# More Examples¶

At this point it would be wise to begin familiarizing yourself more systematically with Aesara’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.

## Logistic Function¶

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 element-wise 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:

```
>>> import aesara
>>> import aesara.tensor as at
>>> x = at.dmatrix('x')
>>> s = 1 / (1 + at.exp(-x))
>>> logistic = aesara.function([x], s)
>>> logistic([[0, 1], [-1, -2]])
array([[ 0.5 , 0.73105858],
[ 0.26894142, 0.11920292]])
```

The reason the logistic is applied element-wise is because all of its operations–division, addition, exponentiation, and division–are themselves element-wise operations.

It is also the case that:

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

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

## Computing More than one Thing at the Same Time¶

Aesara supports functions with multiple outputs. For example, we can
compute the element-wise difference, absolute difference, and
squared difference between two matrices `a`

and `b`

at the same time:

```
>>> a, b = at.dmatrices('a', 'b')
>>> diff = a - b
>>> abs_diff = abs(diff)
>>> diff_squared = diff**2
>>> f = aesara.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.]])]
```

## Setting a Default Value for an Argument¶

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 aesara.compile.io import In
>>> from aesara import function
>>> x, y = at.dscalars('x', 'y')
>>> z = x + y
>>> f = function([x, In(y, value=1)], z)
>>> f(33)
array(34.0)
>>> f(33, 2)
array(35.0)
```

This makes use of the In 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 `In`

instance with
its `value`

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 = at.dscalars('x', 'y', 'w')
>>> z = (x + y) * w
>>> f = function([x, In(y, value=1), In(w, value=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

`In`

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 `In(y, value=1)`

. In the case of ```
In(w,
value=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.

## Copying functions¶

Aesara functions can be copied, which can be useful for creating similar
functions but with different shared variables or updates. This is done using
the `aesara.compile.function.types.Function.copy()`

method of `Function`

objects.
The optimized graph of the original function is copied, so compilation only
needs to be performed once.

Let’s start from the accumulator defined above:

```
>>> import aesara
>>> import aesara.tensor as at
>>> state = aesara.shared(0)
>>> inc = at.iscalar('inc')
>>> accumulator = aesara.function([inc], state, updates=[(state, state+inc)])
```

We can use it to increment the state as usual:

```
>>> accumulator(10)
array(0)
>>> print(state.get_value())
10
```

We can use `copy()`

to create a similar accumulator but with its own internal state
using the `swap`

parameter, which is a dictionary of shared variables to exchange:

```
>>> new_state = aesara.shared(0)
>>> new_accumulator = accumulator.copy(swap={state:new_state})
>>> new_accumulator(100)
[array(0)]
>>> print(new_state.get_value())
100
```

The state of the first function is left untouched:

```
>>> print(state.get_value())
10
```

We now create a copy with updates removed using the `delete_updates`

parameter, which is set to `False`

by default:

```
>>> null_accumulator = accumulator.copy(delete_updates=True)
```

As expected, the shared state is no longer updated:

```
>>> null_accumulator(9000)
[array(10)]
>>> print(state.get_value())
10
```

## Using Random Numbers¶

Because in Aesara 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 Aesara’s computations is
to put random variables in your graph. Aesara 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. Aesara’s random objects are defined and implemented in
RandomStream and, at a lower level,
in RandomVariable.

### Brief Example¶

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

```
from aesara.tensor.random.utils import RandomStream
from aesara import function
srng = RandomStream(seed=234)
rv_u = srng.uniform(0, 1, size=(2,2))
rv_n = srng.normal(0, 1, size=(2,2))
f = function([], rv_u)
g = function([], rv_n, no_default_updates=True)
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 as `RandomVariable`

s
in basic. They only work on CPU.

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)
```

### Seeding Streams¶

You can seed all of the random variables allocated by a `RandomStream`

object by that object’s `RandomStream.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
```

### Sharing Streams Between Functions¶

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.

### Copying Random State Between Aesara Graphs¶

In some use cases, a user might want to transfer the “state” of all random
number generators associated with a given Aesara 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
`aesara.tensor.random.utils.RandomStream`

and
`aesara.sandbox.rng_mrg.MRG_RandomStream`

this can be achieved by copying elements of the `state_updates`

parameter.

Each time a random variable is drawn from a `RandomStream`

object, a tuple is
added to its `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 Aesara graph
corresponding to the random number generation process.

### Other Random Distributions¶

There are other distributions implemented.

## A Real Example: Logistic Regression¶

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

```
import numpy as np
import aesara
import aesara.tensor as at
rng = np.random.default_rng(2882)
N = 400 # training sample size
feats = 784 # number of input variables
# generate a dataset: D = (input_values, target_class)
D = (rng.standard_normal((N, feats)), rng.integers(size=N, low=0, high=2))
training_steps = 10000
# Declare Aesara symbolic variables
x = at.dmatrix("x")
y = at.dvector("y")
# initialize the weight vector w randomly
#
# this and the following bias variable b
# are shared so they keep their values
# between training iterations (updates)
w = aesara.shared(rng.standard_normal(feats), name="w")
# initialize the bias term
b = aesara.shared(0., name="b")
print("Initial model:")
print(w.get_value())
print(b.get_value())
# Construct Aesara expression graph
p_1 = 1 / (1 + at.exp(-at.dot(x, w) - b)) # Probability that target = 1
prediction = p_1 > 0.5 # The prediction thresholded
xent = -y * at.log(p_1) - (1-y) * at.log(1-p_1) # Cross-entropy loss function
cost = xent.mean() + 0.01 * (w ** 2).sum() # The cost to minimize
gw, gb = at.grad(cost, [w, b]) # Compute the gradient of the cost
# w.r.t weight vector w and
# bias term b (we shall
# return to this in a
# following section of this
# tutorial)
# Compile
train = aesara.function(
inputs=[x,y],
outputs=[prediction, xent],
updates=((w, w - 0.1 * gw), (b, b - 0.1 * gb)))
predict = aesara.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())
print(b.get_value())
print("target values for D:")
print(D[1])
print("prediction on D:")
print(predict(D[0]))
```