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python人工智能自定義求導(dǎo)tf_diffs詳解

 更新時間:2022年07月29日 15:11:35   作者:plum_blossom  
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自定義求導(dǎo):(近似求導(dǎo)數(shù)的方法)

讓x向左移動eps得到一個點,向右移動eps得到一個點,這兩個點形成一條直線,這個點的斜率就是x這個位置的近似導(dǎo)數(shù)。

eps足夠小,導(dǎo)數(shù)就足夠真。

def f(x):
    return 3. * x ** 2 + 2. * x - 1
def approximate_derivative(f, x, eps=1e-3):
    return (f(x + eps) - f(x - eps)) / (2. * eps)
print(approximate_derivative(f, 1.))

運行結(jié)果:

7.999999999999119

多元函數(shù)的求導(dǎo)

def g(x1, x2):
    return (x1 + 5) * (x2 ** 2)
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_x2
print(approximate_gradient(g, 2., 3.))

運行結(jié)果:

(8.999999999993236, 41.999999999994486)

在tensorflow中的求導(dǎo)

x1 = tf.Variable(2.0)
x2 = tf.Variable(3.0)
with tf.GradientTape() as tape:
    z = g(x1, x2)
dz_x1 = tape.gradient(z, x1)
print(dz_x1)

運行結(jié)果:

tf.Tensor(9.0, shape=(), dtype=float32)

但是tf.GradientTape()只能使用一次,使用一次之后就會被消解

try:
    dz_x2 = tape.gradient(z, x2)
except RuntimeError as ex:
    print(ex)

運行結(jié)果:

A non-persistent GradientTape can only be used to compute one set of gradients (or jacobians)

解決辦法:設(shè)置persistent = True,記住最后要把tape刪除掉

x1 = tf.Variable(2.0)
x2 = tf.Variable(3.0)
with tf.GradientTape(persistent = True) as tape:
    z = g(x1, x2)
dz_x1 = tape.gradient(z, x1)
dz_x2 = tape.gradient(z, x2)
print(dz_x1, dz_x2)
del tape

運行結(jié)果:

tf.Tensor(9.0, shape=(), dtype=float32) tf.Tensor(42.0, shape=(), dtype=float32)

使用tf.GradientTape()

同時求x1,x2的偏導(dǎo)

x1 = tf.Variable(2.0)
x2 = tf.Variable(3.0)
with tf.GradientTape() as tape:
    z = g(x1, x2)
dz_x1x2 = tape.gradient(z, [x1, x2])
print(dz_x1x2)

運行結(jié)果:

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

對常量求偏導(dǎo)

x1 = tf.constant(2.0)
x2 = tf.constant(3.0)
with tf.GradientTape() as tape:
    z = g(x1, x2)
dz_x1x2 = tape.gradient(z, [x1, x2])
print(dz_x1x2)

運行結(jié)果:

[None, None]

可以使用watch函數(shù)關(guān)注常量上的導(dǎo)數(shù)

x1 = tf.constant(2.0)
x2 = tf.constant(3.0)
with tf.GradientTape() as tape:
    tape.watch(x1)
    tape.watch(x2)
    z = g(x1, x2)
dz_x1x2 = tape.gradient(z, [x1, x2])
print(dz_x1x2)

運行結(jié)果:

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

也可以使用兩個目標函數(shù)對一個變量求導(dǎo):

x = tf.Variable(5.0)
with tf.GradientTape() as tape:
    z1 = 3 * x
    z2 = x ** 2
tape.gradient([z1, z2], x)

運行結(jié)果:

<tf.Tensor: shape=(), dtype=float32, numpy=13.0>

結(jié)果13是z1對x的導(dǎo)數(shù)加上z2對于x的導(dǎo)數(shù)

求二階導(dǎo)數(shù)的方法

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)
    inner_grads = inner_tape.gradient(z, [x1, x2])
outer_grads = [outer_tape.gradient(inner_grad, [x1, x2])
               for inner_grad in inner_grads]
print(outer_grads)
del inner_tape
del outer_tape

運行結(jié)果:

[[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>]]

結(jié)果是一個2x2的矩陣,左上角是z對x1的二階導(dǎo)數(shù),右上角是z先對x1求導(dǎo),在對x2求導(dǎo)

左下角是z先對x2求導(dǎo),在對x1求導(dǎo),右下角是z對x2的二階導(dǎo)數(shù)

學(xué)會自定義求導(dǎo)就可以模擬梯度下降法了,梯度下降就是求導(dǎo),再在導(dǎo)數(shù)的位置前進一點點 模擬梯度下降法:

learning_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(learning_rate * dz_dx)
print(x)

運行結(jié)果:

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

結(jié)合optimizers進行梯度下降法

learning_rate = 0.1
x = tf.Variable(0.0)
optimizer = keras.optimizers.SGD(lr = learning_rate)
for _ in range(100):
    with tf.GradientTape() as tape:
        z = f(x)
    dz_dx = tape.gradient(z, x)
    optimizer.apply_gradients([(dz_dx, x)])
print(x)

運行結(jié)果:

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

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