来自：http://deeplearning.net/software/theano/tutorial/loop.html

loop

一、Scan

• 一个递归的通常的形式，可以用来作为循环语句。
• 约间和映射（在第一个（leading，个人翻译成第一个）维度上进行循环）是scan的特殊情况
• 沿着一些输入序列scan一个函数，然后在每个时间步上生成一个输出。
• 该函数可以查看函数的前K个时间步的结果。
• sum() 可以通过在一个列表上使用 z + x(i) 函数（初始化为Z=0）来得到结果。
• 通常来说，一个for循环可以表示成一个scan（）操作，而且scan是与theano的循环联系最紧密的。
• 使用scan而不是for循环的优势：
  • 迭代的次数是符号graph的一部分。
  • 最小化GPU的迁移（如果用到GPU的话）。
  • 通过连续的步骤计算梯度。
  • 比python中使用theano编译后的for循环稍微快一点。
  • 通过检测实际用到的内存的数量，来降低总的内存使用情况。

所有的文档可以查看库对应的： Scan.

1.1 Scan 例子: 逐元素计算 tanh(x(t).dot(W) + b)

import theano
import theano.tensor as T
import numpy as np# defining the tensor variables
X = T.matrix("X")
W = T.matrix("W")
b_sym = T.vector("b_sym")results, updates = theano.scan(lambda v: T.tanh(T.dot(v, W) + b_sym), sequences=X)
compute_elementwise = theano.function(inputs=[X, W, b_sym], outputs=[results])# test values
x = np.eye(2, dtype=theano.config.floatX)
w = np.ones((2, 2), dtype=theano.config.floatX)
b = np.ones((2), dtype=theano.config.floatX)
b[1] = 2print compute_elementwise(x, w, b)[0]# comparison with numpy
print np.tanh(x.dot(w) + b)

1.2 Scan 例子:计算序列 x(t) = tanh(x(t - 1).dot(W) + y(t).dot(U) + p(T - t).dot(V))

import theano
import theano.tensor as T
import numpy as np# define tensor variables
X = T.vector("X")
W = T.matrix("W")
b_sym = T.vector("b_sym")
U = T.matrix("U")
Y = T.matrix("Y")
V = T.matrix("V")
P = T.matrix("P")results, updates = theano.scan(lambda y, p, x_tm1: T.tanh(T.dot(x_tm1, W) + T.dot(y, U) + T.dot(p, V)),sequences=[Y, P[::-1]], outputs_info=[X])
compute_seq = theano.function(inputs=[X, W, Y, U, P, V], outputs=[results])# test values
x = np.zeros((2), dtype=theano.config.floatX)
x[1] = 1
w = np.ones((2, 2), dtype=theano.config.floatX)
y = np.ones((5, 2), dtype=theano.config.floatX)
y[0, :] = -3
u = np.ones((2, 2), dtype=theano.config.floatX)
p = np.ones((5, 2), dtype=theano.config.floatX)
p[0, :] = 3
v = np.ones((2, 2), dtype=theano.config.floatX)print compute_seq(x, w, y, u, p, v)[0]# comparison with numpy
x_res = np.zeros((5, 2), dtype=theano.config.floatX)
x_res[0] = np.tanh(x.dot(w) + y[0].dot(u) + p[4].dot(v))
for i in range(1, 5):x_res[i] = np.tanh(x_res[i - 1].dot(w) + y[i].dot(u) + p[4-i].dot(v))
print x_res

1.3 Scan 例子: 计算 X的线（指的是按照某一维度方向） 范数

import theano
import theano.tensor as T
import numpy as np# define tensor variable
X = T.matrix("X")
results, updates = theano.scan(lambda x_i: T.sqrt((x_i ** 2).sum()), sequences=[X])
compute_norm_lines = theano.function(inputs=[X], outputs=[results])# test value
x = np.diag(np.arange(1, 6, dtype=theano.config.floatX), 1)
print compute_norm_lines(x)[0]# comparison with numpy
print np.sqrt((x ** 2).sum(1))

1.4 Scan 例子:计算x的列的范数 

import theano
import theano.tensor as T
import numpy as np# define tensor variable
X = T.matrix("X")
results, updates = theano.scan(lambda x_i: T.sqrt((x_i ** 2).sum()), sequences=[X.T])
compute_norm_cols = theano.function(inputs=[X], outputs=[results])# test value
x = np.diag(np.arange(1, 6, dtype=theano.config.floatX), 1)
print compute_norm_cols(x)[0]# comparison with numpy
print np.sqrt((x ** 2).sum(0))

1.5 Scan 例子: 计算x的迹

import theano
import theano.tensor as T
import numpy as np
floatX = "float32"# define tensor variable
X = T.matrix("X")
results, updates = theano.scan(lambda i, j, t_f: T.cast(X[i, j] + t_f, floatX),sequences=[T.arange(X.shape[0]), T.arange(X.shape[1])],outputs_info=np.asarray(0., dtype=floatX))
result = results[-1]
compute_trace = theano.function(inputs=[X], outputs=[result])# test value
x = np.eye(5, dtype=theano.config.floatX)
x[0] = np.arange(5, dtype=theano.config.floatX)
print compute_trace(x)[0]# comparison with numpy
print np.diagonal(x).sum()

1.6 Scan 例子:计算序列 x(t) = x(t - 2).dot(U) + x(t - 1).dot(V) + tanh(x(t - 1).dot(W) + b)

import theano
import theano.tensor as T
import numpy as np# define tensor variables
X = T.matrix("X")
W = T.matrix("W")
b_sym = T.vector("b_sym")
U = T.matrix("U")
V = T.matrix("V")
n_sym = T.iscalar("n_sym")results, updates = theano.scan(lambda x_tm2, x_tm1: T.dot(x_tm2, U) + T.dot(x_tm1, V) + T.tanh(T.dot(x_tm1, W) + b_sym),n_steps=n_sym, outputs_info=[dict(initial=X, taps=[-2, -1])])
compute_seq2 = theano.function(inputs=[X, U, V, W, b_sym, n_sym], outputs=[results])# test values
x = np.zeros((2, 2), dtype=theano.config.floatX) # the initial value must be able to return x[-2]
x[1, 1] = 1
w = 0.5 * np.ones((2, 2), dtype=theano.config.floatX)
u = 0.5 * (np.ones((2, 2), dtype=theano.config.floatX) - np.eye(2, dtype=theano.config.floatX))
v = 0.5 * np.ones((2, 2), dtype=theano.config.floatX)
n = 10
b = np.ones((2), dtype=theano.config.floatX)print compute_seq2(x, u, v, w, b, n)# comparison with numpy
x_res = np.zeros((10, 2))
x_res[0] = x[0].dot(u) + x[1].dot(v) + np.tanh(x[1].dot(w) + b)
x_res[1] = x[1].dot(u) + x_res[0].dot(v) + np.tanh(x_res[0].dot(w) + b)
x_res[2] = x_res[0].dot(u) + x_res[1].dot(v) + np.tanh(x_res[1].dot(w) + b)
for i in range(2, 10):x_res[i] = (x_res[i - 2].dot(u) + x_res[i - 1].dot(v) +np.tanh(x_res[i - 1].dot(w) + b))
print x_res

1.7 Scan 例子:计算 y = tanh(v.dot(A))  关于 x 的jacobian

import theano
import theano.tensor as T
import numpy as np# define tensor variables
v = T.vector()
A = T.matrix()
y = T.tanh(T.dot(v, A))
results, updates = theano.scan(lambda i: T.grad(y[i], v), sequences=[T.arange(y.shape[0])])
compute_jac_t = theano.function([A, v], [results], allow_input_downcast=True) # shape (d_out, d_in)# test values
x = np.eye(5, dtype=theano.config.floatX)[0]
w = np.eye(5, 3, dtype=theano.config.floatX)
w[2] = np.ones((3), dtype=theano.config.floatX)
print compute_jac_t(w, x)[0]# compare with numpy
print ((1 - np.tanh(x.dot(w)) ** 2) * w).T

注意到我们需要对y的索引值而不是y的元素进行迭代。原因在于scan会对它的内部函数创建一个占位符变量，该占位符变量没有和需要替换的那个变量同样的依赖条件。

1.8 Scan 例子: 在scan中累计循环次数

import theano
import theano.tensor as T
import numpy as np# define shared variables
k = theano.shared(0)
n_sym = T.iscalar("n_sym")results, updates = theano.scan(lambda:{k:(k + 1)}, n_steps=n_sym)
accumulator = theano.function([n_sym], [], updates=updates, allow_input_downcast=True)k.get_value()
accumulator(5)
k.get_value()

 1.9 Scan 例子:计算 tanh(v.dot(W) + b) * d ，这里d 是 二项式

import theano
import theano.tensor as T
import numpy as np# define tensor variables
X = T.matrix("X")
W = T.matrix("W")
b_sym = T.vector("b_sym")# define shared random stream
trng = T.shared_randomstreams.RandomStreams(1234)
d=trng.binomial(size=W[1].shape)results, updates = theano.scan(lambda v: T.tanh(T.dot(v, W) + b_sym) * d, sequences=X)
compute_with_bnoise = theano.function(inputs=[X, W, b_sym], outputs=[results],updates=updates, allow_input_downcast=True)
x = np.eye(10, 2, dtype=theano.config.floatX)
w = np.ones((2, 2), dtype=theano.config.floatX)
b = np.ones((2), dtype=theano.config.floatX)print compute_with_bnoise(x, w, b)

注意到如果你想使用一个不会通过scan循环更新的随机变量 d ，你就应该将这个变量作为参数传递给non_sequences 。

1.10 Scan 例子: 计算 pow(A, k)

import theano
import theano.tensor as T
theano.config.warn.subtensor_merge_bug = Falsek = T.iscalar("k")
A = T.vector("A")def inner_fct(prior_result, B):return prior_result * B# Symbolic description of the result
result, updates = theano.scan(fn=inner_fct,outputs_info=T.ones_like(A),non_sequences=A, n_steps=k)# Scan has provided us with A ** 1 through A ** k.  Keep only the last
# value. Scan notices this and does not waste memory saving them.
final_result = result[-1]power = theano.function(inputs=[A, k], outputs=final_result,updates=updates)print power(range(10), 2)
#[  0.   1.   4.   9.  16.  25.  36.  49.  64.  81.]

1.11 Scan 例子: 计算一个多项式

import numpy
import theano
import theano.tensor as T
theano.config.warn.subtensor_merge_bug = Falsecoefficients = theano.tensor.vector("coefficients")
x = T.scalar("x")
max_coefficients_supported = 10000# Generate the components of the polynomial
full_range=theano.tensor.arange(max_coefficients_supported)
components, updates = theano.scan(fn=lambda coeff, power, free_var:coeff * (free_var ** power),outputs_info=None,sequences=[coefficients, full_range],non_sequences=x)polynomial = components.sum()
calculate_polynomial = theano.function(inputs=[coefficients, x],outputs=polynomial)test_coeff = numpy.asarray([1, 0, 2], dtype=numpy.float32)
print calculate_polynomial(test_coeff, 3)
# 19.0

二、Exercise

运行两个例子。

修改并执行多项式的例子，通过scan来进行约间。

答案（Solution）

#!/usr/bin/env python
# Theano tutorial
# Solution to Exercise in section 'Loop'
from __future__ import print_function
import numpyimport theano
import theano.tensor as tt# 1. First exampletheano.config.warn.subtensor_merge_bug = Falsek = tt.iscalar("k")
A = tt.vector("A")def inner_fct(prior_result, A):return prior_result * A# Symbolic description of the result
result, updates = theano.scan(fn=inner_fct,outputs_info=tt.ones_like(A),non_sequences=A, n_steps=k)# Scan has provided us with A ** 1 through A ** k.  Keep only the last
# value. Scan notices this and does not waste memory saving them.
final_result = result[-1]power = theano.function(inputs=[A, k], outputs=final_result,updates=updates)print(power(range(10), 2))
# [  0.   1.   4.   9.  16.  25.  36.  49.  64.  81.]# 2. Second examplecoefficients = tt.vector("coefficients")
x = tt.scalar("x")
max_coefficients_supported = 10000# Generate the components of the polynomial
full_range = tt.arange(max_coefficients_supported)
components, updates = theano.scan(fn=lambda coeff, power, free_var:coeff * (free_var ** power),sequences=[coefficients, full_range],outputs_info=None,non_sequences=x)
polynomial = components.sum()
calculate_polynomial1 = theano.function(inputs=[coefficients, x],outputs=polynomial)test_coeff = numpy.asarray([1, 0, 2], dtype=numpy.float32)
print(calculate_polynomial1(test_coeff, 3))
# 19.0# 3. Reduction performed inside scantheano.config.warn.subtensor_merge_bug = Falsecoefficients = tt.vector("coefficients")
x = tt.scalar("x")
max_coefficients_supported = 10000# Generate the components of the polynomial
full_range = tt.arange(max_coefficients_supported)outputs_info = tt.as_tensor_variable(numpy.asarray(0, 'float64'))components, updates = theano.scan(fn=lambda coeff, power, prior_value, free_var:prior_value + (coeff * (free_var ** power)),sequences=[coefficients, full_range],outputs_info=outputs_info,non_sequences=x)polynomial = components[-1]
calculate_polynomial = theano.function(inputs=[coefficients, x],outputs=polynomial, updates=updates)test_coeff = numpy.asarray([1, 0, 2], dtype=numpy.float32)
print(calculate_polynomial(test_coeff, 3))
# 19.0

参考资料：

[1]官网：http://deeplearning.net/software/theano/tutorial/loop.html