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A weighted average ensemble combines predictions from multiple neural networks by multiplying each model’s output by a coefficient and summing the results. For multiclass classification, combine the models’ probability vectors and choose the class with the highest final score. Select the coefficients on held-out validation data, then compare the tuned ensemble with equal averaging and each model alone; a weighted ensemble is not guaranteed to perform better.
What a weighted average ensemble does
Suppose several neural networks solve the same classification task and output probabilities in the same class order. A weighted ensemble assigns each model a coefficient, scales its probability vector by that coefficient, and adds the scaled vectors together. Models with larger coefficients have more influence on the final prediction.
If the coefficients sum to 1, the result is a weighted average. For example, with two models and weights 0.7 and 0.3, the first model contributes 70% of the combined score and the second contributes 30%. The predicted class is the index with the largest combined score.
This is different from Keras sample weights. Sample weights change how much individual examples contribute to training loss; ensemble weights combine predictions from separate models after training. The Keras guide to built-in training methods describes sample weights in the training context.
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Prepare compatible model predictions
Train multiple networks for the same task, then collect their outputs for examples in a validation set that was not used to fit those models. For multiclass classification, each model should return a probability vector for every example. Confirm that all outputs use the same class ordering and compatible shapes before combining them; otherwise, a class score from one model may be added to a different class score from another.
Probability outputs are the usual input for soft voting, but the probabilities should be meaningfully comparable across models. If one model’s outputs are poorly calibrated or systematically more confident than another’s, its scores may have disproportionate influence even before applying explicit weights. Assess this as an implementation caveat rather than assuming that weights alone correct it.
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Choose weights on validation data
There is no guaranteed analytical formula for the best coefficients. Jason Brownlee’s tutorial says, “There is no analytical solution to finding the weights (we cannot calculate them); instead, the value for the weights can be estimated using either the training dataset or a holdout validation dataset.” Brownlee also warns that using the same data to fit the member models and the ensemble weights is likely to overfit. For a more reliable estimate, select weights on representative held-out validation data, not on the models’ training examples.
Choose a metric that matches the task, then search for coefficients that perform well on the validation set. Brownlee’s weighted average ensemble tutorial demonstrates a grid search with candidate coefficients from 0.0 to 1.0 in steps of 0.1, normalizes each candidate vector by its L1 norm, evaluates the ensemble, and prints the best result. Those are settings from an illustrative example, not universal defaults.
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Exhaustive search becomes expensive as the number of models and candidate values grows: every additional model expands the combinations to evaluate. The tutorial also identifies linear solvers and gradient descent with a unit-sum constraint as alternatives. Whichever approach you use, constrain or regularize the search when appropriate, and treat a small or unrepresentative validation set as a source of uncertainty rather than as definitive evidence of the best weights.
Implement the weighted prediction
For model predictions arranged as an array with shape (models, examples, classes) and weights arranged as (models,), the combination is a weighted sum over the model axis. In NumPy, a tensor contraction expresses that operation directly:
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import numpy as np
# predictions: (number_of_models, number_of_examples, number_of_classes)
# weights: (number_of_models,)
combined_scores = np.tensordot(weights, predictions, axes=(0, 0))
predicted_classes = np.argmax(combined_scores, axis=1)To make the result a weighted average, normalize nonnegative candidate weights so they sum to 1 before combining. If weights are already normalized, the division is unnecessary. Check that the model axis is the one being contracted and that each model’s class dimension has the same ordering.
In scikit-learn, classifiers that provide predict_proba can use VotingClassifier with soft voting and classifier weights. Its official documentation explains that the classifier probabilities are multiplied by their weights and averaged, and the class with the highest average probability is selected.
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Evaluate the result without tuning on the test set
Compare the tuned ensemble, an equal-weight average, and each component model using the same held-out evaluation split and metric. Keep final test data out of weight selection; otherwise, the score used to tune coefficients is no longer an independent final evaluation. This separation is especially important because searching many coefficient combinations can fit peculiarities of the validation examples.
- Record the data split and metric used for evaluation.
- State which model outputs were combined and how their weights were selected.
- Report results for the tuned ensemble, equal-weight average, and individual models.
- Account for the cost of evaluating every member at inference time; an ensemble requires predictions from all included models.
A tuned ensemble may beat an equal average, but it may also tie or underperform the strongest member. Report measured results rather than presenting weighting as a guaranteed improvement.
Version and code considerations
Brownlee’s tutorial is dated August 25, 2020. It notes updates from October 2019 for Keras 2.3 and TensorFlow 2.0, and from January 2020 for changes to scikit-learn v0.22. These are historical version notes, not assurances of compatibility with current releases. Check the code and API behavior against the versions installed in your own environment.
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