# 3.14. Activation Functions¶

This notebook explores the activation functions that are available in Conx.

First, we import a special function plot_f that is designed to plot functions. We also import all of the activation functions:

• softmax
• elu
• selu
• softplus
• softsign
• relu
• tanh
• sigmoid
• hard_sigmoid
• linear

There are additional advanced activation functions not defined and examined here, including:

• ThresholdedReLU - Thresholded Rectified Linear Unit
• LeakyReLU - Leaky version of a Rectified Linear Unit
• PReLU - Parametric Rectified Linear Unit
In [42]:

from conx.activations import *
import conx as cx


## 3.14.1. softmax¶

The softmax activation function takes a vector of input values and returns a vector of output values. This function is unique in that the output values are not computed independently of one another. Mathematically, the outputs of softmax always sum to exactly one, but in practice the sum of the output values will be close to but not exactly equal to one. The outputs of softmax can be interpreted as probabilities, and can be used, for example, with Conx’s choice function.

In [2]:

help(softmax)

Help on function softmax in module conx.activations:

softmax(tensor, axis=-1)
Softmax activation function.

>>> len(softmax([0.1, 0.1, 0.7, 0.0]))
4


In [3]:

softmax([0.1, 0.1, 0.7, 0.0])

Out[3]:

[0.21155263483524323,
0.21155263483524323,
0.38547399640083313,
0.19142073392868042]

In [4]:

sum(softmax([0.1, 0.1, 0.7, 0.0]))

Out[4]:

1.0

In [5]:

softmax([1, 2, 3, 4])

Out[5]:

[0.03205860033631325,
0.08714431524276733,
0.23688282072544098,
0.6439142823219299]

In [6]:

sum(softmax([1, 2, 3, 4]))

Out[6]:

1.0000000186264515

In [7]:

cx.choice(['a', 'b', 'c', 'd'], p=softmax([0.1, 0.1, 0.7, 0.0]))

Out[7]:

'a'


Let’s see how softmax can be used to make probabilistic choices, and see if they match our expectations:

In [8]:

bin = [0] * 4
for x in range(100):
pick = cx.choice([0, 1, 2, 3], p=softmax([0.1, 0.1, 0.7, 0.0]))
bin[pick] += 1
print("softmax:", softmax([0.1, 0.1, 0.7, 0.0]))
print("picks  :", [b/100 for b in bin])

softmax: [0.21155263483524323, 0.21155263483524323, 0.38547399640083313, 0.19142073392868042]
picks  : [0.22, 0.26, 0.35, 0.17]

In [9]:

f = lambda x: softmax([x, 0.5, 0.1, 1])

In [10]:

cx.plot_f(f, frange=(-5, 5, 0.1), xlabel="input", ylabel="output", title="softmax()")


## 3.14.2. elu¶

In [11]:

help(elu)

Help on function elu in module conx.activations:

elu(x, alpha=1.0)
Exponential Linear Unit activation function.

See: https://arxiv.org/abs/1511.07289v1

def elu(x):
if x >= 0:
return x
else:
return alpha * (math.exp(x) - 1.0)

>>> elu(0.0)
0.0
>>> elu(1.0)
1.0
>>> elu(0.5, alpha=0.3)
0.5
>>> round(elu(-1), 1)
-0.6


In [12]:

elu(0.5)

Out[12]:

0.5

In [13]:

cx.plot_f(elu, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="elu()")

In [14]:

elu(0.5, .3)

Out[14]:

0.5

In [15]:

cx.plot_f(lambda x: elu(x, 0.3), frange=(-15, 15, 1), xlabel="input", ylabel="output", title="elu(alpha=0.3)")


## 3.14.3. selu¶

In [16]:

help(selu)

Help on function selu in module conx.activations:

selu(x)
Scaled Exponential Linear Unit activation function.

>>> selu(0)
0.0


In [17]:

selu(0.5)

Out[17]:

0.5253505110740662

In [18]:

cx.plot_f(selu, frange=(-5, 5, 0.5), xlabel="input", ylabel="output", title="selu()")


## 3.14.4. softplus¶

In [19]:

help(softplus)

Help on function softplus in module conx.activations:

softplus(x)
Softplus activation function.

>>> round(softplus(0), 1)
0.7


In [20]:

softplus(0.5)

Out[20]:

0.9740769863128662

In [21]:

cx.plot_f(softplus, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="softplus()")


## 3.14.5. softsign¶

In [22]:

help(softsign)

Help on function softsign in module conx.activations:

softsign(x)
Softsign activation function.

>>> softsign(1)
0.5
>>> softsign(-1)
-0.5


In [23]:

softsign(0.5)

Out[23]:

0.3333333432674408

In [24]:

cx.plot_f(softsign, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="softsign()")


## 3.14.6. relu¶

In [25]:

help(relu)

Help on function relu in module conx.activations:

relu(x, alpha=0.0, max_value=None)
Rectified Linear Unit activation function.

>>> relu(1)
1.0
>>> relu(-1)
0.0


In [26]:

relu(0.5)

Out[26]:

0.5

In [27]:

cx.plot_f(relu, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="relu()")


## 3.14.7. tanh¶

In [28]:

help(tanh)

Help on function tanh in module conx.activations:

tanh(x)
Tanh activation function.

>>> tanh(0)
0.0


In [29]:

tanh(0.5)

Out[29]:

0.46211716532707214

In [30]:

cx.plot_f(tanh, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="tanh()")


## 3.14.8. sigmoid¶

In [31]:

help(sigmoid)

Help on function sigmoid in module conx.activations:

sigmoid(x)
Sigmoid activation function.

>>> sigmoid(0)
0.5


In [32]:

sigmoid(0.5)

Out[32]:

0.622459352016449

In [33]:

cx.plot_f(sigmoid, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="sigmoid()")


## 3.14.9. hard_sigmoid¶

In [34]:

help(hard_sigmoid)

Help on function hard_sigmoid in module conx.activations:

hard_sigmoid(x)
Hard Sigmoid activation function.

>>> round(hard_sigmoid(-1), 1)
0.3


In [35]:

hard_sigmoid(0.5)

Out[35]:

0.6000000238418579

In [36]:

cx.plot_f(hard_sigmoid, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="hard_sigmoid()")


## 3.14.10. linear¶

In [37]:

help(linear)

Help on function linear in module conx.activations:

linear(x)
Linear activation function.

>>> linear(1) == 1
True
>>> linear(-1) == -1
True


In [38]:

linear(0.5)

Out[38]:

0.5

In [39]:

cx.plot_f(linear, frange=(-15, 15, 1), xlabel="input", ylabel="output", title="linear()")


## 3.14.11. Comparison¶

In [40]:

functions = [hard_sigmoid, sigmoid, tanh, relu,
softsign, softplus, elu, selu, linear]
float_range = cx.frange(-2, 2, 0.1)
lines = {}
symbols = {}
for v in float_range:
for f in functions:
if f.__name__ not in lines:
lines[f.__name__] = []
symbols[f.__name__] = "-"
lines[f.__name__].append(f(v))

In [41]:

cx.plot(list(lines.items()), xs=float_range, symbols=symbols, height=6.0, width=8.0,
title="Activation Functions", xlabel="input", ylabel="output")

Out[41]: