Loop¶
Scan¶
- A general form of recurrence, which can be used for looping.
- Reduction and map (loop over the leading dimensions) are special cases of
scan. - You
scana function along some input sequence, producing an output at each time-step. - The function can see the previous K time-steps of your function.
sum()could be computed by scanning the z + x(i) function over a list, given an initial state of z=0.- Often a for loop can be expressed as a
scan()operation, andscanis the closest that Theano comes to looping. - Advantages of using
scanover for loops:- Number of iterations to be part of the symbolic graph.
- Minimizes GPU transfers (if GPU is involved).
- Computes gradients through sequential steps.
- Slightly faster than using a for loop in Python with a compiled Theano function.
- Can lower the overall memory usage by detecting the actual amount of memory needed.
The full documentation can be found in the library: Scan.
Scan Example: Computing tanh(x(t).dot(W) + b) elementwise
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] = 2
print(compute_elementwise(x, w, b)[0])
# comparison with numpy
print(np.tanh(x.dot(w) + b))
[[ 0.96402758 0.99505475]
[ 0.96402758 0.99505475]]
[[ 0.96402758 0.99505475]
[ 0.96402758 0.99505475]]
Scan Example: Computing the sequence 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)
[[-0.99505475 -0.99505475]
[ 0.96471973 0.96471973]
[ 0.99998585 0.99998585]
[ 0.99998771 0.99998771]
[ 1. 1. ]]
[[-0.99505475 -0.99505475]
[ 0.96471973 0.96471973]
[ 0.99998585 0.99998585]
[ 0.99998771 0.99998771]
[ 1. 1. ]]
Scan Example: Computing norms of lines of X
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. 2. 3. 4. 5. 0.]
[ 1. 2. 3. 4. 5. 0.]
Scan Example: Computing norms of columns of X
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)))
[ 0. 1. 2. 3. 4. 5.]
[ 0. 1. 2. 3. 4. 5.]
Scan Example: Computing trace of X
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())
4.0
4.0
Scan Example: Computing the sequence 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)
[array([[ 1.40514825, 1.40514825],
[ 2.88898899, 2.38898899],
[ 4.34018291, 4.34018291],
[ 6.53463142, 6.78463142],
[ 9.82972243, 9.82972243],
[ 14.22203814, 14.09703814],
[ 20.07439936, 20.07439936],
[ 28.12291843, 28.18541843],
[ 39.1913681 , 39.1913681 ],
[ 54.28407732, 54.25282732]])]
[[ 1.40514825 1.40514825]
[ 2.88898899 2.38898899]
[ 4.34018291 4.34018291]
[ 6.53463142 6.78463142]
[ 9.82972243 9.82972243]
[ 14.22203814 14.09703814]
[ 20.07439936 20.07439936]
[ 28.12291843 28.18541843]
[ 39.1913681 39.1913681 ]
[ 54.28407732 54.25282732]]
Scan Example: Computing the Jacobian of y = tanh(v.dot(A)) wrt x
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)
[[ 0.41997434 0. 0.41997434 0. 0. ]
[ 0. 1. 1. 0. 0. ]
[ 0. 0. 1. 0. 0. ]]
[[ 0.41997434 0. 0.41997434 0. 0. ]
[ 0. 1. 1. 0. 0. ]
[ 0. 0. 1. 0. 0. ]]
Note that we need to iterate over the indices of y and not over the elements of y. The reason is that scan create a placeholder variable for its internal function and this placeholder variable does not have the same dependencies than the variables that will replace it.
Scan Example: Accumulate number of loop during a scan
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()
Scan Example: Computing tanh(v.dot(W) + b) * d where d is binomial
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))
[array([[ 0.96402758, 0. ],
[ 0. , 0.96402758],
[ 0. , 0. ],
[ 0.76159416, 0.76159416],
[ 0.76159416, 0. ],
[ 0. , 0.76159416],
[ 0. , 0.76159416],
[ 0. , 0.76159416],
[ 0. , 0. ],
[ 0.76159416, 0.76159416]])]
Note that if you want to use a random variable d that will not be updated through scan loops, you should pass this variable as a non_sequences arguments.
Scan Example: Computing pow(A, k)
import theano
import theano.tensor as T
theano.config.warn.subtensor_merge_bug = False
k = 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.]
Scan Example: Calculating a Polynomial
import numpy
import theano
import theano.tensor as T
theano.config.warn.subtensor_merge_bug = False
coefficients = 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