We identify the marginal polytope, the output space's convex hull,

Ensuring fairness has emerged as one of the primary concerns in AI and its related algorithms. Over time, the field of machine learning fairness has evolved to address these issues. This paper provides an extensive overview of this field and introduces two formal frameworks to tackle open questions in machine learning fairness. In one framework, operator-valued optimisation and min-max objectives are employed to address unfairness in time-series problems. This approach showcases state-of-the-art performance on the notorious COMPAS benchmark dataset, demonstrating its effectiveness in real-world scenarios. In the second framework, the challenge of lacking sensitive attributes, such as gender and race, in commonly used datasets is addressed. This issue is particularly pressing because existing algorithms in this field predominantly rely on the availability or estimations of such attributes to assess and mitigate unfairness. Here, a framework for a group-blind bias-repair is introduced, aiming to mitigate bias without relying on sensitive attributes. The efficacy of this approach is showcased through analyses conducted on the Adult Census Income dataset. Additionally, detailed algorithmic analyses for both frameworks are provided, accompanied by convergence guarantees, ensuring the robustness and reliability of the proposed methodologies.

Resampling from a target measure whose density is unknown is a fundamental problem in mathematical statistics and machine learning. A setting that dominates the machine learning literature consists of learning a map from an easy-to-sample prior, such as the Gaussian distribution, to a target measure. Under this model, samples from the prior are pushed forward to generate a new sample on the target measure, which is often difficult to sample from directly. In this paper, we propose a new model for conditional resampling called mirror Schr\"odinger bridges. Our key observation is that solving the Schr\"odinger bridge problem between a distribution and itself provides a natural way to produce new samples from conditional distributions, giving in-distribution variations of an input data point. We show how to efficiently solve this largely overlooked version of the Schr\"odinger bridge problem. We prove that our proposed method leads to significant algorithmic simplifications over existing alternatives, in addition to providing control over in-distribution variation. Empirically, we demonstrate how these benefits can be leveraged to produce proximal samples in a number of application domains.

Fairness holds a pivotal role in the realm of machine learning, particularly when it comes to addressing groups categorised by sensitive attributes, e.g., gender, race. Prevailing algorithms in fair learning predominantly hinge on accessibility or estimations of these sensitive attributes, at least in the training process. We design a single group-blind projection map that aligns the feature distributions of both groups in the source data, achieving (demographic) group parity, without requiring values of the protected attribute for individual samples in the computation of the map, as well as its use. Instead, our approach utilises the feature distributions of the privileged and unprivileged groups in a boarder population and the essential assumption that the source data are unbiased representation of the population.

We propose in this paper a general framework for deriving loss functions for structured prediction. In our framework, the user chooses a convex set including the output space and provides an oracle for projecting onto that set. Given that oracle, our framework automatically generates a corresponding convex and smooth loss function. As we show, adding a projection as output layer provably makes the loss smaller. We identify the marginal polytope, the output space's convex hull, as the best convex set on which to project. However, because the projection onto the marginal polytope can sometimes be expensive to compute, we allow to use any convex superset instead, with potentially cheaper-to-compute projection. Since efficient projection algorithms are available for numerous convex sets, this allows us to construct loss functions for a variety of tasks. On the theoretical side, when combined with calibrated decoding, we prove that our loss functions can be used as a consistent surrogate for a (potentially non-convex) target loss function of interest. We demonstrate our losses on label ranking, ordinal regression and multilabel classification, confirming the improved accuracy enabled by projections.