Being Properly Improper
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In today's ML, data can be twisted (changed) in various ways, either for bad or good intent. Such twisted data challenges the founding theory of properness for supervised losses which form the basis for many popular losses for class probability estimation. Unfortunately, at its core, properness ensures that the optimal models also learn the twist. In this paper, we analyse such class probability-based losses when they are stripped off the mandatory properness; we define twist-proper losses as losses formally able to retrieve the optimum (untwisted) estimate off the twists, and show that a natural extension of a half-century old loss introduced by S. Arimoto is twist proper. We then turn to a theory that has provided some of the best off-the-shelf algorithms for proper losses, boosting. Boosting can require access to the derivative of the convex conjugate of a loss to compute examples weights. Such a function can be hard to get, for computational or mathematical reasons; this turns out to be the case for Arimoto's loss. We bypass this difficulty by inverting the problem as follows: suppose a blueprint boosting algorithm is implemented with a general weight update function. What are the losses for which boosting-compliant minimisation happens? Our answer comes as a general boosting algorithm which meets the optimal boosting dependence on the number of calls to the weak learner; when applied to Arimoto's loss, it leads to a simple optimisation algorithm whose performances are showcased on several domains and twists.
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