Theano - 导数

计算梯度

计算x^2的梯度

import numpy
import theano
import theano.tensor as T
from theano import pp
x = T.dscalar('x')
y = x ** 2
gy = T.grad(y, x)
pp(gy)
f = theano.function([x], gy)
pp(f.maker.fgraph.outputs[0])
f(4)
numpy.allclose(f(94.2), 188.4)

计算逻辑函数的梯度

x = T.dmatrix('x')
s = T.sum(1 / (1 + T.exp(-x)))
gs = T.grad(s, x)
dlogistic = theano.function([x], gs)
dlogistic([[0, 1], [-1, -2]])

计算Jacobian

x = T.dvector('x')
y = x ** 2
J, updates = theano.scan(lambda i, y, x: T.grad(y[i], x), sequences=T.arange(y.shape[0]), non_sequences=[y,x])
f = theano.function([x], J, updates=updates)
f([4, 4])

计算Hessian矩阵

x = T.dvector('x')
y = x ** 2
cost = y.sum()
gy = T.grad(cost, x)
H, updates = theano.scan(lambda i, gy, x: T.grad(gy[i], x), sequences=T.arange(gy.shape[0]), non_sequences=[gy,x])
f = theano.function([x], H, updates=updates)
f([4,4])

Jacobian times a Vector

右算子(R-operator)

W = T.dmatrix('W')
V = T.dmatrix('V')
x = T.dvector('x')
y = T.dot(x, W)
JV = T.Rop(y, W, V)
f = theano.function([W, V, x], JV)
f([[1,1], [1,1]], [[2,2], [2,2]], [0,1])

左算子（L-operator)

W = T.dmatrix('W')
v = T.dvector('v')
x = T.dvector('x')
y = T.dot(x, W)
VJ = T.Lop(y, W, v)
f = theano.function([v, x], VJ)
f([2,2], [0,1])

Hessian times a Vector

x = T.dvector('x')
v = T.dvector('v')
y = T.sum(x ** 2)
gy = T.grad(y, x)
vH = T.grad(T.sum(gy * v), x)
f = theano.function([x,v], vH)
f([4,4], [2,2])

右算子

x = T.dvector('x')
v = T.dvector('v')
y = T.sum(x ** 2)
gy = T.grad(y, x)
Hv = T.Rop(gy, x, v)
f = theano.function([x,v], Hv)
f([4,4], [2,2])