Learning through Monotone Links with Representation Losses: the Case of Sharpened Fenchel-Young Losses

Recently, Fenchel-Young losses have been introduced to learn links used in generalized linear models: they are nonnegative convex functions that vanish on the link; more recently introduced Fitzpatrick losses share the same properties. In this paper, we introduce a broader class of losses that we call representation losses, as they come from the theory of representations of maximal monotone operators. Each representation loss comes with its own enlargement of the link, from which one can deduce geometrical minimizers guarantees to the fitted data. Within representation losses, we distinguish the so-called sharpened Fenchel-Young losses. This subclass of losses are target-dependent Fenchel-Young losses whose link enlargement is controlled by means of a sharpener. Our approach offers perspectives to control geometrical minimizer guarantees to the fitted data when designing a loss.