常常会面临这样的需要：为某种特定的情形寻找最佳模型。“最佳”常常会被解读为某种类似于“最小化模型残差”或者“最大化数据的可能性”。换句话说，它代表了优化某种问题的解决方案。
这意味着我们需要解决一连串的最优化问题。尤其是，我们需要从零开始解决问题。我们
采用的方法是一种叫作梯度下降（gradient descent）的技术，适合从零开始逐步解决问题。
用梯度下降法计算最小点
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通过差商来求近似导数
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差商近似值的拟合度
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from collections import Counter
from linear_algebra import distance, vector_subtract, scalar_multiply
from functools import reduce
import math, random
import matplotlib.pyplot as plt
from pylab import mpl
mpl.rcParams['font.sans-serif'] = ['SimHei']
def sum_of_squares(v):
"""computes the sum of squared elements in v"""
return sum(v_i ** 2 for v_i in v)
def difference_quotient(f, x, h):
return (f(x + h) - f(x)) / h
def plot_estimated_derivative():
def square(x):
return x * x
def derivative(x):
return 2 * x
derivative_estimate = lambda x: difference_quotient(square, x, h=0.00001)
# plot to show they're basically the same
import matplotlib.pyplot as plt
x = range(-10,10)
plt.plot(x, list(map(derivative, x)), 'rx') # red x
plt.plot(x, list(map(derivative_estimate, x)), 'b+') # blue +
plt.show() # purple *, hopefully
def partial_difference_quotient(f, v, i, h):
# add h to just the i-th element of v
w = [v_j + (h if j == i else 0)
for j, v_j in enumerate(v)]
return (f(w) - f(v)) / h
def estimate_gradient(f, v, h=0.00001):
return [partial_difference_quotient(f, v, i, h)
for i, _ in enumerate(v)]
def step(v, direction, step_size):
"""move step_size in the direction from v"""
return [v_i + step_size * direction_i
for v_i, direction_i in zip(v, direction)]
def sum_of_squares_gradient(v):
return [2 * v_i for v_i in v]
def safe(f):
"""define a new function that wraps f and return it"""
def safe_f(*args, **kwargs):
try:
return f(*args, **kwargs)
except:
return float('inf') # this means "infinity" in Python
return safe_f
#
#
# minimize / maximize batch
#
#
def minimize_batch(target_fn, gradient_fn, theta_0, tolerance=0.000001):
"""use gradient descent to find theta that minimizes target function"""
step_sizes = [100, 10, 1, 0.1, 0.01, 0.001, 0.0001, 0.00001]
theta = theta_0 # set theta to initial value
target_fn = safe(target_fn) # safe version of target_fn
value = target_fn(theta) # value we're minimizing
while True:
gradient = gradient_fn(theta)
next_thetas = [step(theta, gradient, -step_size)
for step_size in step_sizes]
# choose the one that minimizes the error function
next_theta = min(next_thetas, key=target_fn)
next_value = target_fn(next_theta)
# stop if we're "converging"
if abs(value - next_value) < tolerance:
return theta
else:
theta, value = next_theta, next_value
def negate(f):
"""return a function that for any input x returns -f(x)"""
return lambda *args, **kwargs: -f(*args, **kwargs)
def negate_all(f):
"""the same when f returns a list of numbers"""
return lambda *args, **kwargs: [-y for y in f(*args, **kwargs)]
def maximize_batch(target_fn, gradient_fn, theta_0, tolerance=0.000001):
return minimize_batch(negate(target_fn),
negate_all(gradient_fn),
theta_0,
tolerance)
#
# minimize / maximize stochastic
#
def in_random_order(data):
"""generator that returns the elements of data in random order"""
indexes = [i for i, _ in enumerate(data)] # create a list of indexes
random.shuffle(indexes) # shuffle them
for i in indexes: # return the data in that order
yield data[i]
def minimize_stochastic(target_fn, gradient_fn, x, y, theta_0, alpha_0=0.01):
data = list(zip(x, y))
theta = theta_0 # initial guess
alpha = alpha_0 # initial step size
min_theta, min_value = None, float("inf") # the minimum so far
iterations_with_no_improvement = 0
# if we ever go 100 iterations with no improvement, stop
while iterations_with_no_improvement < 100:
value = sum( target_fn(x_i, y_i, theta) for x_i, y_i in data )
if value < min_value:
# if we've found a new minimum, remember it
# and go back to the original step size
min_theta, min_value = theta, value
iterations_with_no_improvement = 0
alpha = alpha_0
else:
# otherwise we're not improving, so try shrinking the step size
iterations_with_no_improvement += 1
alpha *= 0.9
# and take a gradient step for each of the data points
for x_i, y_i in in_random_order(data):
gradient_i = gradient_fn(x_i, y_i, theta)
theta = vector_subtract(theta, scalar_multiply(alpha, gradient_i))
return min_theta
def maximize_stochastic(target_fn, gradient_fn, x, y, theta_0, alpha_0=0.01):
return minimize_stochastic(negate(target_fn),
negate_all(gradient_fn),
x, y, theta_0, alpha_0)
if __name__ == "__main__":
print("using the gradient")
v = [random.randint(-10,10) for i in range(3)]
tolerance = 0.0000001
while True:
#print v, sum_of_squares(v)
gradient = sum_of_squares_gradient(v) # compute the gradient at v
next_v = step(v, gradient, -0.01) # take a negative gradient step
if distance(next_v, v) < tolerance: # stop if we're converging
break
v = next_v # continue if we're not
print("minimum v", v)
print("minimum value", sum_of_squares(v))
print()
print("using minimize_batch")
v = [random.randint(-10,10) for i in range(3)]
v = minimize_batch(sum_of_squares, sum_of_squares_gradient, v)
print("minimum v", v)
print("minimum value", sum_of_squares(v))
plot_estimated_derivative()
执行结果
using the gradient
minimum v [1.6985844650062676e-06, 1.1323896433375117e-06, -4.529558573350047e-06]
minimum value 2.4684396358507597e-11
using minimize_batch
minimum v [-0.000519229685853483, -0.001038459371706966, -0.001038459371706966]
minimum value 2.42639520004356e-06
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