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]:
_images/Learning_4_0.svg
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")
net.compile(error="mae", optimizer="adam")
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")
_images/Learning_14_0.png
In [14]:
if net.saved():
    net.load()
    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)
_images/Learning_15_0.svg
========================================================
       |  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")
_images/Learning_16_0.png
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"))
_images/Learning_20_1.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]:
_images/Learning_27_0.svg
In [32]:
net = cx.Network("Non-Linearly Separable", 2, 5, 1, activation="sigmoid")
net.compile(error="mae", optimizer="adam")
In [33]:
net.picture()
Out[33]:
Layer: output (output) output range: (0, 1) shape = (1,) Keras class = Dense activation = sigmoidoutputWeights from hidden to output output/kernel has shape (5, 1) output/bias has shape (1,)Layer: hidden (hidden) output range: (0, 1) shape = (5,) Keras class = Dense activation = sigmoidhiddenWeights from input to hidden hidden/kernel has shape (2, 5) hidden/bias has shape (5,)Layer: input (input) output range: (-Infinity, +Infinity) shape = (2,) Keras class = InputinputNon-Linearly Separable
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")
_images/Learning_37_0.png

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.load()
    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)
_images/Learning_39_0.svg
========================================================
       |  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")
_images/Learning_40_0.png
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]:
_images/Learning_42_0.png
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]:
_images/Learning_44_0.png
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"))
_images/Learning_46_1.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]: