Communications - October 2012

system ( http://netflixprize.com). As the competition progressed, teams found they obtained the best results by combining their learners with other teams’, and merged into larger and larger teams. The winner and runner-up were both stacked ensembles of over 100 learners, and combining the two ensembles further improved the results. Doubtless we will see even larger ones in the future.

Model ensembles should not be confused with Bayesian model averaging (BMA)—the theoretically optimal approach to learning. 4 In BMA, predictions on new examples are made by averaging the individual predictions of all classifiers in the hypothesis space, weighted by how well the classifiers explain the training data and how much we believe in them a priori. Despite their superficial similarities, ensembles and BMA are very different. Ensembles change the hypothesis space (for example, from single decision trees to linear combinations of them), and can take a wide variety of forms. BMA assigns weights to the hypotheses in the original space according to a fixed formula. BMA weights are extremely different from those produced by (say) bagging or boosting: the latter are fairly even, while the former are extremely skewed, to the point where the single highest-weight classifier usually dominates, making BMA effectively equivalent to just selecting it. 8 A practical consequence of this is that, while model ensembles are a key part of the machine learning toolkit, BMA is seldom worth the trouble.

Simplicity Does not
imply Accuracy

Occam’s razor famously states that entities should not be multiplied beyond necessity. In machine learning, this is often taken to mean that, given two classifiers with the same training error, the simpler of the two will likely have the lowest test error. Purported proofs of this claim appear regularly in the literature, but in fact there are many counterexamples to it, and the “no free lunch” theorems imply it cannot be true.

We saw one counterexample previously: model ensembles. The generalization error of a boosted ensemble

Just because
a function can
be represented
does not mean
it can be learned.

continues to improve by adding classifiers even after the training error has reached zero. Another counterexample is support vector machines, which can effectively have an infinite number of parameters without overfitting. Conversely, the function sign(sin(ax)) can discriminate an arbitrarily large, arbitrarily labeled set of points on the x axis, even though it has only one parameter. 23 Thus, contrary to intuition, there is no necessary connection between the number of parameters of a model and its tendency to overfit.

A more sophisticated view instead equates complexity with the size of the hypothesis space, on the basis that smaller spaces allow hypotheses to be represented by shorter codes. Bounds like the one in the section on theoretical guarantees might then be viewed as implying that shorter hypotheses generalize better. This can be further refined by assigning shorter codes to the hypotheses in the space we have some a priori preference for. But viewing this as “proof” of a trade-off between accuracy and simplicity is circular reasoning: we made the hypotheses we prefer simpler by design, and if they are accurate it is because our preferences are accurate, not because the hypotheses are “simple” in the representation we chose.

A further complication arises from the fact that few learners search their hypothesis space exhaustively. A learner with a larger hypothesis space that tries fewer hypotheses from it is less likely to overfit than one that tries more hypotheses from a smaller space. As Pearl18 points out, the size of the hypothesis space is only a rough guide to what really matters for relating training and test error: the procedure by which a hypothesis is chosen.

Domingos7 surveys the main arguments and evidence on the issue of Occam’s razor in machine learning. The conclusion is that simpler hypotheses should be preferred because simplicity is a virtue in its own right, not because of a hypothetical connection with accuracy. This is probably what Occam meant in the first place.