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]:
from conx import *
import random
Using Theano backend.
conx, version 3.5.3

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]:
scatter([
         ["Positive", positives],
         ["Negative", negatives],
        ],
    height=8.0,
    width=8.0,
    symbols={"Positive": "bo", "Negative": "ro"})
_images/Learning_4_0.png
In [4]:
ds = 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()
Input Summary:
   count  : 1000 (900 for training, 100 for testing)
   shape  : [(2,)]
   range  : (0.0, 0.998)
Target Summary:
   count  : 1000 (900 for training, 100 for testing)
   shape  : [(1,)]
   range  : (0.0, 1.0)
In [9]:
net = 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: 0
      incorrect: 900
Total percentage correct: 0.0
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]:
net.train(epochs=10000, accuracy=1.0, report_rate=50,
         tolerance=0.4, batch_size=len(net.dataset.train_inputs),
         plot=True, record=100)
_images/Learning_15_0.svg
========================================================================
       |  Training |  Training |  Validate |  Validate
Epochs |     Error |  Accuracy |     Error |  Accuracy
------ | --------- | --------- | --------- | ---------
# 7729 |   0.26827 |   1.00000 |   0.26607 |   1.00000
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]:
[[[3.3333942890167236], [-6.821496486663818]], [1.604598879814148]]
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 frange(0, 1.1, .1):
    for x in 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  0.6
 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  0.1  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
 0.1  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  0.9
 0.1  0.2  0.3  0.5  0.6  0.8  0.9  0.9  1.0
 0.1  0.2  0.4  0.5  0.7  0.8  0.9  0.9  1.0
In [19]:
net.playback(lambda net, epoch: net.plot_activation_map(title="Epoch %s" % epoch,
                                                        scatter=net.test(tolerance=0.4, interactive=False),
                                                        symbols=symbols,
                                                        interactive=False))
_images/Learning_20_1.svg

3.2. Non-Linearly Separable

In [20]:
import math
In [21]:
def distance(x1, y1, x2, y2):
    return math.sqrt((x1 - x2) ** 2 + (y1 - y2) ** 2)
In [22]:
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 [23]:
scatter([
         ["Positive", positives],
         ["Negative", negatives],
        ],
    height=8.0,
    width=8.0,
    symbols={"Positive": "bo", "Negative": "ro"})
_images/Learning_25_0.png
In [24]:
net = Network("Non-Linearly Separable", 2, 5, 1, activation="sigmoid")
net.compile(error="mae", optimizer="adam")
In [25]:
net
Out[25]:
Non-Linearly SeparableLayer: output (output) 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) 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) shape = (2,) Keras class = Inputinput
In [26]:
ds = Dataset()
In [27]:
ds.load([(p, [ 1.0], "Positive") for p in positives] +
        [(n, [ 0.0], "Negative") for n in negatives])
In [28]:
ds.shuffle()
In [29]:
ds.split(.1)
In [30]:
net.set_dataset(ds)
In [31]:
net.test(tolerance=0.4)
========================================================
Testing validation dataset with tolerance 0.4...
Total count: 900
      correct: 0
      incorrect: 900
Total percentage correct: 0.0
In [32]:
net.dashboard()
In [33]:
net.plot_activation_map(scatter=net.test(interactive=False), symbols=symbols, title="Before Training")
_images/Learning_35_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 [35]:
net.train(epochs=10000, accuracy=1.0, report_rate=50,
          tolerance=0.4, batch_size=len(net.dataset.train_inputs),
          plot=True, record=100)
_images/Learning_37_0.svg
========================================================================
       |  Training |  Training |  Validate |  Validate
Epochs |     Error |  Accuracy |     Error |  Accuracy
------ | --------- | --------- | --------- | ---------
#14856 |   0.02812 |   1.00000 |   0.01998 |   1.00000
In [36]:
net.plot_activation_map(scatter=net.test(interactive=False), symbols=symbols, title="After Training")
_images/Learning_38_0.png
In [37]:
net.get_weights("hidden")
Out[37]:
[[[-10.24807071685791,
   -3.3982436656951904,
   -5.170109748840332,
   2.7826385498046875,
   -13.015877723693848],
  [-3.794675827026367,
   -7.523599624633789,
   -0.7802948951721191,
   8.869245529174805,
   12.65350341796875]],
 [4.493064880371094,
  8.14928913116455,
  -4.387434482574463,
  -8.400566101074219,
  4.322581768035889]]
In [38]:
net.get_weights_as_image("hidden").resize((400, 200))
Out[38]:
_images/Learning_40_0.png
In [39]:
net.get_weights("output")
Out[39]:
[[[-15.507240295410156],
  [8.740614891052246],
  [0.7526621222496033],
  [-13.142610549926758],
  [12.83068561553955]],
 [-11.701411247253418]]
In [40]:
net.get_weights_as_image("output").resize((500, 100))
Out[40]:
_images/Learning_42_0.png
In [41]:
for y in frange(0, 1.1, .1):
    for x in 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  0.0  0.1  0.3  0.8  0.8  0.3  0.0
 0.0  0.0  0.1  0.4  1.0  1.0  1.0  0.7  0.1
 0.0  0.0  0.2  0.9  1.0  1.0  1.0  0.9  0.2
 0.0  0.0  0.7  1.0  1.0  1.0  1.0  1.0  0.4
 0.0  0.0  0.7  1.0  1.0  1.0  1.0  1.0  0.5
 0.0  0.0  0.2  0.6  0.9  1.0  1.0  0.9  0.5
 0.0  0.0  0.0  0.0  0.1  0.3  0.5  0.5  0.2
 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 [42]:
net.playback(lambda net, epoch: net.plot_activation_map(title="Epoch: %s" % epoch,
                                                        scatter=net.test(interactive=False),
                                                        symbols=symbols,
                                                        interactive=False))
_images/Learning_44_1.svg

3.2.1. Non-Linearly Separable - Deeper

In [43]:
net = Network("Non-Linearly Separable", 2, 5, 2, 1, activation="sigmoid")
net.compile(error="mae", optimizer="adam")
In [44]:
net.set_dataset(ds)
In [45]:
net.dashboard()
In [46]:
net.train(epochs=25000, accuracy=1.0, report_rate=50,
          tolerance=0.4, batch_size=len(net.dataset.train_inputs),
          plot=True, record=100)
_images/Learning_49_0.svg
========================================================================
       |  Training |  Training |  Validate |  Validate
Epochs |     Error |  Accuracy |     Error |  Accuracy
------ | --------- | --------- | --------- | ---------
#11268 |   0.01345 |   1.00000 |   0.00938 |   1.00000
In [48]:
net.plot_activation_map()
net.plot_activation_map("hidden2")
_images/Learning_50_0.png
_images/Learning_50_1.png
In [49]:
net.playback(lambda net, epoch: net.plot_activation_map(title="Epoch %s" % epoch,
                                                        scatter=net.test(interactive=False),
                                                        symbols=symbols,
                                                        interactive=False))
_images/Learning_51_1.svg
_images/Learning_51_2.png
In [54]:
%%time
net.movie(lambda net, epoch: net.plot_activation_map(title="Epoch %s" % epoch,
                                                        scatter=net.test(interactive=False),
                                                        symbols=symbols,
                                                        interactive=False, format="pil"),
         step=1, duration=200)
CPU times: user 46.6 s, sys: 34 s, total: 1min 20s
Wall time: 41.6 s
Out[54]: