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前言

在TensorFlow中，微分是个非常重要的概念。它们分别用于自动求导（计算梯度）和高效地处理数据。下面我将分别介绍这两个主题。

微分(Automatic Differentiation)
TensorFlow提供了强大的自动求导功能，这对于训练机器学习模型尤其重要，因为需要通过反向传播算法来更新模型参数。自动求导允许 TensorFlow 自动计算损失函数相对于模型参数的梯度，从而简化了模型训练过程中的优化步骤。

使用 tf.GradientTape
tf.GradientTape是TensorFlow 2.x 中实现自动求导的核心工具。它会记录在其上下文内的所有操作，并可以在需要时计算这些操作相对于输入张量的梯度。

示例

手动微分实现

from tensorflow import keras
import numpy as np
import pandas
import matplotlib.pyplot as plt# 3 * x^2 + 2 * x - 1
# 手动微分
def f(x):return 3. * x ** 2 + 2. * x - 1.# 近似求导
def approximate_derivative(f, x, eps=1e-3):return (f(x + eps) - f(x - eps)) / (2. * eps)print(approximate_derivative(f, 1.))

结果如下：

7.999999999999119

两个未知数, 求偏导

from tensorflow import keras
import numpy as np
import pandas
import matplotlib.pyplot as plt# 两个未知数, 求偏导
def g(x1, x2):return (x1 + 5) * (x2 ** 2)# 近似求导
def approximate_derivative(f, x, eps=1e-3):return (f(x + eps) - f(x - eps)) / (2. * eps)# 分别求g对x1,和x2的偏导.
def approximate_gradient(g, x1, x2, eps=1e-3):dg_x1 = approximate_derivative(lambda x: g(x, x2), x1, eps)dg_x2 = approximate_derivative(lambda x: g(x1, x), x2, eps)return dg_x1, dg_x2print(approximate_gradient(g, 2, 3))

结果如下：

(8.999999999993236, 41.999999999994486)

tf.GradientTape常量求导

from tensorflow import keras
import numpy as np
import pandas
import matplotlib.pyplot as plt
import tensorflow as tf x1 = tf.Variable(2.0)
x2 = tf.Variable(3.0)def g(x1, x2):return (x1 + 5) * (x2 ** 2)with tf.GradientTape() as tape:z = g(x1, x2)dz_x1x2 = tape.gradient(z, [x1, x2])
print(dz_x1x2)

结果如下：

[<tf.Tensor: shape=(), dtype=float32, numpy=9.0>, <tf.Tensor: shape=(), dtype=float32, numpy=42.0>]

tf.GradientTape二阶导数

from tensorflow import keras
import numpy as np
import pandas
import matplotlib.pyplot as plt
import tensorflow as tf x1 = tf.Variable(2.0)
x2 = tf.Variable(3.0)def g(x1, x2):return (x1 + 5) * (x2 ** 2)# 二阶导数
# 嵌套tf.GradientTape
x1 = tf.Variable(2.0)
x2 = tf.Variable(3.0)with tf.GradientTape(persistent=True) as outer_tape:with tf.GradientTape(persistent=True) as inner_tape:z = g(x1, x2)# 一阶导    inner_grads = inner_tape.gradient(z, [x1, x2])
# 对一阶导的结果再求导
outer_grads = [outer_tape.gradient(inner_grad, [x1, x2]) for inner_grad in inner_grads]
print(outer_grads)
del inner_tape
del outer_tape

结果如下：

[[None, <tf.Tensor: shape=(), dtype=float32, numpy=6.0>], [<tf.Tensor: shape=(), dtype=float32, numpy=6.0>, <tf.Tensor: shape=(), dtype=float32, numpy=14.0>]]

tf.GradientTape实现梯度下降

from tensorflow import keras
import numpy as np
import pandas
import matplotlib.pyplot as plt
import tensorflow as tf def f(x):return 3. * x ** 2 + 2. * x - 1.# 使用tf.GradientTape实现梯度下降
learing_rate = 0.1
x = tf.Variable(0.0)for _ in range(100):with tf.GradientTape() as tape:z = f(x)dz_dx = tape.gradient(z, x)x.assign_sub(learing_rate * dz_dx) # x -= learning_rate * dz_dx
print(x)

结果如下：

<tf.Variable 'Variable:0' shape=() dtype=float32, numpy=-0.3333333>

结合optimizer实现梯度下降

from tensorflow import keras
import numpy as np
import pandas
import matplotlib.pyplot as plt
import tensorflow as tf def f(x):return 3. * x ** 2 + 2. * x - 1.from tensorflow import keras# 结合optimizer去实现梯度下降
learing_rate = 0.1
x = tf.Variable(0.0)
optimizer = keras.optimizers.SGD(lr=learing_rate)for _ in range(100):with tf.GradientTape() as tape:z = f(x)dz_dx = tape.gradient(z, x)
#     x.assign_sub(learing_rate * dz_dx) # x -= learning_rate * dz_dxoptimizer.apply_gradients([(dz_dx, x)])
print(x)

结果如下

<tf.Variable 'Variable:0' shape=() dtype=float32, numpy=-0.3333333>