How to develop a Stacking Ensemble for Deep Learning Neural Networks in Python with Keras

Model averaging is an ensemble technique where multiple sub-models contribute equally to a combined prediction

Model averaging can be improved by weighting the contributions of each sub-model to the combined prediction by the expected performance of the submodel. This can be extended further by training an entirely new model to learn how to best combine the contributions from each submodel. This approach is called stacked generalization, or stacking for short, and can result in better predictive performance than any single contributing model.

In this tutorial, you will discover how to develop a stacked generalization ensemble for deep learning neural networks.

After completing this tutorial, you will know:

  • Stacked generalization is an ensemble method where a new model learns how to best combine the predictions from multiple existing models.

  • How to develop a stacking model using neural networks as a submodel and a scikit-learn classifier as the meta-learner.

  • How to develop a stacking model where neural network sub-models are embedded in a larger stacking ensemble model for training and prediction.

Tutorial Overview

This tutorial is divided into six parts, they are:

  1. Stacked Generalization ensemble
  2. Multi-Class Classification Problem
  3. Multilayer Perceptron Model
  4. Train and Save Sub-model
  5. Separate Stacking Model
  6. Integrated Stacking Model

Stacked Generalization Ensemble

A model averaging ensemble combines the predictions from multiple trained models.

A limitation of this approach is that each model contributes the same amount to the ensemble prediction, regardless of how well the model performed. A variation of this approach, called a weighted average ensemble, weighs the contribution of each ensemble member by the trust or expected performance of the model on a holdout dataset. This allows well-performing models to contribute more and less-well-performing models to contribute less. The weighted average ensemble provides an improvement over the model average ensemble.

A further generalization of this approach is replacing the linear weighted sum (e.g. linear regression) model used to combine the predictions of the sub-models with any learning algorithm. This approach is called stacked generalization, or stacking for short.

In stacking, an algorithm takes the outputs of sub-models as input and attempts to learn how to best combine the input predictions to make a better output prediction.

It may be helpful to think of the stacking procedure as having two levels: level 0 and level 1

  • Level 0: The level 0 data is the training dataset inputs and level 0 models learn to make predictions from this data
  • Level 1: The level 1 data takes the output of the level 0 model s as input and the single level 1 model, or meta-learner, learns to make predictions from this data

Unlike a weighted average ensemble, a stacked generalization ensemble can use the set of predictions as a context and conditionally decide to weigh the input predictions differently, potentially resulting in better performance.

Interestingly, although stacking is described as an ensemble learning method with two or more level 0 models, it can be used in the case where there is only a single level 0 model. In this case, the level 1, or meta-learner, model learns to correct the predictions from the level 0 model.

It is important that the meta-learner is trained on a separate dataset to the examples used to train the level 0 models to avoid overfitting.

A simple way that this can be achieved is by splitting the training dataset into a train and validation set. The level 0 models are then trained on the train set. The level 1 model is then trained using the validation set, where the raw inputs are first fed through the level 0 models to get predictions that are used as inputs to the level 1 model.

A limitation of the hold-out validation set approach to training a stacking model is that level 0 and level 1 models are not trained on the full dataset.

A more sophisticated approach to training a stacked model involves using k-fold cross-validation to develop the training dataset for the meta-learner model. Each level 0 model is trained using k-fold cross-validation (or even leave-one-out cross-validation for maximum effect); the models are then discarded, but the predictions are retained. This means for each model, there are predictions made by a version of the model that was not trained on those examples, e.g. like having holdout examples, but in this case for the entire training dataset.

The predictions are then used as inputs to train the meta-learner. Level 0 models are then trained on the entire training dataset and together with the meta-learner, the stacked model can be used to make predictions on new data.

In practice, it is common to use different algorithms to prepare each of the level 0 models, to provide a diverse set of predictions.

It is also common to use a simple linear model to combine the predictions. Because use of a linear model is common, stacking is more recently referred to as “model blending” or simply “blending,” especially in machine learning competitions.

A stacked generalization ensemble can be developed for regression and classification problems. In the case of classification problems, better results have been seen when using the prediction of class probabilities as input to the meta-learner instead of class labels.

Now that we are familiar with stacked generalization, we can work through a case study of developing a stacked deep learning model.

Multi-Class Classfication Problem

We will use a small multi-class classification problem as the basis to demonstrate the stacking ensemble.

The Scikit-learn class provides the make_blobs() function

The problem has two input variables (to represent the x and y coordinates of the points) and a standard deviation of 2.0 for points within each group. We will use the same random state (seed for the pseudorandom number generator to ensue that we always get the same data points

from sklearn.datasets import make_blobs
from matplotlib import pyplot as plt
import pandas as pd

# generate 2d classification dataset
X, y = make_blobs(n_samples=1000, centers=3, n_features=2, cluster_std=2, random_state=2)

# scatter plot, dots colored by class value
df = pd.DataFrame(dict(x=X[:, 0], y=X[:, 1], label=y))

The results are the input and output elements of a dataset that we can model.

In order to get a feeling for the complexity of the problem, we can graph each point on a two-dimensional scatter plot and color each point by class value.

colors = {0:'red', 1:'blue', 2:'green'}
fig, ax = plt.subplots()
grouded = df.groupby('label')
for key, group in grouded:
    group.plot(ax=ax, kind='scatter', x='x', y='y', label=key, color=colors[key])
plt.show()

plot

Running the example creates a scatter plot of the entire dataset. We can see that the standard deviation of 2.0 means that the classes are not linearly separable (separable by a line) causing many ambiguous points.

This is desirable as it means that the problem is non-trivial and will allow a neural network model to find many different “good enough” candidate solutions, resulting in a high variance.

Multilayer Perceptron Model

Before we define a model, we need to contrive a problem that is appropriate for the stacking ensemble.

In our problem, the training dataset is relatively small. Specifically, there is a 10:1 ratio of examples in the training dataset to the holdout dataset. This mimics a situation where we may have a vast number of unlabeled examples and a small number of labeled examples with which to train a model.

We will create 1,100 data points from the blobs problem. The model will be trained on the first 100 points and the remaining 1,000 will be held back in a test dataset, unavailable to the model.

The problem is a multi-class classification problem, and we will model it using a softmax activation function on the output layer. This means that the model will predict a vector with three elements with the probability that the sample belongs to each of the three classes. Therefore, we must one hot encode the class values before we split the rows into the train and test datasets. We can do this using the Keras to_categorical() function.

# use PlaidML as backend intead of default TensorFlow, 
# so that can utilize the power of MacBook Pro's AMD GPU
import os
os.environ["KERAS_BACKEND"] = "plaidml.keras.backend"

# import the modules
from keras.utils import to_categorical
from keras.models import Sequential
from keras.layers import Dense

Using plaidml.keras.backend backend.

# generate 2d clasification dataset
X, y = make_blobs(n_samples=1100, centers=3, n_features=2, cluster_std=2, random_state=2)

# apply one-hot encoding
y = to_categorical(y)

# split train test set
n_train = 100
X_train, X_test = X[:n_train, :], X[n_train:, :]
y_train, y_test = y[:n_train], y[n_train:]
print(f"The shape of X train set is {X_train.shape}, the shape of X test set is {X_test.shape}.") 

The shape of X train set is (100, 2), the shape of X test set is (1000, 2).

Next, we can define and combine the model.

The model will expect samples with two input variables. The model then has a single hidden layer with 25 nodes and a rectified linear activation function, then an output layer with three nodes to predict the probability of each of the three classes and a softmax activation function.

Because the problem is multi-class, we will use the categorical cross entropy loss function to optimize the model and the efficient Adam flavor of stochastic gradient desent

# define model
model = Sequential()
model.add(Dense(25, input_dim=2, activation='relu'))
model.add(Dense(3, activation='softmax'))
model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy'])

INFO:plaidml:Opening device "metal_amd_radeon_pro_560x.0"

The model is fit for 500 training epochs and we will evaluate the model each epoch on the test set, using the test set as validation set

# fit the model
history = model.fit(X_train, y_train, validation_data=(X_test, y_test), epochs=150, verbose=1)

Epoch 150/150
100/100 [==============================] - 0s 1ms/step - loss: 0.4755 - acc: 0.7900 - val_loss: 0.5328 - val_acc: 0.7800

At the end of the run, we will evaluate the performance of the model on the train and test sets.

# evaluate the model
_, train_acc = model.evaluate(X_train, y_train, verbose=1)

100/100 [==============================] - 0s 450us/step

# evaluate the model
_, test_acc = model.evaluate(X_test, y_test, verbose=1)

1000/1000 [==============================] - 0s 112us/step

print(f"The train accuracy is {train_acc}, and the test accuracy is {test_acc}.")

The train accuracy is 0.79, and the test accuracy is 0.78.

Then finally, we will plot learning curves of the model accuracy over each training epoch on both the training and validation datasets.

# learning curves of model accuracy
plt.plot(history.history['acc'], label='train')
plt.plot(history.history['val_acc'], label='test')
plt.legend()
plt.show()

plot_2

Running the example first prints the shape of each dataset for confirmation, then the performance of the final model on the train and test datasets.

Your specific results will vary (by design!) given the high variance nature of the model.

In this case, we can see that the model achieved about 78% accuracy on the training dataset, which we know is optimistic, and about 73.9% on the test dataset, which we would expect to be more realistic.

We can now look at using instances of this model as part of a stacking ensemble.

Train and Save Sub-Models

To keep this example simple, we will use multiple instances of the same model as level-0 or sub-models in the stacking ensemble.

We will also use a holdout validation dataset to train the level-1 or meta-learner in the ensemble.

A more advanced example may use different types of MLP models (deeper, wider, etc.) as sub-models and train the meta-learner using k-fold cross-validation

In this section, we will train multiple sub-models and save them to file for later use in our stacking ensembles.

The first step is to create a function that will define and fit an MLP model on the training dataset.

# fit model on dataset
def fit_model(X_train, y_train):
    # define the model
    model = Sequential()
    model.add(Dense(25, input_dim=2, activation='relu'))
    model.add(Dense(3, activation='softmax'))
    model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy'])
    
    # fit the model
    model.fit(X_train, y_train, epochs=150, verbose=0)

    return model

Next, we can create a sub-directory to store the models.

Note, if the directory already exists, you may have to delete it when re-running this code.

from os import makedirs
makedirs('models')

Finally, we can create multiple instances of the MLP and save each to the “models/” subdirectory with a unique filename.

In this case, we will create five sub-models, but you can experiment with a different number of models and see how it impacts model performance.

# fit and save models:
n_members = 5
for i in range(n_members):
    # fit model
    model = fit_model(X_train, y_train)

    # save model
    filename = 'models/model_' + str(i + 1) + '.h5'
    model.save(filename)
    print(f"[INFO]>>Save {filename}.")

INFO:plaidml:Opening device "metal_amd_radeon_pro_560x.0"
[INFO]>>Save models/model_1.h5.
[INFO]>>Save models/model_2.h5.
[INFO]>>Save models/model_3.h5.
[INFO]>>Save models/model_4.h5.
[INFO]>>Save models/model_5.h5.

!ls -l models

total 280
-rw-r--r--  1 johnnylu  staff  27936 Jan 16 12:40 model_1.h5
-rw-r--r--  1 johnnylu  staff  27936 Jan 16 12:40 model_2.h5
-rw-r--r--  1 johnnylu  staff  27936 Jan 16 12:40 model_3.h5
-rw-r--r--  1 johnnylu  staff  27936 Jan 16 12:41 model_4.h5
-rw-r--r--  1 johnnylu  staff  27936 Jan 16 12:41 model_5.h5

Separate Stacking Model

We can now train a meta-learner that will best combine the predictions from the sub-models and ideally perform better than any single sub-model.

The first step is to load the saved models.

We can use the load_model() Keras function and create a Python list of loaded models.

n_members = 5
from keras.models import load_model

# load models from file
def load_all_model(n_models):
    all_models = []
    for i in range(n_models):
        # define filename for this ensemble
        filename = 'models/model_' + str(i + 1) + '.h5'
        model = load_model(filename)

        # add to list of members
        all_models.append(model)
        print(f"[INFO]>>loaded {filename}.")
    return all_models


# load all models: 
members = load_all_model(n_members)
print(f"Loaded {len(members)} models.")

[INFO]>>loaded models/model_1.h5.
[INFO]>>loaded models/model_2.h5.
[INFO]>>loaded models/model_3.h5.
[INFO]>>loaded models/model_4.h5.
[INFO]>>loaded models/model_5.h5.
Loaded 5 models.

It would be useful to know how well the single models perform on the test dataset as we would expect a stacking model to perform better.

We can easily evaluate each single model on the training dataset and establish a baseline of performance.

# evaluate standalone models on test dataset
for model in members:
    _, acc = model.evaluate(X_test, y_test, verbose=1)
    print(f"Model Test Set Accuracy: {acc}.")

1000/1000 [==============================] - 0s 402us/step
Model Test Set Accuracy: 0.78.
1000/1000 [==============================] - 0s 98us/step
Model Test Set Accuracy: 0.724.
1000/1000 [==============================] - 0s 100us/step
Model Test Set Accuracy: 0.755.
1000/1000 [==============================] - 0s 103us/step
Model Test Set Accuracy: 0.778.
1000/1000 [==============================] - 0s 101us/step
Model Test Set Accuracy: 0.752.

Next, we can train our meta-learner. This requires two steps:

  • Prepare a training dataset for the meta-learner.
  • Use the prepared training set to fit a meta-learner model

We will prepare a training dataset for the meta-learner by providing examples from the test set to each of the submodels and collecting the predictions. In this case, each model will output three predictions for each example for the probabilities that a given example belongs to each of the three classes. Therefore, the 1,000 examples in the test set will result in five arrays with the shape [1000, 3].

We can combine these arrays into a three-dimensional array with the shape [1000, 5, 3] by using the dstack() numpy functionthat will stack each new set of predictions

As input for new model, we will require 1,000 examples with some number of features. Given that we have 5 models and each model makes three predictions per example, then we would have 15 (3 x 5) features for each example provided to the submodels. We can transform the [1000, 5, 3] shaped predictions from the sub-models into a [1000, 15] shaped array to be used to train a meta-learner using the reshape() numpy function and flattening the final two dimensions. The stacked_dataset() function implements this steps

# check the sub-model prediction output shape
test = model.predict(X_test)
print(test)
print("\n")
print(test.shape)
print("\n")
print(y_test)

[[0.87745166 0.00568948 0.11685885]
 [0.00420502 0.08186033 0.91393465]
 [0.0560521  0.27464437 0.66930354]
 ...
 [0.22192474 0.524442   0.25363323]
 [0.8633721  0.05750959 0.0791183 ]
 [0.66364163 0.11609959 0.2202588 ]]


(1000, 3)


[[1. 0. 0.]
 [0. 0. 1.]
 [0. 0. 1.]
 ...
 [0. 1. 0.]
 [1. 0. 0.]
 [0. 0. 1.]]

# numpy dstack example
import numpy as np
from numpy import dstack

a = np.array((1, 2, 3))
print(f"The array a is {a}")
print("\n")
b = np.array((4, 5, 6))
print(f"The array b is {b}")
print("\n")

c = dstack((a, b))
print(f"dstack: {c}")
print(f"Shape of dstack: {c.shape}")

The array a is [1 2 3]


The array b is [4 5 6]


dstack: [[[1 4]
  [2 5]
  [3 6]]]
Shape of dstack: (1, 3, 2)

# create stacked model input dataset as output from the ensemble
def stacked_dataset(members, inputX):
    stackX = None
    for model in members:
        # make prediction
        yhat = model.predict(inputX, verbose=0)
        # stack predictions into [rows, members, probabilities]
        if stackX is None:
            stackX = yhat
        else:
            stackX = dstack((stackX, yhat))
        # flatten predictions to [rows, members x probabilities]
    stackX = stackX.reshape((stackX.shape[0], stackX.shape[1]*stackX.shape[2]))
    return stackX


stackX = stacked_dataset(members, X_test)

stackX.shape

(1000, 15)

Once prepared, we can use this input dataset along with the output, or y part, of the test set to train a new meta-learner

In this case, we will train a simple logistic regression algorithm from the scikit-learn library

Logistic Regression only supports binary classification, although the implementation of logistic regression in scikit-learn in the LogisticRegression class support multi-class classification (more than two classes) using a one-vs-rest scheme. The function fit_stacked_model() below will prepare the training dataset for meta-learner by calling the stacked_dataset() function, then fit a logistic regression model that is then returned.

# import Logistic Regression class
from sklearn.linear_model import LogisticRegression
# fit a model based on the outputs from the ensemble members
def fit_stacked_model(members, inputX, inputy):
    # create dataset using ensemble
    stackedX = stacked_dataset(members, inputX)
    # fit standalone mode
    model = LogisticRegression()
    model.fit(stackedX, inputy)
    
    return model


# make a prediction with the stacked model
def stacked_prediction(members, model, inputX):
    # create dataset using ensemble
    stackedX = stacked_dataset(members, inputX)
    # make a prediction
    yhat = model.predict(stackedX)
    return yhat

We can call this function and pass in the list of loaded models and the training dataset

from sklearn.datasets import make_blobs
# reset the X, y and X_test, y_test variable

X, y = make_blobs(n_samples=1100, centers=3, n_features=2, cluster_std=2, random_state=2)
# split into train and test
n_train = 100
X_train, X_test = X[:n_train, :], X[n_train:, :]
y_train, y_test = y[:n_train], y[n_train:]
print(X_train.shape, X_test.shape)

(100, 2) (1000, 2)

from sklearn.metrics import accuracy_score
# load all models
n_members = 5
members = load_all_model(n_members)
print('Loaded %d models' % len(members))
# evaluate standalone models on test dataset
for model in members:
	testy_enc = to_categorical(y_test)
	_, acc = model.evaluate(X_test, testy_enc, verbose=0)
	print('Model Accuracy: %.3f' % acc)
# fit stacked model using the ensemble
model = fit_stacked_model(members, X_test, y_test)
# evaluate model on test set
yhat = stacked_prediction(members, model, X_test)
acc = accuracy_score(y_test, yhat)
print('Stacked Test Accuracy: %.3f' % acc)

[INFO]>>loaded models/model_1.h5.
[INFO]>>loaded models/model_2.h5.
[INFO]>>loaded models/model_3.h5.
[INFO]>>loaded models/model_4.h5.
[INFO]>>loaded models/model_5.h5.
Loaded 5 models
Model Accuracy: 0.780
Model Accuracy: 0.724
Model Accuracy: 0.755
Model Accuracy: 0.778
Model Accuracy: 0.752
Stacked Test Accuracy: 0.826

Integrated Stacking Model

When using neural networks as sub-models, it may be desirable to use a neural network as a meta-learner.

Specifically, the sub-networks can be embedded in a larger multi-headed neural network that then learns how to best combine the predictions from each input sub-model. It allows the stacking ensemble to be treated as a single large model.

The benefit of this approach is that the outputs of the submodels are provided directly to the meta-learner. Further, it is also possible to update the weights of the submodels in conjunction with the meta-learner model, if this is desirable.

This can be achieved using the Keras functional interface for developing models.

After the models are loaded as a list, a larger stacking ensemble model can be defined where each of the loaded models is used as a separate input-head to the model. This requires that all of the layers in each of the loaded models be marked as not trainable so the weights cannot be updated when the new larger model is being trained. Keras also requires that each layer has a unique name, therefore the names of each layer in each of the loaded models will have to be updated to indicate to which ensemble member they belong.

# import modules
from sklearn.datasets import make_blobs
from sklearn.metrics import accuracy_score
from keras.models import load_model
from keras.utils import to_categorical
from keras.utils import plot_model
from keras.models import Model
from keras.layers import Input
from keras.layers import Dense
from keras.layers.merge import concatenate
from numpy import argmax

Once the sub-models have been prepared, we can define the stacking ensemble model.

The input layer for each of the sub-models will be used as separete input head to this new model. This means that k copies of any input data will have to be provided to the model, where k is the number of input models, in this case, k = 5.

The outputs of each of the models can then be merged. In this case, we will use a simple concatenation merge, where a single 15-elememts vector will be created from the 3 class-probabilities predicted by each of the 5 models.

We will then define a hidden layer to interpret this input to the meta-learner and an output layer that will make its own probabilistic prediction. The define_stacked_model() function below implements this and will return a stacked generalization neural network model given a list of trained sub-models.

# define stacked model from multiple member input models
def define_stacked_model(members):
    # update all layers in all models to not be trainable
    for i in range(len(members)):
        model = members[i]
        for layer in model.layers:
            # make not trainable
            layer.trainable = False
            # rename to avoid 'unique layer name' issue
            layer.name = 'ensemble_' + str(i + 1) + '_' + layer.name
    # define multi-headed input
    ensemble_visible = [model.input for model in members]
    # concatenate merge output from each model
    ensemble_outputs = [model.output for model in members]
    merge = concatenate(ensemble_outputs)
    hidden = Dense(10, activation='relu')(merge)
    output = Dense(3, activation='softmax')(hidden)
    model = Model(inputs=ensemble_visible, outputs=output)
    # plot graph of ensemble
    plot_model(model, show_shapes=True, to_file='model_graph.png')
    # complie
    model.compile(loss='categorical_crossentropy', optimizer='adam', metrics=['accuracy'])

    return model

stacked_model = define_stacked_model(members)

A plot of the network graph is created when this function is called to give an idea of how the ensemble model fits together.

Once the model is defined, it can be fit. We can fit it directly on the holdout test dataset.

Because the sub-models are not trainable, their weights wil not be updated during training and only the weights of the new hidden and output layer will be updated. The fit_stacked_model() function below will fit the stacking neural network model on for 300 epochs.

# fit a stacked model
def fit_stacked_model(model, inputX, inputy):
    # prepare input data
    X = [inputX for _ in range(len(model.input))]
    # encode y variable
    inputy_enc = to_categorical(inputy)
    # fit the mode
    model.fit(X, inputy_enc, epochs=300, verbose=0)

Once fit, we can use the new stacked model to make a prediction on new data.

This is as simple as calling the predict() function on the model. One minor change is that we require k copies of the input data in a list to be provided to the model for each of the k sub-models. the predict_stacked_model() function below simplifies this process of making a prediction with the stacking model.

# make a prediction with a stacked model
def predict_stacked_model(model, inputX):
    # prepare input data
    X = [inputX for _ in range(len(model.input))]
    # make prediction
    return model.predict(X, verbose=1)

We can call this function to make a prediction for the test dataset and report the accuracy

We would expect the performance of the neural network learner to be better than any individual submodel and perhaps competitive with the linear meta-learner used in the previous section.

# generate 2d classification dataset
from sklearn.model_selection import train_test_split
X, y = make_blobs(n_samples=1100, centers=3, n_features=2, cluster_std=2, random_state=2)
# split into train and test
X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.5, random_state=3)
# load all models
n_members = 5
members = load_all_model(n_members)
print('Loaded %d models' % len(members))
# define ensemble model
stacked_model = define_stacked_model(members)
# fit stacked model on test dataset
fit_stacked_model(stacked_model, X_test, y_test)
# make predictions and evaluate
yhat = predict_stacked_model(stacked_model, X_test)
yhat = argmax(yhat, axis=1)
acc = accuracy_score(y_test, yhat)
print('Stacked Test Accuracy: %.3f' % acc)

[INFO]>>loaded models/model_1.h5.
[INFO]>>loaded models/model_2.h5.
[INFO]>>loaded models/model_3.h5.
[INFO]>>loaded models/model_4.h5.
[INFO]>>loaded models/model_5.h5.
Loaded 5 models
550/550 [==============================] - 0s 448us/step
Stacked Test Accuracy: 0.820

Running the example first loads the five sub-models.

A larger stacking ensemble neural network is defined and fit on the test dataset, then the new model is used to make a prediction on the test dataset. We can see that in this case, the model achieved an higher accuracy. out-performing the linear model from the previous section.

Extensions

This section lists some ideas for extending the tutorial that you may wish to explor

  • Alternate Meta-Learner. Update the example to use an alternate meta-learner classifier model to the logistic regression model
  • Single Level 0 Models. Update the example to use a single level-0 model and compare the results.
  • ** Vary Level0 Models**. Develop a study that demostrates the relationship between test classification accuracy and the number of sub-models used in the stacked ensemble.
  • Cross-Validation Stacking Ensemble. Update the example to use k-fold cross-validation to prepare the training dataset for the meta-learner model.
  • Use Raw Input in Meta-Learner. Update the example so that the meta-learner algorithms take the raw input data for the sample as well as the output from the sub-models and compare performance.

Further Reading

Books

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