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This paper addresses multi-class classification with possibilistic supervision, where labels are possibility distributions expressing graded plausibility over classes. They construct a convex set of admissible probability distributions based on probabilistic compatibility and linear shape constraints derived from the possibility distribution. The authors then compute the Kullback-Leibler projection of a model's output onto this set using Dykstra's algorithm, minimizing the divergence between the prediction and its projection for training.
Learning from graded plausibility labels is now tractable: a novel KL-projection method efficiently transforms possibility distributions into probability distributions, improving classification performance on noisy NLI data.
We consider learning with possibilistic supervision for multi-class classification. For each training instance, the supervision is a normalized possibility distribution that expresses graded plausibility over the classes. From this possibility distribution, we construct a non-empty closed convex set of admissible probability distributions by combining two requirements: probabilistic compatibility with the possibility and necessity measures induced by the possibility distribution, and linear shape constraints that must be satisfied to preserve the qualitative structure of the possibility distribution. Thus, classes with the same possibility degree receive equal probabilities, and if a class has a strictly larger possibility degree than another class, then it receives a strictly larger probability. Given a strictly positive probability vector output by a model for an instance, we compute its Kullback-Leibler projection onto the admissible set. This projection yields the closest admissible probability distribution in Kullback-Leibler sense. We can then train the model by minimizing the divergence between the prediction and its projection, which quantifies the smallest adjustment needed to satisfy the induced dominance and shape constraints. The projection is computed with Dykstra's algorithm using Bregman projections associated with the negative entropy, and we provide explicit formulas for the projections onto each constraint set. Experiments conducted on synthetic data and on a real-world natural language inference task, based on the ChaosNLI dataset, show that the proposed projection algorithm is efficient enough for practical use, and that the resulting projection-based learning objective can improve predictive performance.
