Entity Embedding

ACTL3143 & ACTL5111 Deep Learning for Actuaries

Patrick Laub

Entity Embedding

Lecture Outline

• Entity Embedding

• Categorical Variables & Entity Embeddings

• Keras’ Functional API

• French Motor Dataset with Embeddings

• Scale By Exposure

Revisit the French motor dataset

from pathlib import Path
from sklearn.datasets import fetch_openml

if not Path("french-motor.csv").exists():
    freq = fetch_openml(data_id=41214, as_frame=True).frame
    freq.to_csv("french-motor.csv", index=False)
else:
    freq = pd.read_csv("french-motor.csv")

freq

	IDpol	ClaimNb	Exposure	Area	VehPower	VehAge	DrivAge	BonusMalus	VehBrand	VehGas	Density	Region
0	1.0	1	0.10000	D	5	0	55	50	B12	'Regular'	1217	R82
1	3.0	1	0.77000	D	5	0	55	50	B12	'Regular'	1217	R82
...	...	...	...	...	...	...	...	...	...	...	...	...
678011	6114329.0	0	0.00274	B	4	0	60	50	B12	'Regular'	95	R26
678012	6114330.0	0	0.00274	B	7	6	29	54	B12	'Diesel'	65	R72

678013 rows × 12 columns

Data dictionary

Variable	Description	Preprocessing
IDpol	Policy number (unique identifier)	Dropped
ClaimNb	Number of claims on the given policy	Target
Exposure*	Total exposure in yearly units	Normalised
Area	Area code (ordinal)	Ordinal Encode
VehPower	Power of the car (ordinal encoded)	Normalised
VehAge	Age of the car in years	Normalised
DrivAge	Age of the (most common) driver in years	Normalised
BonusMalus	Bonus–malus level between 50 and 230 (with reference level 100)	Normalised
VehBrand*	Car brand (nominal)	One-hot
VehGas	Diesel or regular fuel car (binary)	One-hot
Density	Density of inhabitants per km2 in the city of the living place of the driver	Normalised
Region*	Regions in France (prior to 2016)	One-hot

The model

Have \{ (\mathbf{x}_i, y_i) \}_{i=1, \dots, n} for \mathbf{x}_i \in \mathbb{R}^{47} and y_i \in \mathbb{N}_0.

Assume the distribution Y_i \sim \mathsf{Poisson}(\lambda(\mathbf{x}_i))

We have \mathbb{E} Y_i = \lambda(\mathbf{x}_i). The NN takes \mathbf{x}_i & predicts \mathbb{E} Y_i.

Note

For insurance, this is a bit weird. The exposures are different for each policy.

\lambda(\mathbf{x}_i) is the expected number of claims for the duration of policy i’s contract.

Normally, \text{Exposure}_i \not\in \mathbf{x}_i, and \lambda(\mathbf{x}_i) is the expected rate per year, then Y_i \sim \mathsf{Poisson}(\text{Exposure}_i \times \lambda(\mathbf{x}_i)).

What values do we see in the data?

freq = freq.drop("IDpol", axis=1).head(25_000)

X_train, X_test, y_train, y_test = train_test_split(
  freq.drop("ClaimNb", axis=1), freq["ClaimNb"], random_state=36861)

# Reset each index to start at 0 again.
X_train_raw = X_train.reset_index(drop=True)
X_test_raw = X_test.reset_index(drop=True)

X_train_raw["Area"].value_counts()
X_train_raw["VehBrand"].value_counts()
X_train_raw["VehGas"].value_counts()
X_train_raw["Region"].value_counts()

Area
C    5514
D    4116
     ... 
B    2387
F     444
Name: count, Length: 6, dtype: int64

VehBrand
B1     4998
B2     4906
       ... 
B11     283
B14     140
Name: count, Length: 11, dtype: int64

VehGas
'Regular'    10658
'Diesel'      8092
Name: count, dtype: int64

Region
R24    6493
R82    2112
       ... 
R42      48
R43      26
Name: count, Length: 22, dtype: int64

How we preprocessed last time

from sklearn.compose import make_column_transformer

ct = make_column_transformer(
  (OneHotEncoder(sparse_output=False, drop="first"), ["VehGas", "VehBrand", "Region"]),
  (OrdinalEncoder(), ["Area"]),
  remainder=StandardScaler(),
  verbose_feature_names_out=False
)
X_train = ct.fit_transform(X_train_raw)

X_train_raw.head(3)

	Exposure	Area	VehPower	VehAge	DrivAge	BonusMalus	VehBrand	VehGas	Density	Region
0	1.00	A	7	8	50	52	B2	'Diesel'	13	R24
1	0.79	B	7	7	28	80	B12	'Diesel'	65	R21
2	1.00	C	6	13	30	50	B1	'Regular'	133	R53

X_train.head(3)

	VehGas_'Regular'	VehBrand_B10	VehBrand_B11	VehBrand_B12	VehBrand_B13	VehBrand_B14	VehBrand_B2	VehBrand_B3	VehBrand_B4	VehBrand_B5	...	Region_R91	Region_R93	Region_R94	Area	Exposure	VehPower	VehAge	DrivAge	BonusMalus	Density
0	0.0	0.0	0.0	0.0	0.0	0.0	1.0	0.0	0.0	0.0	...	0.0	0.0	0.0	0.0	1.129272	0.366510	0.223226	0.374405	-0.524020	-0.394690
1	0.0	0.0	0.0	1.0	0.0	0.0	0.0	0.0	0.0	0.0	...	0.0	0.0	0.0	1.0	0.566087	0.366510	0.046100	-1.131699	1.122382	-0.381092
2	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	2.0	1.129272	-0.167408	1.108854	-0.994781	-0.641620	-0.363309

3 rows × 39 columns

Categorical Variables & Entity Embeddings

Lecture Outline

• Entity Embedding

• Categorical Variables & Entity Embeddings

• Keras’ Functional API

• French Motor Dataset with Embeddings

• Scale By Exposure

Region column

French Administrative Regions

One-hot encoding

oh = OneHotEncoder(sparse_output=False)
X_train_oh = oh.fit_transform(X_train_raw[["Region"]])
X_test_oh = oh.transform(X_test_raw[["Region"]])
print(list(X_train_raw["Region"][:5]))
X_train_oh.head()

['R24', 'R21', 'R53', 'R24', 'R82']

	Region_R11	Region_R21	Region_R22	Region_R23	Region_R24	Region_R25	Region_R26	Region_R31	Region_R41	Region_R42	...	Region_R53	Region_R54	Region_R72	Region_R73	Region_R74	Region_R82	Region_R83	Region_R91	Region_R93	Region_R94
0	0.0	0.0	0.0	0.0	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
1	0.0	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
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
3	0.0	0.0	0.0	0.0	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
4	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	0.0	0.0	0.0	0.0

5 rows × 22 columns

Train on one-hot inputs

num_regions = len(oh.categories_[0])

random.seed(12)
model = Sequential([
  Dense(2, input_dim=num_regions),
  Dense(1, activation="exponential")
])

model.compile(optimizer="adam", loss="poisson")

es = EarlyStopping(verbose=True)
hist = model.fit(X_train_oh, y_train, epochs=100, verbose=0,
    validation_split=0.2, callbacks=[es])                       
hist.history["val_loss"][-1]

Epoch 7: early stopping

0.7678562998771667

Make a fake batch of data

X = np.eye(num_regions)
pd.DataFrame(X, columns=oh.categories_[0])

	R11	R21	R22	R23	R24	R25	R26	R31	R41	R42	...	R53	R54	R72	R73	R74	R82	R83	R91	R93	R94
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	0.0
1	0.0	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
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
20	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	0.0
21	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

22 rows × 22 columns

model.layers[0](X)

<tf.Tensor: shape=(22, 2), dtype=float32, numpy=
array([[-0.2 , -0.12],
       [ 0.18, -0.19],
       [-0.21,  0.11],
       [-0.82,  0.11],
       [-0.03, -0.68],
       [-0.69, -0.17],
       [-0.32, -0.37],
       [ 0.24, -0.  ],
       [-0.9 , -0.54],
       [ 0.25, -0.36],
       [-0.26, -0.06],
       [-1.11, -0.32],
       [ 0.17, -0.68],
       [-0.94, -0.58],
       [-0.14,  0.05],
       [ 0.1 ,  0.  ],
       [-0.46, -0.37],
       [-0.59, -0.34],
       [-0.4 , -0.46],
       [-0.19,  0.18],
       [ 0.32, -0.14],
       [-0.3 ,  0.35]], dtype=float32)>

The first layer

layer = model.layers[0]
W, b = layer.get_weights()
X.shape, W.shape, b.shape

((22, 22), (22, 2), (2,))

X @ W + b

array([[-0.2 , -0.12],
       [ 0.18, -0.19],
       [-0.21,  0.11],
       [-0.82,  0.11],
       [-0.03, -0.68],
       [-0.69, -0.17],
       [-0.32, -0.37],
       [ 0.24, -0.  ],
       [-0.9 , -0.54],
       [ 0.25, -0.36],
       [-0.26, -0.06],
       [-1.11, -0.32],
       [ 0.17, -0.68],
       [-0.94, -0.58],
       [-0.14,  0.05],
       [ 0.1 ,  0.  ],
       [-0.46, -0.37],
       [-0.59, -0.34],
       [-0.4 , -0.46],
       [-0.19,  0.18],
       [ 0.32, -0.14],
       [-0.3 ,  0.35]])

W + b

array([[-0.2 , -0.12],
       [ 0.18, -0.19],
       [-0.21,  0.11],
       [-0.82,  0.11],
       [-0.03, -0.68],
       [-0.69, -0.17],
       [-0.32, -0.37],
       [ 0.24, -0.  ],
       [-0.9 , -0.54],
       [ 0.25, -0.36],
       [-0.26, -0.06],
       [-1.11, -0.32],
       [ 0.17, -0.68],
       [-0.94, -0.58],
       [-0.14,  0.05],
       [ 0.1 ,  0.  ],
       [-0.46, -0.37],
       [-0.59, -0.34],
       [-0.4 , -0.46],
       [-0.19,  0.18],
       [ 0.32, -0.14],
       [-0.3 ,  0.35]], dtype=float32)

Just a look-up operation

display(list(oh.categories_[0]))

['R11',
 'R21',
 'R22',
 'R23',
 'R24',
 'R25',
 'R26',
 'R31',
 'R41',
 'R42',
 'R43',
 'R52',
 'R53',
 'R54',
 'R72',
 'R73',
 'R74',
 'R82',
 'R83',
 'R91',
 'R93',
 'R94']

W + b

array([[-0.2 , -0.12],
       [ 0.18, -0.19],
       [-0.21,  0.11],
       [-0.82,  0.11],
       [-0.03, -0.68],
       [-0.69, -0.17],
       [-0.32, -0.37],
       [ 0.24, -0.  ],
       [-0.9 , -0.54],
       [ 0.25, -0.36],
       [-0.26, -0.06],
       [-1.11, -0.32],
       [ 0.17, -0.68],
       [-0.94, -0.58],
       [-0.14,  0.05],
       [ 0.1 ,  0.  ],
       [-0.46, -0.37],
       [-0.59, -0.34],
       [-0.4 , -0.46],
       [-0.19,  0.18],
       [ 0.32, -0.14],
       [-0.3 ,  0.35]], dtype=float32)

Turn the region into an index

oe = OrdinalEncoder()
X_train_reg = oe.fit_transform(X_train_raw[["Region"]])
X_test_reg = oe.transform(X_test_raw[["Region"]])

for i, reg in enumerate(oe.categories_[0][:3]):
  print(f"The Region value {reg} gets turned into {i}.")

The Region value R11 gets turned into 0.
The Region value R21 gets turned into 1.
The Region value R22 gets turned into 2.

Use an Embedding layer

from keras.layers import Embedding
num_regions = X_train_raw["Region"].nunique()

random.seed(12)
model = Sequential([
  Embedding(input_dim=num_regions, output_dim=2),
  Dense(1, activation="exponential")
])

model.compile(optimizer="adam", loss="poisson")

es = EarlyStopping(verbose=True)
hist = model.fit(X_train_reg, y_train, epochs=100, verbose=0,
    validation_split=0.2, callbacks=[es])
hist.history["val_loss"][-1]

Epoch 7: early stopping

0.7678869366645813

model.layers

[<Embedding name=embedding, built=True>, <Dense name=dense_2, built=True>]

Keras’ Embedding Layer

model.layers[0].get_weights()[0]

array([[-0.11, -0.1 ],
       [ 0.04,  0.  ],
       [-0.01,  0.02],
       [-0.24, -0.13],
       [-0.31, -0.35],
       [-0.33, -0.25],
       [-0.28, -0.25],
       [ 0.12,  0.08],
       [-0.6 , -0.5 ],
       [-0.01, -0.06],
       [-0.09, -0.06],
       [-0.58, -0.44],
       [-0.24, -0.29],
       [-0.67, -0.56],
       [-0.01,  0.01],
       [ 0.07,  0.05],
       [-0.35, -0.31],
       [-0.38, -0.32],
       [-0.3 , -0.28],
       [ 0.04,  0.07],
       [ 0.09,  0.03],
       [ 0.07,  0.13]], dtype=float32)

X_train_raw["Region"].head(4)

0    R24
1    R21
2    R53
3    R24
Name: Region, dtype: object

X_sample = X_train_reg[:4].to_numpy()
X_sample

array([[ 4.],
       [ 1.],
       [12.],
       [ 4.]])

enc_tensor = model.layers[0](X_sample)
keras.ops.convert_to_numpy(enc_tensor).squeeze()

array([[-0.31, -0.35],
       [ 0.04,  0.  ],
       [-0.24, -0.29],
       [-0.31, -0.35]], dtype=float32)

The learned embeddings

points = model.layers[0].get_weights()[0]
plt.scatter(points[:,0], points[:,1])
for i in range(num_regions):
  plt.text(points[i,0]+0.01, points[i,1] , s=oe.categories_[0][i])

Entity embeddings

Embeddings will gradually improve during training.

Embeddings & other inputs

Illustration of a neural network with both continuous and categorical inputs.

We can’t do this with Sequential models…

Keras’ Functional API

Lecture Outline

• Entity Embedding

• Categorical Variables & Entity Embeddings

• Keras’ Functional API

• French Motor Dataset with Embeddings

• Scale By Exposure

Converting Sequential models

from keras.models import Model
from keras.layers import Input

random.seed(12)

model = Sequential([
  Dense(30, "leaky_relu"),
  Dense(1, "exponential")
])

model.compile(
  optimizer="adam",
  loss="poisson")

hist = model.fit(
  X_train_oh, y_train,
  epochs=1, verbose=0,
  validation_split=0.2)
hist.history["val_loss"][-1]

0.7700941562652588

random.seed(12)

inputs = Input(shape=(X_train_oh.shape[1],))
x = Dense(30, "leaky_relu")(inputs)
out = Dense(1, "exponential")(x)
model = Model(inputs, out)

model.compile(
  optimizer="adam",
  loss="poisson")

hist = model.fit(
  X_train_oh, y_train,
  epochs=1, verbose=0,
  validation_split=0.2)
hist.history["val_loss"][-1]

0.7700941562652588

See one-length tuples.

Wide & Deep network

An illustration of the wide & deep network architecture.

Add a skip connection from input to output layers.

from keras.layers \
    import Concatenate

inp = Input(shape=X_train.shape[1:])
hidden1 = Dense(30, "leaky_relu")(inp)
hidden2 = Dense(30, "leaky_relu")(hidden1)
concat = Concatenate()(
  [inp, hidden2])
output = Dense(1)(concat)
model = Model(
    inputs=[inp],
    outputs=[output])

Naming the layers

For complex networks, it is often useful to give meaningful names to the layers.

input_ = Input(shape=X_train.shape[1:], name="input")
hidden1 = Dense(30, activation="leaky_relu", name="hidden1")(input_)
hidden2 = Dense(30, activation="leaky_relu", name="hidden2")(hidden1)
concat = Concatenate(name="combined")([input_, hidden2])
output = Dense(1, name="output")(concat)
model = Model(inputs=[input_], outputs=[output])

Inspecting a complex model

from keras.utils import plot_model

plot_model(model, show_layer_names=True)

model.summary(line_length=75)

Model: "functional_5"

┏━━━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┳━━━━━━━━━━━┳━━━━━━━━━━━━━━━━━━━┓
┃ Layer (type)        ┃ Output Shape      ┃   Param # ┃ Connected to      ┃
┡━━━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━╇━━━━━━━━━━━╇━━━━━━━━━━━━━━━━━━━┩
│ input (InputLayer)  │ (None, 39)        │         0 │ -                 │
├─────────────────────┼───────────────────┼───────────┼───────────────────┤
│ hidden1 (Dense)     │ (None, 30)        │     1,200 │ input[0][0]       │
├─────────────────────┼───────────────────┼───────────┼───────────────────┤
│ hidden2 (Dense)     │ (None, 30)        │       930 │ hidden1[0][0]     │
├─────────────────────┼───────────────────┼───────────┼───────────────────┤
│ combined            │ (None, 69)        │         0 │ input[0][0],      │
│ (Concatenate)       │                   │           │ hidden2[0][0]     │
├─────────────────────┼───────────────────┼───────────┼───────────────────┤
│ output (Dense)      │ (None, 1)         │        70 │ combined[0][0]    │
└─────────────────────┴───────────────────┴───────────┴───────────────────┘

 Total params: 2,200 (8.59 KB)

 Trainable params: 2,200 (8.59 KB)

 Non-trainable params: 0 (0.00 B)

French Motor Dataset with Embeddings

Lecture Outline

• Entity Embedding

• Categorical Variables & Entity Embeddings

• Keras’ Functional API

• French Motor Dataset with Embeddings

• Scale By Exposure

The desired architecture

Illustration of a neural network with both continuous and categorical inputs.

Preprocess all French motor inputs

Transform the categorical variables to integers:

num_brands, num_regions = X_train_raw[["VehBrand", "Region"]].nunique()

ct = make_column_transformer(
  (OrdinalEncoder(), ["VehBrand", "Region", "Area", "VehGas"]),
  remainder=StandardScaler(),
  verbose_feature_names_out=False
)
X_train = ct.fit_transform(X_train_raw)
X_test = ct.transform(X_test_raw)

Split the brand and region data apart from the rest:

X_train_brand = X_train["VehBrand"]
X_train_region = X_train["Region"]
X_train_rest = X_train.drop(["VehBrand", "Region"], axis=1)

X_test_brand = X_test["VehBrand"]
X_test_region = X_test["Region"]
X_test_rest = X_test.drop(["VehBrand", "Region"], axis=1)

Organise the inputs

Make a Keras Input for: vehicle brand, region, & others.

veh_brand = Input(shape=(1,), name="veh_brand")
region = Input(shape=(1,), name="region")
other_inputs = Input(shape=X_train_rest.shape[1:], name="other_inputs")

Create embeddings and join them with the other inputs.

from keras.layers import Reshape

random.seed(1337)
veh_brand_ee = Embedding(input_dim=num_brands, output_dim=2,
    name="veh_brand_ee")(veh_brand)                                
veh_brand_ee = Reshape(target_shape=(2,))(veh_brand_ee)

region_ee = Embedding(input_dim=num_regions, output_dim=2,
    name="region_ee")(region)
region_ee = Reshape(target_shape=(2,))(region_ee)

x = Concatenate(name="combined")([veh_brand_ee, region_ee, other_inputs])

Complete the model and fit it

Feed the combined embeddings & continuous inputs to some normal dense layers.

x = Dense(30, "relu", name="hidden")(x)
out = Dense(1, "exponential", name="out")(x)

model = Model([veh_brand, region, other_inputs], out)
model.compile(optimizer="adam", loss="poisson")

hist = model.fit((X_train_brand, X_train_region, X_train_rest),
    y_train, epochs=100, verbose=0,
    callbacks=[EarlyStopping(patience=5)], validation_split=0.2)
np.min(hist.history["val_loss"])

np.float64(0.6845806837081909)

Plotting this model

plot_model(model, show_layer_names=True)

Why we need to reshape

plot_model(model, show_layer_names=True, show_shapes=True)

Scale By Exposure

Lecture Outline

• Entity Embedding

• Categorical Variables & Entity Embeddings

• Keras’ Functional API

• French Motor Dataset with Embeddings

• Scale By Exposure

Two different models

Have \{ (\mathbf{x}_i, y_i) \}_{i=1, \dots, n} for \mathbf{x}_i \in \mathbb{R}^{47} and y_i \in \mathbb{N}_0.

Model 1: Say Y_i \sim \mathsf{Poisson}(\lambda(\mathbf{x}_i)).

But, the exposures are different for each policy. \lambda(\mathbf{x}_i) is the expected number of claims for the duration of policy i’s contract.

Model 2: Say Y_i \sim \mathsf{Poisson}(\text{Exposure}_i \times \lambda(\mathbf{x}_i)).

Now, \text{Exposure}_i \not\in \mathbf{x}_i, and \lambda(\mathbf{x}_i) is the rate per year.

Just take continuous variables

ct = make_column_transformer(
  ("passthrough", ["Exposure"]),
  ("drop", ["VehBrand", "Region", "Area", "VehGas"]),
  remainder=StandardScaler(),
  verbose_feature_names_out=False
)
X_train = ct.fit_transform(X_train_raw)
X_test = ct.transform(X_test_raw)

Split exposure apart from the rest:

X_train_exp = X_train["Exposure"]
X_test_exp = X_test["Exposure"]
X_train_rest = X_train.drop("Exposure", axis=1)
X_test_rest = X_test.drop("Exposure", axis=1)

Organise the inputs:

exposure = Input(shape=(1,), name="exposure")
other_inputs = Input(shape=X_train_rest.shape[1:], name="other_inputs")

Make & fit the model

Feed the continuous inputs to some normal dense layers.

random.seed(1337)
x = Dense(30, "relu", name="hidden1")(other_inputs)
x = Dense(30, "relu", name="hidden2")(x)
lambda_ = Dense(1, "exponential", name="lambda")(x)

out = lambda_ * exposure # In past, need keras.layers.Multiply()[lambda_, exposure]
model = Model([exposure, other_inputs], out)
model.compile(optimizer="adam", loss="poisson")

es = EarlyStopping(patience=10, restore_best_weights=True, verbose=1)
hist = model.fit((X_train_exp, X_train_rest),
    y_train, epochs=100, verbose=0,
    callbacks=[es], validation_split=0.2)
np.min(hist.history["val_loss"])

Epoch 74: early stopping
Restoring model weights from the end of the best epoch: 64.

np.float64(0.9126634001731873)

Plot the model

plot_model(model, show_layer_names=True)

Package Versions

from watermark import watermark
print(watermark(python=True, packages="keras,matplotlib,numpy,pandas,seaborn,scipy,torch,tensorflow,tf_keras"))

Python implementation: CPython
Python version       : 3.11.12
IPython version      : 9.3.0

keras     : 3.8.0
matplotlib: 3.10.0
numpy     : 2.0.2
pandas    : 2.2.2
seaborn   : 0.13.2
scipy     : 1.15.3
torch     : 2.6.0+cu124
tensorflow: 2.18.0
tf_keras  : 2.18.0

Glossary

• entity embeddings
• Input layer
• Keras functional API

• Reshape layer
• skip connection
• wide & deep network