Suppose you want to predict binary y given x. You fit a conditional probability model to data and form a classifier by thresholding on 0.5. How should you fit that distribution?

Traditionally people do it by minimizing log-loss on data, which is equivalent to maximum likelihood estimation, but that has the property of recovering the conditional distribution exactly with enough data/modelling freedom. We don't care about exact probabilities, so in some sense it's doing too much work.

Additionally, log-loss minimization may sacrifice classification accuracy if it allows it to model probabilities better.

Here's an example, consider predicting binary y from real valued x. The 4 points give the possible 4 possible x values and their true probabilities. If you model p(y=1|x) as 1/(1+Exp(f_n(x))) where f_n is any n'th degree polynomial.

Take n=2, then minimizing log loss and thresholding on 1/2 produces Bayes-optimal classifier

However, for n=3, the model that minimizes log loss will have suboptimal decision rules for half the data.

Hinge loss is less sensitive to exact probabilities. In particular, minimizer of hinge loss over probability densities will be a function that returns returns 1 over the region where true p(y=1|x) is greater than 0.5, and 0 otherwise. If we are fitting functions of the form above, then once hinge-loss minimizer attains the minimum, adding extra degrees of freedom will never increase approximation error.

Here's example suggested by Olivier Bousquet, suppose your decision boundaries are simple, but the actual probabilities are complicated, how well will hinge loss vs log loss do? Consider the following conditional density

Now use the conditional density functions of the same form as before, find minimizers of both log-loss and hinge loss. Hinge-loss minimization always produces Bayes optimal model for all n>1

When minimizing log-loss, the approximation error starts to increase as the fitter tries to match the exact oscillations in the true probability density function, and ends up overshooting.

Here's the plot of the area on which log loss minimizer produces suboptimal decision rule

Mathematica notebook (web version)

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