Recently, I've become interested in model stacking as a form of ensemble learning. In particular, I've experimented a bit with some toy datasets for regression problems. I've basically implemented individual "level 0" regressors, stored each regressor's output predictions as a new feature for a "meta-regressor" to take as its input, and fit this meta-regressor on these new features (the predictions from the level 0 regressors). I was extremely surprised to see even modest improvements over the individual regressors when testing the meta-regressor against a validation set.

So, here's my question: why is model stacking effective? Intuitively, I would expect the model doing the stacking to perform poorly since it appears to have an impoverished feature representation compared to each of the level 0 models. That is, if I train 3 level 0 regressors on a dataset with 20 features, and use these level 0 regressors' predictions as input to my meta-regressor, this means my meta-regressor has only 3 features to learn from. It just seems like there is more information encoded in the 20 original features that the level 0 regressors have for training than the 3 output features that the meta-regressor uses for training.


2 Answers 2


Think of ensembling as basically an exploitation of the central limit theorem.

The central limit theorem loosely says that, as the sample size increases, the mean of the sample will become an increasingly accurate estimate of the actual location of the population mean (assuming that's the statistic you're looking at), and the variance will tighten.

If you have one model and it produces one prediction for your dependent variable, that prediction will likely be high or low to some degree. But if you have 3 or 5 or 10 different models that produce different predictions, for any given observation, the high predictions from some models will tend to offset the low errors from some other models, and the net effect will be a convergence of the average (or other combination) of the predictions towards "the truth." Not on every observation, but in general that's the tendency. And so, generally, an ensemble will outperform the best single model.

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    $\begingroup$ Another way of saying what you said in your good answer is that part of the error comes from the model itself outside of the data, and so having multiple models and making a intelligently waited some of them reduces that error. $\endgroup$ Jan 11, 2021 at 0:57

Late answer, but some key points can be added. I personnally think of model stacking as "the natural sequel" of model averaging. And there is a reason why model averages are often better than single models.

Model averaging

Usually, two (different) models with similar performance often perform better than the best model when the average of the prediction is used. This is particularly true when the penalty is a convex function (MSE, RMSE...) and is a consequence of Jensen's inequality

Model (weighted) averaging

Model averaging can be seen as a particular case of model stacking. If you use a linear model on top on the "first stage" models, you are simply optimizing the weights given to each model (while model averaging was just giving the same weight to each model).

Model stacking

Then, when you drop the linear assumption on the model you train on the "stage 1 features" you simply increase the size of the space in which you "search" your model. The larger this space is, the more likely you are to find better performance.

I presented this in more details in model stacking: a tutorial

You may also be interested in this article: Stacked generalization by David H. Wolpert, which is one of the first (as far as I know) scholar publication about this method.


I could not find many other references online so I detailed the arguments above on my blog: why does model staging works ?


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