# Multi-Class Single-Label classification¶

The natural extension of binary classification is a multi-class classification task. We first approach multi-class single-label classification, which makes the assumption that each example is assigned to one and only one label.

We use the Iris flower data set, which consists of a classification into three mutually-exclusive classes; call these $A$, $B$ and $C$.

While one could train three unary predicates $A(x)$, $B(x)$ and $C(x)$, it turns out to be more effective if this problem is modelled by a single binary predicate $P(x,l)$, where $l$ is a variable denoting a multi-class label, in this case classes $A$, $B$ or $C$.

• This syntax allows one to write statements quantifying over the classes, e.g. $\forall x ( \exists l ( P(x,l)))$.
• Since the classes are mutually-exclusive in this case, the output layer of the $\mathtt{MLP}$ representing $P(x,l)$ will be a $\mathtt{softmax}$ layer, instead of a $\mathtt{sigmoid}$ function, to learn the probability of $A$, $B$ and $C$. This avoids writing additional constraints $\lnot (A(x) \land B(x))$, $\lnot (A(x) \land C(x))$, ...
In [1]:
import logging; logging.basicConfig(level=logging.INFO)
import tensorflow as tf
import pandas as pd
import logictensornetworks as ltn

Init Plugin
Init Graph Optimizer
Init Kernel


# Data¶

Load the iris dataset: 50 samples from each of three species of iris flowers (setosa, virginica, versicolor), measured with four features.

In [2]:
df_train = pd.read_csv("iris_training.csv")

   sepal_length  sepal_width  petal_length  petal_width  species
0           6.4          2.8           5.6          2.2        2
1           5.0          2.3           3.3          1.0        1
2           4.9          2.5           4.5          1.7        2
3           4.9          3.1           1.5          0.1        0
4           5.7          3.8           1.7          0.3        0

In [3]:
labels_train = df_train.pop("species")
labels_test = df_test.pop("species")
batch_size = 64
ds_train = tf.data.Dataset.from_tensor_slices((df_train,labels_train)).batch(batch_size)
ds_test = tf.data.Dataset.from_tensor_slices((df_test,labels_test)).batch(batch_size)

Metal device set to: Apple M1

systemMemory: 16.00 GB
maxCacheSize: 5.33 GB


2021-08-30 14:38:15.642262: I tensorflow/core/common_runtime/pluggable_device/pluggable_device_factory.cc:305] Could not identify NUMA node of platform GPU ID 0, defaulting to 0. Your kernel may not have been built with NUMA support.
2021-08-30 14:38:15.642359: I tensorflow/core/common_runtime/pluggable_device/pluggable_device_factory.cc:271] Created TensorFlow device (/job:localhost/replica:0/task:0/device:GPU:0 with 0 MB memory) -> physical PluggableDevice (device: 0, name: METAL, pci bus id: <undefined>)


# LTN¶

Predicate with softmax P(x,class)

In [4]:
class MLP(tf.keras.Model):
"""Model that returns logits."""
def __init__(self, n_classes, hidden_layer_sizes=(16,16,8)):
super(MLP, self).__init__()
self.denses = [tf.keras.layers.Dense(s, activation="elu") for s in hidden_layer_sizes]
self.dense_class = tf.keras.layers.Dense(n_classes)
self.dropout = tf.keras.layers.Dropout(0.2)

def call(self, inputs, training=False):
x = inputs
for dense in self.denses:
x = dense(x)
x = self.dropout(x, training=training)
return self.dense_class(x)

logits_model = MLP(4)
p = ltn.Predicate(ltn.utils.LogitsToPredicateModel(logits_model,single_label=True))


Constants to index/iterate on the classes

In [5]:
class_A = ltn.Constant(0, trainable=False)
class_B = ltn.Constant(1, trainable=False)
class_C = ltn.Constant(2, trainable=False)


Operators and axioms

In [6]:
Not = ltn.Wrapper_Connective(ltn.fuzzy_ops.Not_Std())
And = ltn.Wrapper_Connective(ltn.fuzzy_ops.And_Prod())
Or = ltn.Wrapper_Connective(ltn.fuzzy_ops.Or_ProbSum())
Implies = ltn.Wrapper_Connective(ltn.fuzzy_ops.Implies_Reichenbach())
Forall = ltn.Wrapper_Quantifier(ltn.fuzzy_ops.Aggreg_pMeanError(p=2),semantics="forall")

In [7]:
formula_aggregator = ltn.Wrapper_Formula_Aggregator(ltn.fuzzy_ops.Aggreg_pMeanError(p=2))

@tf.function
def axioms(features, labels, training=False):
x_A = ltn.Variable("x_A",features[labels==0])
x_B = ltn.Variable("x_B",features[labels==1])
x_C = ltn.Variable("x_C",features[labels==2])
axioms = [
Forall(x_A,p([x_A,class_A],training=training)),
Forall(x_B,p([x_B,class_B],training=training)),
Forall(x_C,p([x_C,class_C],training=training))
]
sat_level = formula_aggregator(axioms).tensor
return sat_level


Initialize all layers and the static graph

In [8]:
for features, labels in ds_test:
print("Initial sat level %.5f"%axioms(features,labels))
break

Initial sat level 0.25581

2021-08-30 14:38:20.990753: I tensorflow/compiler/mlir/mlir_graph_optimization_pass.cc:176] None of the MLIR Optimization Passes are enabled (registered 2)
2021-08-30 14:38:20.992807: W tensorflow/core/platform/profile_utils/cpu_utils.cc:128] Failed to get CPU frequency: 0 Hz
2021-08-30 14:38:20.992905: I tensorflow/core/grappler/optimizers/custom_graph_optimizer_registry.cc:112] Plugin optimizer for device_type GPU is enabled.


# Training¶

Define the metrics. While training, we measure:

1. The level of satisfiability of the Knowledge Base of the training data.
2. The level of satisfiability of the Knowledge Base of the test data.
3. The training accuracy.
4. The test accuracy.
In [9]:
metrics_dict = {
'train_sat_kb': tf.keras.metrics.Mean(name='train_sat_kb'),
'test_sat_kb': tf.keras.metrics.Mean(name='test_sat_kb'),
'train_accuracy': tf.keras.metrics.CategoricalAccuracy(name="train_accuracy"),
'test_accuracy': tf.keras.metrics.CategoricalAccuracy(name="test_accuracy")
}


Define the training and test step

In [10]:
optimizer = tf.keras.optimizers.Adam(learning_rate=0.001)
@tf.function
def train_step(features, labels):
# sat and update
sat = axioms(features, labels, training=True)
loss = 1.-sat
sat = axioms(features, labels) # compute sat without dropout
metrics_dict['train_sat_kb'](sat)
# accuracy
predictions = logits_model(features)
metrics_dict['train_accuracy'](tf.one_hot(labels,3),predictions)

@tf.function
def test_step(features, labels):
# sat
sat = axioms(features, labels)
metrics_dict['test_sat_kb'](sat)
# accuracy
predictions = logits_model(features)
metrics_dict['test_accuracy'](tf.one_hot(labels,3),predictions)


Train

In [11]:
import commons

EPOCHS = 500

commons.train(
EPOCHS,
metrics_dict,
ds_train,
ds_test,
train_step,
test_step,
csv_path="iris_results.csv",
track_metrics=20
)

2021-08-30 14:39:58.964336: I tensorflow/core/grappler/optimizers/custom_graph_optimizer_registry.cc:112] Plugin optimizer for device_type GPU is enabled.
2021-08-30 14:39:59.951405: I tensorflow/core/grappler/optimizers/custom_graph_optimizer_registry.cc:112] Plugin optimizer for device_type GPU is enabled.
2021-08-30 14:40:00.487437: I tensorflow/core/grappler/optimizers/custom_graph_optimizer_registry.cc:112] Plugin optimizer for device_type GPU is enabled.

Epoch 0, train_sat_kb: 0.2620, test_sat_kb: 0.2640, train_accuracy: 0.3000, test_accuracy: 0.4667
Epoch 20, train_sat_kb: 0.4088, test_sat_kb: 0.4085, train_accuracy: 0.7333, test_accuracy: 0.5667
Epoch 40, train_sat_kb: 0.5422, test_sat_kb: 0.5404, train_accuracy: 0.9417, test_accuracy: 0.9000
Epoch 60, train_sat_kb: 0.6432, test_sat_kb: 0.6381, train_accuracy: 0.9417, test_accuracy: 0.9000
Epoch 80, train_sat_kb: 0.7105, test_sat_kb: 0.7041, train_accuracy: 0.9583, test_accuracy: 0.9000
Epoch 100, train_sat_kb: 0.7486, test_sat_kb: 0.7443, train_accuracy: 0.9667, test_accuracy: 0.9333
Epoch 120, train_sat_kb: 0.7888, test_sat_kb: 0.7884, train_accuracy: 0.9667, test_accuracy: 0.9667
Epoch 140, train_sat_kb: 0.8182, test_sat_kb: 0.8197, train_accuracy: 0.9750, test_accuracy: 0.9667
Epoch 160, train_sat_kb: 0.8356, test_sat_kb: 0.8374, train_accuracy: 0.9750, test_accuracy: 1.0000
Epoch 180, train_sat_kb: 0.8525, test_sat_kb: 0.8457, train_accuracy: 0.9750, test_accuracy: 0.9667
Epoch 200, train_sat_kb: 0.8561, test_sat_kb: 0.8563, train_accuracy: 0.9833, test_accuracy: 0.9667
Epoch 220, train_sat_kb: 0.8706, test_sat_kb: 0.8541, train_accuracy: 0.9833, test_accuracy: 0.9667
Epoch 240, train_sat_kb: 0.8739, test_sat_kb: 0.8587, train_accuracy: 0.9833, test_accuracy: 0.9667
Epoch 260, train_sat_kb: 0.8694, test_sat_kb: 0.8635, train_accuracy: 0.9750, test_accuracy: 0.9667
Epoch 280, train_sat_kb: 0.8709, test_sat_kb: 0.8625, train_accuracy: 0.9750, test_accuracy: 0.9667
Epoch 300, train_sat_kb: 0.8782, test_sat_kb: 0.8429, train_accuracy: 0.9833, test_accuracy: 0.9667
Epoch 320, train_sat_kb: 0.8780, test_sat_kb: 0.8387, train_accuracy: 0.9833, test_accuracy: 0.9667
Epoch 340, train_sat_kb: 0.8791, test_sat_kb: 0.8614, train_accuracy: 0.9750, test_accuracy: 0.9667
Epoch 360, train_sat_kb: 0.8880, test_sat_kb: 0.8497, train_accuracy: 0.9833, test_accuracy: 0.9333
Epoch 380, train_sat_kb: 0.8894, test_sat_kb: 0.8541, train_accuracy: 0.9750, test_accuracy: 0.9333
Epoch 400, train_sat_kb: 0.8870, test_sat_kb: 0.8401, train_accuracy: 0.9917, test_accuracy: 0.9667
Epoch 420, train_sat_kb: 0.8894, test_sat_kb: 0.8402, train_accuracy: 0.9917, test_accuracy: 0.9667
Epoch 440, train_sat_kb: 0.8912, test_sat_kb: 0.8557, train_accuracy: 0.9750, test_accuracy: 0.9667
Epoch 460, train_sat_kb: 0.8953, test_sat_kb: 0.8519, train_accuracy: 0.9750, test_accuracy: 0.9333
Epoch 480, train_sat_kb: 0.8810, test_sat_kb: 0.8593, train_accuracy: 0.9750, test_accuracy: 0.9667

In [ ]: