.. _sec_gd:
Gradiente descendente
=====================
Nesta seção, apresentaremos os conceitos básicos da descida gradiente.
Isso é breve por necessidade. Veja, por exemplo,
:cite:`Boyd.Vandenberghe.2004` para uma introdução detalhada à
otimização convexa. Embora o último raramente seja usado diretamente no
aprendizado profundo, uma compreensão da descida gradiente é a chave
para entender os algoritmos de descida gradiente estocásticos. Por
exemplo, o problema de otimização pode divergir devido a uma taxa de
aprendizado excessivamente grande. Este fenômeno já pode ser visto na
descida do gradiente. Da mesma forma, o pré-condicionamento é uma
técnica comum na descida de gradiente e é transportado para algoritmos
mais avançados. Comecemos com um caso especial simples.
Gradiente descendente em uma dimensão
-------------------------------------
A descida gradiente em uma dimensão é um excelente exemplo para explicar
por que o algoritmo de descida gradiente pode reduzir o valor da função
objetivo. Considere alguma função de valor real continuamente
diferenciável :math:`f: \mathbb{R} \rightarrow \mathbb{R}`. Usando uma
expansão de Taylor (:numref:`sec_single_variable_calculus`), obtemos
que
.. math:: f(x + \epsilon) = f(x) + \epsilon f'(x) + \mathcal{O}(\epsilon^2).
:label: gd-taylor
Ou seja, na primeira aproximação :math:`f(x+\epsilon)` é dado pelo valor
da função :math:`f(x)` e a primeira derivada :math:`f'(x)` em :math:`x`.
Não é irracional supor que, para pequenos :math:`\epsilon` movendo-se na
direção do gradiente negativo, :math:`f` diminuirá. Para manter as
coisas simples, escolhemos um tamanho de passo fixo :math:`\eta > 0` e
escolhemos :math:`\epsilon = -\eta f'(x)`. Conectando isso à expansão
Taylor acima, obtemos
.. math:: f(x - \eta f'(x)) = f(x) - \eta f'^2(x) + \mathcal{O}(\eta^2 f'^2(x)).
Se a derivada :math:`f'(x) \neq 0` não desaparecer, fazemos progresso,
pois :math:`\eta f'^2(x)>0`. Além disso, podemos sempre escolher
:math:`\eta` pequeno o suficiente para que os termos de ordem superior
se tornem irrelevantes. Daí chegamos a
.. math:: f(x - \eta f'(x)) \lessapprox f(x).
Isso significa que, se usarmos
.. math:: x \leftarrow x - \eta f'(x)
para iterar :math:`x`, o valor da função :math:`f(x)` pode diminuir.
Portanto, na descida do gradiente, primeiro escolhemos um valor inicial
:math:`x` e uma constante :math:`\eta > 0` e, em seguida, os usamos para
iterar continuamente :math:`x` até que a condição de parada seja
alcançada, por exemplo, quando a magnitude do gradiente :math:`|f'(x)|`
é pequeno o suficiente ou o número de iterações atingiu um determinado
valor.
Para simplificar, escolhemos a função objetivo :math:`f(x)=x^2` para
ilustrar como implementar a descida gradiente. Embora saibamos que
:math:`x=0` é a solução para minimizar :math:`f(x)`, ainda usamos esta
função simples para observar como :math:`x` muda. Como sempre, começamos
importando todos os módulos necessários.
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.. code:: python
%matplotlib inline
from mxnet import np, npx
from d2l import mxnet as d2l
npx.set_np()
f = lambda x: x**2 # Objective function
gradf = lambda x: 2 * x # Its derivative
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.. code:: python
%matplotlib inline
import numpy as np
import torch
from d2l import torch as d2l
f = lambda x: x**2 # Objective function
gradf = lambda x: 2 * x # Its derivative
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.. code:: python
%matplotlib inline
import numpy as np
import tensorflow as tf
from d2l import tensorflow as d2l
f = lambda x: x**2 # Objective function
gradf = lambda x: 2 * x # Its derivative
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.. code:: python
def gd(eta):
x = 10.0
results = [x]
for i in range(10):
x -= eta * gradf(x)
results.append(float(x))
print('epoch 10, x:', x)
return results
res = gd(0.2)
.. parsed-literal::
:class: output
epoch 10, x: 0.06046617599999997
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.. code:: python
def gd(eta):
x = 10.0
results = [x]
for i in range(10):
x -= eta * gradf(x)
results.append(float(x))
print('epoch 10, x:', x)
return results
res = gd(0.2)
.. parsed-literal::
:class: output
epoch 10, x: 0.06046617599999997
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.. code:: python
def gd(eta):
x = 10.0
results = [x]
for i in range(10):
x -= eta * gradf(x)
results.append(float(x))
print('epoch 10, x:', x)
return results
res = gd(0.2)
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:class: output
epoch 10, x: 0.06046617599999997
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.. code:: python
def show_trace(res):
n = max(abs(min(res)), abs(max(res)))
f_line = np.arange(-n, n, 0.01)
d2l.set_figsize()
d2l.plot([f_line, res], [[f(x) for x in f_line], [f(x) for x in res]],
'x', 'f(x)', fmts=['-', '-o'])
show_trace(res)
.. figure:: output_gd_79c039_27_0.svg
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.. code:: python
def show_trace(res):
n = max(abs(min(res)), abs(max(res)))
f_line = torch.arange(-n, n, 0.01)
d2l.set_figsize()
d2l.plot([f_line, res], [[f(x) for x in f_line], [f(x) for x in res]],
'x', 'f(x)', fmts=['-', '-o'])
show_trace(res)
.. figure:: output_gd_79c039_30_0.svg
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.. code:: python
def show_trace(res):
n = max(abs(min(res)), abs(max(res)))
f_line = tf.range(-n, n, 0.01)
d2l.set_figsize()
d2l.plot([f_line, res], [[f(x) for x in f_line], [f(x) for x in res]],
'x', 'f(x)', fmts=['-', '-o'])
show_trace(res)
.. figure:: output_gd_79c039_33_0.svg
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.. code:: python
show_trace(gd(0.05))
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:class: output
epoch 10, x: 3.4867844009999995
.. figure:: output_gd_79c039_39_1.svg
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.. code:: python
show_trace(gd(0.05))
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:class: output
epoch 10, x: 3.4867844009999995
.. figure:: output_gd_79c039_42_1.svg
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.. code:: python
show_trace(gd(0.05))
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:class: output
epoch 10, x: 3.4867844009999995
.. figure:: output_gd_79c039_45_1.svg
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.. code:: python
show_trace(gd(1.1))
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:class: output
epoch 10, x: 61.917364224000096
.. figure:: output_gd_79c039_51_1.svg
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.. code:: python
show_trace(gd(1.1))
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:class: output
epoch 10, x: 61.917364224000096
.. figure:: output_gd_79c039_54_1.svg
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.. code:: python
show_trace(gd(1.1))
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:class: output
epoch 10, x: 61.917364224000096
.. figure:: output_gd_79c039_57_1.svg
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.. code:: python
c = np.array(0.15 * np.pi)
f = lambda x: x * np.cos(c * x)
gradf = lambda x: np.cos(c * x) - c * x * np.sin(c * x)
show_trace(gd(2))
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:class: output
epoch 10, x: -1.5281651
.. figure:: output_gd_79c039_63_1.svg
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.. code:: python
c = torch.tensor(0.15 * np.pi)
f = lambda x: x * torch.cos(c * x)
gradf = lambda x: torch.cos(c * x) - c * x * torch.sin(c * x)
show_trace(gd(2))
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:class: output
epoch 10, x: tensor(-1.5282)
.. figure:: output_gd_79c039_66_1.svg
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c = tf.constant(0.15 * np.pi)
f = lambda x: x * tf.cos(c * x)
gradf = lambda x: tf.cos(c * x) - c * x * tf.sin(c * x)
show_trace(gd(2))
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:class: output
epoch 10, x: tf.Tensor(-1.5281651, shape=(), dtype=float32)
.. figure:: output_gd_79c039_69_1.svg
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def train_2d(trainer, steps=20): #@save
"""Optimize a 2-dim objective function with a customized trainer."""
# s1 and s2 are internal state variables and will
# be used later in the chapter
x1, x2, s1, s2 = -5, -2, 0, 0
results = [(x1, x2)]
for i in range(steps):
x1, x2, s1, s2 = trainer(x1, x2, s1, s2)
results.append((x1, x2))
return results
def show_trace_2d(f, results): #@save
"""Show the trace of 2D variables during optimization."""
d2l.set_figsize()
d2l.plt.plot(*zip(*results), '-o', color='#ff7f0e')
x1, x2 = np.meshgrid(np.arange(-5.5, 1.0, 0.1),
np.arange(-3.0, 1.0, 0.1))
d2l.plt.contour(x1, x2, f(x1, x2), colors='#1f77b4')
d2l.plt.xlabel('x1')
d2l.plt.ylabel('x2')
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.. code:: python
def train_2d(trainer, steps=20): #@save
"""Optimize a 2-dim objective function with a customized trainer."""
# s1 and s2 are internal state variables and will
# be used later in the chapter
x1, x2, s1, s2 = -5, -2, 0, 0
results = [(x1, x2)]
for i in range(steps):
x1, x2, s1, s2 = trainer(x1, x2, s1, s2)
results.append((x1, x2))
return results
def show_trace_2d(f, results): #@save
"""Show the trace of 2D variables during optimization."""
d2l.set_figsize()
d2l.plt.plot(*zip(*results), '-o', color='#ff7f0e')
x1, x2 = torch.meshgrid(torch.arange(-5.5, 1.0, 0.1),
torch.arange(-3.0, 1.0, 0.1))
d2l.plt.contour(x1, x2, f(x1, x2), colors='#1f77b4')
d2l.plt.xlabel('x1')
d2l.plt.ylabel('x2')
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.. code:: python
def train_2d(trainer, steps=20): #@save
"""Optimize a 2-dim objective function with a customized trainer."""
# s1 and s2 are internal state variables and will
# be used later in the chapter
x1, x2, s1, s2 = -5, -2, 0, 0
results = [(x1, x2)]
for i in range(steps):
x1, x2, s1, s2 = trainer(x1, x2, s1, s2)
results.append((x1, x2))
return results
def show_trace_2d(f, results): #@save
"""Show the trace of 2D variables during optimization."""
d2l.set_figsize()
d2l.plt.plot(*zip(*results), '-o', color='#ff7f0e')
x1, x2 = tf.meshgrid(tf.range(-5.5, 1.0, 0.1),
tf.range(-3.0, 1.0, 0.1))
d2l.plt.contour(x1, x2, f(x1, x2), colors='#1f77b4')
d2l.plt.xlabel('x1')
d2l.plt.ylabel('x2')
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f = lambda x1, x2: x1 ** 2 + 2 * x2 ** 2 # Objective
gradf = lambda x1, x2: (2 * x1, 4 * x2) # Gradient
def gd(x1, x2, s1, s2):
(g1, g2) = gradf(x1, x2) # Compute gradient
return (x1 - eta * g1, x2 - eta * g2, 0, 0) # Update variables
eta = 0.1
show_trace_2d(f, train_2d(gd))
.. figure:: output_gd_79c039_87_0.svg
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.. code:: python
f = lambda x1, x2: x1 ** 2 + 2 * x2 ** 2 # Objective
gradf = lambda x1, x2: (2 * x1, 4 * x2) # Gradient
def gd(x1, x2, s1, s2):
(g1, g2) = gradf(x1, x2) # Compute gradient
return (x1 - eta * g1, x2 - eta * g2, 0, 0) # Update variables
eta = 0.1
show_trace_2d(f, train_2d(gd))
.. figure:: output_gd_79c039_90_0.svg
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.. code:: python
f = lambda x1, x2: x1 ** 2 + 2 * x2 ** 2 # Objective
gradf = lambda x1, x2: (2 * x1, 4 * x2) # Gradient
def gd(x1, x2, s1, s2):
(g1, g2) = gradf(x1, x2) # Compute gradient
return (x1 - eta * g1, x2 - eta * g2, 0, 0) # Update variables
eta = 0.1
show_trace_2d(f, train_2d(gd))
.. figure:: output_gd_79c039_93_0.svg
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.. code:: python
c = np.array(0.5)
f = lambda x: np.cosh(c * x) # Objective
gradf = lambda x: c * np.sinh(c * x) # Derivative
hessf = lambda x: c**2 * np.cosh(c * x) # Hessian
def newton(eta=1):
x = 10.0
results = [x]
for i in range(10):
x -= eta * gradf(x) / hessf(x)
results.append(float(x))
print('epoch 10, x:', x)
return results
show_trace(newton())
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:class: output
epoch 10, x: 0.0
.. figure:: output_gd_79c039_99_1.svg
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.. code:: python
c = torch.tensor(0.5)
f = lambda x: torch.cosh(c * x) # Objective
gradf = lambda x: c * torch.sinh(c * x) # Derivative
hessf = lambda x: c**2 * torch.cosh(c * x) # Hessian
def newton(eta=1):
x = 10.0
results = [x]
for i in range(10):
x -= eta * gradf(x) / hessf(x)
results.append(float(x))
print('epoch 10, x:', x)
return results
show_trace(newton())
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:class: output
epoch 10, x: tensor(0.)
.. figure:: output_gd_79c039_102_1.svg
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.. code:: python
c = tf.constant(0.5)
f = lambda x: tf.cosh(c * x) # Objective
gradf = lambda x: c * tf.sinh(c * x) # Derivative
hessf = lambda x: c**2 * tf.cosh(c * x) # Hessian
def newton(eta=1):
x = 10.0
results = [x]
for i in range(10):
x -= eta * gradf(x) / hessf(x)
results.append(float(x))
print('epoch 10, x:', x)
return results
show_trace(newton())
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:class: output
epoch 10, x: tf.Tensor(0.0, shape=(), dtype=float32)
.. figure:: output_gd_79c039_105_1.svg
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.. code:: python
c = np.array(0.15 * np.pi)
f = lambda x: x * np.cos(c * x)
gradf = lambda x: np.cos(c * x) - c * x * np.sin(c * x)
hessf = lambda x: - 2 * c * np.sin(c * x) - x * c**2 * np.cos(c * x)
show_trace(newton())
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:class: output
epoch 10, x: 26.834133
.. figure:: output_gd_79c039_111_1.svg
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.. code:: python
c = torch.tensor(0.15 * np.pi)
f = lambda x: x * torch.cos(c * x)
gradf = lambda x: torch.cos(c * x) - c * x * torch.sin(c * x)
hessf = lambda x: - 2 * c * torch.sin(c * x) - x * c**2 * torch.cos(c * x)
show_trace(newton())
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:class: output
epoch 10, x: tensor(26.8341)
.. figure:: output_gd_79c039_114_1.svg
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c = tf.constant(0.15 * np.pi)
f = lambda x: x * tf.cos(c * x)
gradf = lambda x: tf.cos(c * x) - c * x * tf.sin(c * x)
hessf = lambda x: - 2 * c * tf.sin(c * x) - x * c**2 * tf.cos(c * x)
show_trace(newton())
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:class: output
epoch 10, x: tf.Tensor(26.834133, shape=(), dtype=float32)
.. figure:: output_gd_79c039_117_1.svg
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.. code:: python
show_trace(newton(0.5))
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:class: output
epoch 10, x: 7.26986
.. figure:: output_gd_79c039_123_1.svg
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.. code:: python
show_trace(newton(0.5))
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:class: output
epoch 10, x: tensor(7.2699)
.. figure:: output_gd_79c039_126_1.svg
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show_trace(newton(0.5))
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:class: output
epoch 10, x: tf.Tensor(7.26986, shape=(), dtype=float32)
.. figure:: output_gd_79c039_129_1.svg
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