-0.07522691]
 [-0.00509935]
 [-0.79988172]
 [-0.1247629 ]
 [-0.83384197]
 [ 0.0070327 ]
 [ 0.916285  ]
 [ 1.40066481]
 [ 0.02468298]
 [-1.16986177]
 [-0.17584702]
 [-0.22990252]
 [-0.18561223]
 [ 0.89437478]
 [-0.02239539]]
linear2   bias:  [[0.47546356]]

五、参考

本项目github地址：https://github.com/justld/LDDL

参考：
1、PaddlePaddle

六、致谢

感谢 萌新~、曾焯淇 两位同学的宝贵建议。

七、其他

深度学习理论简单，实现也很简单，但是想要能够实现组网搭建模型，却并不是一件容易的事情，需要注意的细节太多了，实现上述的功能耗时一周左右，但是代码跑通的时候真的很开心。

其实在参加黑客马拉松活动的时候，就有自己实现一个简单模型的想法，但是因为事情太多，一直没有实现。这次抽出2周时间参加飞桨论文复现挑战赛，还剩余几天时间，终于把这个事情给解决了。

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