# 3.1. Learning¶

The shallowest network is one that has no hidden layers at all. But this type of network can only solve one type of problem: those that are linearly separable. This notebook explores learning linearly and non-lineraly separable datasets.

## 3.1.1. Linearly Separable¶

In [1]:

import conx as cx
import random

Using Theano backend.
Conx, version 3.6.0


First, let’s construct a fake linearly-separable dataset.

In [2]:

count = 500

positives = [(i/count, i/(count * 2) + random.random()/6) for i in range(count)]
negatives = [(i/count, 0.3 + i/(count * 2) + random.random()/6) for i in range(count)]

In [3]:

cx.scatter([
["Positive", positives],
["Negative", negatives],
],
height=8.0,
width=8.0,
symbols={"Positive": "bo", "Negative": "ro"})

Out[3]:

In [4]:

ds = cx.Dataset()

In [5]:

ds.load([(p, [ 1.0], "Positive") for p in positives] +
[(n, [ 0.0], "Negative") for n in negatives])

In [6]:

ds.shuffle()

In [7]:

ds.split(.1)

In [8]:

ds.summary()

_________________________________________________________________
Unnamed Dataset:
Patterns    Shape                 Range
=================================================================
inputs      (2,)                  (0.0, 0.998)
targets     (1,)                  (0.0, 1.0)
=================================================================
Total patterns: 1000
Training patterns: 900
Testing patterns: 100
_________________________________________________________________

In [9]:

net = cx.Network("Linearly Separable", 2, 1, activation="sigmoid")

In [10]:

net.set_dataset(ds)

In [11]:

net.dashboard()

In [12]:

net.test(tolerance=0.4)

========================================================
Testing validation dataset with tolerance 0.4...
Total count: 900
correct: 246
incorrect: 654
Total percentage correct: 0.2733333333333333

In [13]:

symbols = {
"Positive (correct)": "w+",
"Positive (wrong)": "k+",
"Negative (correct)": "w_",
"Negative (wrong)": "k_",
}

net.plot_activation_map(scatter=net.test(tolerance=0.4, interactive=False),
symbols=symbols, title="Before Training")

In [14]:

if net.saved():
net.plot_results()
else:
net.train(epochs=10000, accuracy=1.0, report_rate=50,
tolerance=0.4, batch_size=len(net.dataset.train_inputs),
plot=True, record=100, save=True)

========================================================
|  Training |  Training |  Validate |  Validate
Epochs |     Error |  Accuracy |     Error |  Accuracy
------ | --------- | --------- | --------- | ---------
# 6775 |   0.26114 |   1.00000 |   0.26946 |   1.00000
Saving network... Saved!

In [15]:

net.plot_activation_map(scatter=net.test(tolerance=0.4, interactive=False),
symbols=symbols, title="After Training")

In [16]:

net.get_weights("output")

Out[16]:

[[[3.3727526664733887], [-7.073390007019043]], [1.7067580223083496]]

In [17]:

from conx.activations import sigmoid

def output(x, y):
wts = net.get_weights("output")
return sigmoid(x * wts[0][1][0] + y * wts[0][0][0] + wts[1][0])

def ascii(f):
return "%4.1f" % f

In [18]:

for y in cx.frange(0, 1.1, .1):
for x in cx.frange(1.0, 0.1, -0.1):
print(ascii(output(x, y)), end=" ")
print()

 0.0  0.0  0.0  0.0  0.1  0.1  0.2  0.4  0.6
0.0  0.0  0.0  0.1  0.1  0.2  0.3  0.5  0.7
0.0  0.0  0.0  0.1  0.1  0.2  0.4  0.6  0.7
0.0  0.0  0.1  0.1  0.2  0.3  0.5  0.6  0.8
0.0  0.0  0.1  0.1  0.2  0.4  0.6  0.7  0.8
0.0  0.0  0.1  0.2  0.3  0.5  0.6  0.8  0.9
0.0  0.1  0.1  0.2  0.4  0.5  0.7  0.8  0.9
0.0  0.1  0.2  0.3  0.5  0.6  0.8  0.9  0.9
0.1  0.1  0.2  0.4  0.5  0.7  0.8  0.9  1.0
0.1  0.2  0.3  0.4  0.6  0.8  0.9  0.9  1.0
0.1  0.2  0.4  0.5  0.7  0.8  0.9  1.0  1.0

In [23]:

net.playback(lambda net, epoch: net.plot_activation_map(title="Epoch %s" % epoch,
scatter=net.test(tolerance=0.4, interactive=False),
symbols=symbols,
format="svg"))

In [24]:

net.set_weights_from_history(-1)

In [27]:

net.movie(lambda net, epoch: net.plot_activation_map(title="Epoch %s" % epoch,
scatter=net.test(tolerance=0.4, interactive=False),
symbols=symbols,
format="image"))

Out[27]:


## 3.1.2. Non-Linearly Separable¶

In [28]:

import math

In [29]:

def distance(x1, y1, x2, y2):
return math.sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2)

In [30]:

negatives = []
while len(negatives) < 500:
x = random.random()
y = random.random()
d = distance(x, y, 0.5, 0.5)
if d > 0.375 and d < 0.5:
negatives.append([x, y])
positives = []
while len(positives) < 500:
x = random.random()
y = random.random()
d = distance(x, y, 0.5, 0.5)
if d < 0.25:
positives.append([x, y])

In [31]:

cx.scatter([
["Positive", positives],
["Negative", negatives],
],
height=8.0,
width=8.0,
symbols={"Positive": "bo", "Negative": "ro"})

Out[31]:

In [32]:

net = cx.Network("Non-Linearly Separable", 2, 5, 1, activation="sigmoid")

In [33]:

net.picture()

Out[33]:

In [34]:

ds = cx.Dataset()

In [35]:

ds.load([(p, [ 1.0], "Positive") for p in positives] +
[(n, [ 0.0], "Negative") for n in negatives])

In [36]:

ds.shuffle()

In [37]:

ds.split(.1)

In [38]:

net.set_dataset(ds)

In [39]:

net.test(tolerance=0.4)

========================================================
Testing validation dataset with tolerance 0.4...
Total count: 900
correct: 449
incorrect: 451
Total percentage correct: 0.4988888888888889

In [40]:

net.dashboard()

In [41]:

net.plot_activation_map(scatter=net.test(interactive=False), symbols=symbols, title="Before Training")


You may want to either net.reset() or net.retrain() if the following cell doesn’t complete with 100% accuracy. Calling net.reset() may be needed if the network has landed in a local maxima; net.retrain() may be necessary if the network just needs additional training.

In [44]:

if net.saved():
net.plot_results()
else:
net.train(epochs=10000, accuracy=1.0, report_rate=50,
tolerance=0.4, batch_size=256,
plot=True, record=100, save=True)

========================================================
|  Training |  Training |  Validate |  Validate
Epochs |     Error |  Accuracy |     Error |  Accuracy
------ | --------- | --------- | --------- | ---------
#17746 |   0.02727 |   1.00000 |   0.02608 |   1.00000
Saving network... Saved!

In [45]:

net.plot_activation_map(scatter=net.test(interactive=False), symbols=symbols, title="After Training")

In [46]:

net.get_weights("hidden")

Out[46]:

[[[-6.404449939727783,
7.37056827545166,
-12.947518348693848,
7.471460819244385,
-7.8771443367004395],
[-11.006869316101074,
11.604531288146973,
-1.4833985567092896,
-14.498926162719727,
-12.94011116027832]],
[5.999775409698486,
9.866920471191406,
10.296037673950195,
7.081101894378662,
7.790738582611084]]

In [47]:

net.get_weights_as_image("hidden").resize((400, 200))

Out[47]:

In [48]:

net.get_weights("output")

Out[48]:

[[[-26.79941177368164],
[-7.317086696624756],
[14.472185134887695],
[12.987898826599121],
[9.788432121276855]],
[-10.76565933227539]]

In [49]:

net.get_weights_as_image("output").resize((500, 100))

Out[49]:

In [50]:

for y in cx.frange(0, 1.1, .1):
for x in cx.frange(1.0, 0.1, -0.1):
print(ascii(net.propagate([x, y])[0]), end=" ")
print()

 0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
0.0  0.0  0.1  0.6  0.7  0.3  0.1  0.0  0.0
0.0  0.0  0.3  0.9  1.0  1.0  0.8  0.3  0.0
0.0  0.0  0.3  1.0  1.0  1.0  1.0  1.0  0.6
0.0  0.0  0.3  1.0  1.0  1.0  1.0  1.0  0.9
0.0  0.0  0.2  0.9  1.0  1.0  1.0  1.0  0.7
0.0  0.0  0.1  0.8  1.0  1.0  0.9  0.5  0.1
0.0  0.0  0.0  0.1  0.4  0.4  0.2  0.1  0.0
0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0
0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0  0.0

In [51]:

net.playback(lambda net, epoch: net.plot_activation_map(title="Epoch: %s" % epoch,
scatter=net.test(interactive=False),
symbols=symbols,
format="svg"))

In [52]:

net.movie(lambda net, epoch: net.plot_activation_map(title="Epoch %s" % epoch,
scatter=net.test(tolerance=0.4, interactive=False),
symbols=symbols,
format="image"))

Out[52]: