本文将首先推导SVM的主要公式，接着基于Platt-SMO算法，从零开始实现支持多种核函数的SVM，然后基于One-Versus-One策略实现多分类，最后在MNIST和CIFAR-10数据集上进行性能测试。从零开始实现支持向量机
这篇文章诞生于机器学习课程无聊的大作业，既然已经为此浪费了不少时间，不妨再多花点时间写一篇文章，借此记录一下实现过程。支持向量机的数学形式简约而直观，但一旦涉及具体实现，各种问题就会接踵而来。在本文中，笔者将首先推导SVM的主要公式，接着基于Platt-SMO算法，从零开始实现支持多种核函数的SVM，然后基于One-Versus-One策略实现多分类，最后在MNIST和CIFAR-10数据集上进行性能测试
数学推导
基本形式
这是一个二次规划问题，我们可以采用梯度下降或者坐标下降等方法求解。
核函数与核技巧
Platt-SMO算法
算法实现
核函数
我们对上述四种核函数进行实现，这里将核函数封装成类，通过实现__call__方法，使其实例可以像函数一样被调用
线性核代码如下
class LinearKernel(object):
def __init__(self):
self.name = 'linear'
def __call__(self, X, y):
return X @ y.T
多项式核代码如下
class PolynomialKernel(object):
def __init__(self, gamma=1.0, degree=3):
self.name = 'polynomial'
self.gamma = gamma
self.degree = degree
def __call__(self, X, y):
return np.power(self.gamma * (X @ y.T) + 1, self.degree)
高斯核代码如下
class GaussianKernel(object):
def __init__(self, gamma=1.0):
self.name = 'gaussian'
self.gamma = gamma
def __call__(self, X, y):
return np.exp(-self.gamma * np.sum(np.square(X - y), axis=1))
Sigmoid核代码如下
class SigmoidKernel(object):
def __init__(self, gamma=1.0, bias=0.0):
self.name = 'sigmoid'
self.gamma = gamma
self.bias = bias
def __call__(self, X, y):
return np.tanh(self.gamma * (X @ y.T) + self.bias)
另外，我们定义一个工具函数，方便核函数的创建
def CreateKernel(entry):
if entry['name'] == 'linear':
return LinearKernel()
elif entry['name'] == 'polynomial':
return PolynomialKernel(entry['gamma'], entry['degree'])
elif entry['name'] == 'gaussian':
return GaussianKernel(entry['gamma'])
elif entry['name'] == 'sigmoid':
return SigmoidKernel(entry['gamma'], entry['bias'])
raise AttributeError('invalid kernel')
支持向量机
参考scikit-learn的封装，我们定义一个类，提供fit和predict两种方法，参数包括最大迭代次数、惩罚系数、误差精度和核函数类型，利用私有函数实现 和 的选择和单步更新，对于线性核，我们提供weight属性，用于获取线性核的分类超平面参数，除了一些简化以外，代码基本按照Platt-SMO算法进行实现
class SupportVectorMachine(object):
def __init__(self, iteration=100, penalty=1.0, epsilon=1e-6, kernel=None):
self.iteration = iteration
self.penalty = penalty
self.epsilon = epsilon
if kernel is None:
kernel = {'name': 'linear'}
self.kernel = CreateKernel(kernel)
def __compute_w(self):
return (self.a * self.y) @ self.X
def __compute_e(self, i):
return (self.a * self.y) @ self.K[:, i] + self.b - self.y[i]
def __select_j(self, i):
j = np.random.randint(1, self.m)
return j if j > i else j - 1
def __step_forward(self, i):
e_i = self.__compute_e(i)
if ((self.a[i] > 0) and (e_i * self.y[i] > self.epsilon)) or ((self.a[i] < self.penalty) and (e_i * self.y[i] < -self.epsilon)):
j = self.__select_j(i)
e_j = self.__compute_e(j)
a_i, a_j = np.copy(self.a[i]), np.copy(self.a[j])
if self.y[i] == self.y[j]:
L = max(0, a_i + a_j - self.penalty)
H = min(self.penalty, a_i + a_j)
else:
L = max(0, a_j - a_i)
H = min(self.penalty, self.penalty + a_j - a_i)
if L == H:
return False
d = 2 * self.K[i, j] - self.K[i, i] - self.K[j, j]
if d >= 0:
return False
self.a[j] = np.clip(a_j - self.y[j] * (e_i - e_j) / d, L, H)
if np.abs(self.a[j] - a_j) < self.epsilon:
return False
self.a[i] = a_i + self.y[i] * self.y[j] * (a_j - self.a[j])
b_i = self.b - e_i - self.y[i] * self.K[i, i] * (self.a[i] - a_i) - self.y[j] * self.K[j, i] * (self.a[j] - a_j)
b_j = self.b - e_j - self.y[i] * self.K[i, j] * (self.a[i] - a_i) - self.y[j] * self.K[j, j] * (self.a[j] - a_j)
if 0 < self.a[i] < self.penalty:
self.b = b_i
elif 0 < self.a[j] < self.penalty:
self.b = b_j
else:
self.b = (b_i + b_j) / 2
return True
return False
def setup(self, X, y):
self.X, self.y = X, y
self.m, self.n = X.shape
self.b = 0.0
self.a = np.zeros(self.m)
self.K = np.zeros((self.m, self.m))
for i in range(self.m):
self.K[:, i] = self.kernel(X, X[i, :])
def fit(self, X, y):
self.setup(X, y)
entire = True
for _ in range(self.iteration):
change = 0
if entire:
for i in range(self.m):
change += self.__step_forward(i)
else:
index = np.nonzero((0 < self.a) * (self.a < self.penalty))[0]
for i in index:
change += self.__step_forward(i)
if entire:
entire = False
elif change == 0:
entire = True
def predict(self, X):
m = X.shape[0]
y = np.zeros(m)
for i in range(m):
y[i] = np.sign((self.a * self.y) @ self.kernel(self.X, X[i, :]) + self.b)
return y
@property
def weight(self):
if self.kernel.name != 'linear':
raise AttributeError('non-linear kernel')
return self.__compute_w(), self.b
多分类
基于One-Versus-One策略, 我们构造 个SVM, 其中 为类别数, 训练每个分类器时, 选取相应类别的样本作为训练集, 并将标签映射到 -1 和 1 , 在预测时, 用每个分类器的预测结果进行投票, 从而得到最终结果
我们采用与支持向量机完全相同的封装，提供fit和predict两种方法，使该类成为通用的分类模型
class SupportVectorClassifier(object):
def __init__(self, iteration=100, penalty=1.0, epsilon=1e-6, kernel=None):
self.iteration = iteration
self.penalty = penalty
self.epsilon = epsilon
self.kernel = kernel
self.classifier = []
def __build_model(self, y):
self.label = np.unique(y)
for i in range(len(self.label)):
for j in range(i+1, len(self.label)):
model = SupportVectorMachine(self.iteration, self.penalty, self.epsilon, self.kernel)
self.classifier.append((i, j, model))
def fit(self, X, y):
self.__build_model(y)
for i, j, model in tqdm(self.classifier):
index = np.where((y == self.label[i]) | (y == self.label[j]))[0]
X_ij, y_ij = X[index], np.where(y[index] == self.label[i], -1, 1)
model.fit(X_ij, y_ij)
def predict(self, X):
vote = np.zeros((X.shape[0], len(self.label)))
for i, j, model in tqdm(self.classifier):
y = model.predict(X)
vote[np.where(y == -1)[0], i] += 1
vote[np.where(y == 1)[0], j] += 1
return self.label[np.argmax(vote, axis=1)]
性能测试
首先，我们在二维平面上构造两组简单的正态分布数据，用于可视化支持向量机的分类效果，首先构造数据并训练模型
X = np.concatenate((np.random.randn(500, 2) - 2, np.random.randn(500, 2) + 2))
y = np.concatenate((np.ones(500), -np.ones(500)))
C = SupportVectorMachine(iteration=100)
C.fit(X, y)
w, b = C.weight
u = np.linspace(-3, 3, 100)
v = (-b - w[0] * u) / w[1]
然后根据模型参数绘制分类效果
plt.scatter(X[:500, 0], X[:500, 1], label='Positive')
plt.scatter(X[500:, 0], X[500:, 1], label='Negative')
plt.plot(u, v, label='Separation', c='g')
plt.xlabel('$x$')
plt.ylabel('$y$')
plt.title('Separation Sample')
plt.grid()
plt.legend()
plt.tight_layout()
plt.savefig('./figure/separation.png')
plt.show()
可以看到，我们实现的SVM可以很好地将两组数据分开
def MNIST(path, group='train'):
if group == 'train':
with gzip.open(os.path.join(path, 'train-images-idx3-ubyte.gz'), 'rb') as file:
image = np.frombuffer(file.read(), np.uint8, offset=16).reshape(-1, 1, 28, 28) / 255.0
with gzip.open(os.path.join(path, 'train-labels-idx1-ubyte.gz'), 'rb') as file:
label = np.frombuffer(file.read(), np.uint8, offset=8)
elif group == 'test':
with gzip.open(os.path.join(path, 't10k-images-idx3-ubyte.gz'), 'rb') as file:
image = np.frombuffer(file.read(), np.uint8, offset=16).reshape(-1, 1, 28, 28) / 255.0
with gzip.open(os.path.join(path, 't10k-labels-idx1-ubyte.gz'), 'rb') as file:
label = np.frombuffer(file.read(), np.uint8, offset=8)
remain = 500 if group == 'train' else 100
image_list, label_list = [], []
for value in range(10):
index = np.where(label == value)[0][:remain]
image_list.append(image[index])
label_list.append(label[index])
image, label = np.concatenate(image_list), np.concatenate(label_list)
index = np.random.permutation(len(label))
return image[index], label[index]
对于CIFAR10数据集，我们做同样的处理
def CIFAR10(path, group='train'):
if group == 'train':
image_list, label_list = [], []
for i in range(1, 6):
filename = os.path.join(path, 'data_batch_{}'.format(i))
with open(filename, 'rb') as file:
data = pickle.load(file, encoding='bytes')
image_list.append(np.array(data[b'data'], dtype=np.float32).reshape(-1, 3, 32, 32) / 255.0)
label_list.append(np.array(data[b'labels'], dtype=np.int32))
image, label = np.concatenate(image_list), np.concatenate(label_list)
elif group == 'test':
filename = os.path.join(path, 'test_batch')
with open(filename, 'rb') as file:
data = pickle.load(file, encoding='bytes')
image = np.array(data[b'data'], dtype=np.float32).reshape(-1, 3, 32, 32) / 255.0
label = np.array(data[b'labels'], dtype=np.int32)
remain = 500 if group == 'train' else 100
image_list, label_list = [], []
for value in range(10):
index = np.where(label == value)[0][:remain]
image_list.append(image[index])
label_list.append(label[index])
image, label = np.concatenate(image_list), np.concatenate(label_list)
index = np.random.permutation(len(label))
return image[index], label[index]
由于CIFAR10数据集较为困难，我们考虑利用CV方法进行特征提取，这里我们使用HOG特征提高分类效果，首先将彩色图像转换为灰度图像
def RGB2Gray(image):
image = 0.299 * image[0] + 0.587 * image[1] + 0.114 * image[2]
return image.reshape(1, *image.shape)
然后实现一个简单的HOG特征提取函数，这里我们没有实现区块重叠，对该函数进行改进应该可以进一步提高分类效果 whaosoft aiot http://143ai.com
def HOG(image, block=4, partition=8):
image = RGB2Gray(image).squeeze(axis=0)
height, width = image.shape
gradient = np.zeros((2, height, width), dtype=np.float32)
for i in range(1, height-1):
for j in range(1, width-1):
delta_x = image[i, j-1] - image[i, j+1]
delta_y = image[i+1, j] - image[i-1, j]
gradient[0, i, j] = np.sqrt(delta_x ** 2 + delta_y ** 2)
gradient[1, i, j] = np.degrees(np.arctan2(delta_y, delta_x))
if gradient[1, i, j] < 0:
gradient[1, i, j] += 180
unit = 360 / partition
vertical, horizontal = height // block, width // block
feature = np.zeros((vertical, horizontal, partition), dtype=np.float32)
for i in range(vertical):
for j in range(horizontal):
for k in range(block):
for l in range(block):
rho = gradient[0, i*block+k, j*block+l]
theta = gradient[1, i*block+k, j*block+l]
index = int(theta // unit)
feature[i, j, index] += rho
feature[i, j] /= np.linalg.norm(feature[i, j]) + 1e-6
return feature.reshape(-1)
基于这些工具函数，我们可以优雅地完成图像分类任务，对于MNIST数据集，一个基于线性核的分类示例如下
X_train, y_train = MNIST('./dataset/mnist_data/', group='train')
X_test, y_test = MNIST('./dataset/mnist_data/', group='test')
X_train, X_test = X_train.reshape(-1, 28*28), X_test.reshape(-1, 28*28)
model = SupportVectorClassifier(iteration=100, penalty=0.05)
model.fit(X_train, y_train)
p_train, p_test = model.predict(X_train), model.predict(X_test)
r_train, r_test = ComputeAccuracy(p_train, y_train), ComputeAccuracy(p_test, y_test)
print('Kernel: Linear, Train: {:.2%}, Test: {:.2%}'.format(r_train, r_test))
对于CIFAR10数据集，一个基于HOG特征和高斯核的分类示例如下
X_train, y_train = CIFAR10('./dataset/cifar-10-batches-py/', group='train')
X_test, y_test = CIFAR10('./dataset/cifar-10-batches-py/', group='test')
X_train, X_test = BatchHOG(X_train, partition=16), BatchHOG(X_test, partition=16)
kernel = {'name': 'gaussian', 'gamma': 0.03}
model = SupportVectorClassifier(iteration=100, kernel=kernel)
model.fit(X_train, y_train)
p_train, p_test = model.predict(X_train), model.predict(X_test)
r_train, r_test = ComputeAccuracy(p_train, y_train), ComputeAccuracy(p_test, y_test)
print('Kernel: Gaussian, Train: {:.2%}, Test: {:.2%}'.format(r_train, r_test))
经过测试，我们实现的SVM分类器在MNIST和CIFAR10数据集上的分类精度如下表所示
此外，我们对模型的收敛性和各个核函数的参数选择进行了测试，模型精度与迭代次数的关系如下图所示
上述结果揭示了各个参数对模型性能的影响，可以为调参提供一定的指导作用
写在最后
SVM从过去的炙手可热到如今的日薄西山，仅仅过去了十年的时间，无论是精度还是效率，SVM都完败于当下随处可见的神经网络，关于从零开始实现SVM的意义，我也感到迷茫，但这一过程或多或少改变了我对机器学习的认知，一个简洁优雅的多项式时间精确算法，也许只能满足理论研究者的洁癖，而优化复杂模型的近似算法，在工程上赢得了未来。