At this point it would be wise to begin familiarizing yourself more
systematically with Theano’s fundamental objects and operations by
browsing this section of the library: *Basic Tensor Functionality*.

As the tutorial unfolds, you should also gradually acquaint yourself with the other relevant areas of the library and with the relevant subjects of the documentation entrance page.

Here’s another straightforward example, though a bit more elaborate than adding two numbers together. Let’s say that you want to compute the logistic curve, which is given by:

You want to compute the function *elementwise* on matrices of doubles, which means that
you want to apply this function to each individual element of the
matrix.

Well, what you do is this:

```
>>> x = T.dmatrix('x')
>>> s = 1 / (1 + T.exp(-x))
>>> logistic = function([x], s)
>>> logistic([[0, 1], [-1, -2]])
array([[ 0.5 , 0.73105858],
[ 0.26894142, 0.11920292]])
```

The reason logistic is performed elementwise is because all of its operations—division, addition, exponentiation, and division—are themselves elementwise operations.

It is also the case that:

We can verify that this alternate form produces the same values:

```
>>> s2 = (1 + T.tanh(x / 2)) / 2
>>> logistic2 = function([x], s2)
>>> logistic2([[0, 1], [-1, -2]])
array([[ 0.5 , 0.73105858],
[ 0.26894142, 0.11920292]])
```

Theano supports functions with multiple outputs. For example, we can
compute the *elementwise* difference, absolute difference, and
squared difference between two matrices *a* and *b* at the same time:

```
>>> a, b = T.dmatrices('a', 'b')
>>> diff = a - b
>>> abs_diff = abs(diff)
>>> diff_squared = diff**2
>>> f = function([a, b], [diff, abs_diff, diff_squared])
```

Note

dmatrices produces as many outputs as names that you provide. It is a shortcut for allocating symbolic variables that we will often use in the tutorials.

When we use the function f, it returns the three variables (the printing was reformatted for readability):

```
>>> f([[1, 1], [1, 1]], [[0, 1], [2, 3]])
[array([[ 1., 0.],
[-1., -2.]]),
array([[ 1., 0.],
[ 1., 2.]]),
array([[ 1., 0.],
[ 1., 4.]])]
```

Let’s say you want to define a function that adds two numbers, except that if you only provide one number, the other input is assumed to be one. You can do it like this:

```
>>> from theano import Param
>>> x, y = T.dscalars('x', 'y')
>>> z = x + y
>>> f = function([x, Param(y, default=1)], z)
>>> f(33)
array(34.0)
>>> f(33, 2)
array(35.0)
```

This makes use of the *Param* class which allows
you to specify properties of your function’s parameters with greater detail. Here we
give a default value of 1 for *y* by creating a `Param` instance with
its `default` field set to 1.

Inputs with default values must follow inputs without default values (like Python’s functions). There can be multiple inputs with default values. These parameters can be set positionally or by name, as in standard Python:

```
>>> x, y, w = T.dscalars('x', 'y', 'w')
>>> z = (x + y) * w
>>> f = function([x, Param(y, default=1), Param(w, default=2, name='w_by_name')], z)
>>> f(33)
array(68.0)
>>> f(33, 2)
array(70.0)
>>> f(33, 0, 1)
array(33.0)
>>> f(33, w_by_name=1)
array(34.0)
>>> f(33, w_by_name=1, y=0)
array(33.0)
```

Note

`Param` does not know the name of the local variables *y* and *w*
that are passed as arguments. The symbolic variable objects have name
attributes (set by `dscalars` in the example above) and *these* are the
names of the keyword parameters in the functions that we build. This is
the mechanism at work in `Param(y, default=1)`. In the case of `Param(w,
default=2, name='w_by_name')`. We override the symbolic variable’s name
attribute with a name to be used for this function.

You may like to see *Function* in the library for more detail.

Because in Theano you first express everything symbolically and afterwards compile this expression to get functions, using pseudo-random numbers is not as straightforward as it is in NumPy, though also not too complicated.

The way to think about putting randomness into Theano’s computations is
to put random variables in your graph. Theano will allocate a NumPy
RandomStream object (a random number generator) for each such
variable, and draw from it as necessary. We will call this sort of
sequence of random numbers a *random stream*. *Random streams* are at
their core shared variables, so the observations on shared variables
hold here as well. Theanos’s random objects are defined and implemented in
*RandomStreams* and, at a lower level,
in *RandomStreamsBase*.

Here’s a brief example. The setup code is:

```
from theano.tensor.shared_randomstreams import RandomStreams
from theano import function
srng = RandomStreams(seed=234)
rv_u = srng.uniform((2,2))
rv_n = srng.normal((2,2))
f = function([], rv_u)
g = function([], rv_n, no_default_updates=True) #Not updating rv_n.rng
nearly_zeros = function([], rv_u + rv_u - 2 * rv_u)
```

Here, ‘rv_u’ represents a random stream of 2x2 matrices of draws from a uniform
distribution. Likewise, ‘rv_n’ represents a random stream of 2x2 matrices of
draws from a normal distribution. The distributions that are implemented are
defined in `RandomStreams` and, at a lower level,
in *raw_random*. They only work on CPU.
See Other Implementations for GPU version.

Now let’s use these objects. If we call f(), we get random uniform numbers. The internal state of the random number generator is automatically updated, so we get different random numbers every time.

```
>>> f_val0 = f()
>>> f_val1 = f() #different numbers from f_val0
```

When we add the extra argument `no_default_updates=True` to
`function` (as in *g*), then the random number generator state is
not affected by calling the returned function. So, for example, calling
*g* multiple times will return the same numbers.

```
>>> g_val0 = g() # different numbers from f_val0 and f_val1
>>> g_val1 = g() # same numbers as g_val0!
```

An important remark is that a random variable is drawn at most once during any
single function execution. So the *nearly_zeros* function is guaranteed to
return approximately 0 (except for rounding error) even though the *rv_u*
random variable appears three times in the output expression.

```
>>> nearly_zeros = function([], rv_u + rv_u - 2 * rv_u)
```

Random variables can be seeded individually or collectively.

You can seed just one random variable by seeding or assigning to the
`.rng` attribute, using `.rng.set_value()`.

```
>>> rng_val = rv_u.rng.get_value(borrow=True) # Get the rng for rv_u
>>> rng_val.seed(89234) # seeds the generator
>>> rv_u.rng.set_value(rng_val, borrow=True) # Assign back seeded rng
```

You can also seed *all* of the random variables allocated by a `RandomStreams`
object by that object’s `seed` method. This seed will be used to seed a
temporary random number generator, that will in turn generate seeds for each
of the random variables.

```
>>> srng.seed(902340) # seeds rv_u and rv_n with different seeds each
```

As usual for shared variables, the random number generators used for random
variables are common between functions. So our *nearly_zeros* function will
update the state of the generators used in function *f* above.

For example:

```
>>> state_after_v0 = rv_u.rng.get_value().get_state()
>>> nearly_zeros() # this affects rv_u's generator
>>> v1 = f()
>>> rng = rv_u.rng.get_value(borrow=True)
>>> rng.set_state(state_after_v0)
>>> rv_u.rng.set_value(rng, borrow=True)
>>> v2 = f() # v2 != v1
>>> v3 = f() # v3 == v1
```

In some use cases, a user might want to transfer the “state” of all random
number generators associated with a given theano graph (e.g. g1, with compiled
function f1 below) to a second graph (e.g. g2, with function f2). This might
arise for example if you are trying to initialize the state of a model, from
the parameters of a pickled version of a previous model. For
`theano.tensor.shared_randomstreams.RandomStreams` and
`theano.sandbox.rng_mrg.MRG_RandomStreams`
this can be achieved by copying elements of the state_updates parameter.

Each time a random variable is drawn from a RandomStreams object, a tuple is
added to the state_updates list. The first element is a shared variable,
which represents the state of the random number generator associated with this
*particular* variable, while the second represents the theano graph
corresponding to the random number generation process (i.e. RandomFunction{uniform}.0).

An example of how “random states” can be transferred from one theano function to another is shown below.

```
import theano
import numpy
import theano.tensor as T
from theano.sandbox.rng_mrg import MRG_RandomStreams
from theano.tensor.shared_randomstreams import RandomStreams
class Graph():
def __init__(self, seed=123):
self.rng = RandomStreams(seed)
self.y = self.rng.uniform(size=(1,))
g1 = Graph(seed=123)
f1 = theano.function([], g1.y)
g2 = Graph(seed=987)
f2 = theano.function([], g2.y)
print 'By default, the two functions are out of sync.'
print 'f1() returns ', f1()
print 'f2() returns ', f2()
def copy_random_state(g1, g2):
if isinstance(g1.rng, MRG_RandomStreams):
g2.rng.rstate = g1.rng.rstate
for (su1, su2) in zip(g1.rng.state_updates, g2.rng.state_updates):
su2[0].set_value(su1[0].get_value())
print 'We now copy the state of the theano random number generators.'
copy_random_state(g1, g2)
print 'f1() returns ', f1()
print 'f2() returns ', f2()
```

This gives the following output:

```
# By default, the two functions are out of sync.
f1() returns [ 0.72803009]
f2() returns [ 0.55056769]
# We now copy the state of the theano random number generators.
f1() returns [ 0.59044123]
f2() returns [ 0.59044123]
```

There are *other distributions implemented*.

The preceding elements are featured in this more realistic example. It will be used repeatedly.

```
import numpy
import theano
import theano.tensor as T
rng = numpy.random
N = 400
feats = 784
D = (rng.randn(N, feats), rng.randint(size=N, low=0, high=2))
training_steps = 10000
# Declare Theano symbolic variables
x = T.matrix("x")
y = T.vector("y")
w = theano.shared(rng.randn(feats), name="w")
b = theano.shared(0., name="b")
print "Initial model:"
print w.get_value(), b.get_value()
# Construct Theano expression graph
p_1 = 1 / (1 + T.exp(-T.dot(x, w) - b)) # Probability that target = 1
prediction = p_1 > 0.5 # The prediction thresholded
xent = -y * T.log(p_1) - (1-y) * T.log(1-p_1) # Cross-entropy loss function
cost = xent.mean() + 0.01 * (w ** 2).sum()# The cost to minimize
gw, gb = T.grad(cost, [w, b]) # Compute the gradient of the cost
# (we shall return to this in a
# following section of this tutorial)
# Compile
train = theano.function(
inputs=[x,y],
outputs=[prediction, xent],
updates=((w, w - 0.1 * gw), (b, b - 0.1 * gb)))
predict = theano.function(inputs=[x], outputs=prediction)
# Train
for i in range(training_steps):
pred, err = train(D[0], D[1])
print "Final model:"
print w.get_value(), b.get_value()
print "target values for D:", D[1]
print "prediction on D:", predict(D[0])
```