  Properness for supervised losses stipulates that the loss function shapes the
learning algorithm towards the true posterior of the data generating
distribution. Unfortunately, data in modern machine learning can be corrupted
or twisted in many ways. Hence, optimizing a proper loss function on twisted
data could perilously lead the learning algorithm towards the twisted
posterior, rather than to the desired clean posterior. Many papers cope with
specific twists (e.g., label/feature/adversarial noise), but there is a growing
need for a unified and actionable understanding atop properness. Our chief
theoretical contribution is a generalization of the properness framework with a
notion called twist-properness, which delineates loss functions with the
ability to "untwist" the twisted posterior into the clean posterior. Notably,
we show that a nontrivial extension of a loss function called $\alpha$-loss,
which was first introduced in information theory, is twist-proper. We study the
twist-proper $\alpha$-loss under a novel boosting algorithm, called PILBoost,
and provide formal and experimental results for this algorithm. Our overarching
practical conclusion is that the twist-proper $\alpha$-loss outperforms the
proper $\log$-loss on several variants of twisted data.
