## 2 Definitions and Related Work
Reference [20] demonstrated that these embeddings reflect societal biases. There are also methods to show how these embeddings can be unbiased by aligning them with a set of neutral reference vectors [21; 22]. These de-biasing methods have, however, strong limitations, as explained in [23], where the authors showed that, although the de-biasing embedding methods can reduce the visibility of gender bias, they do not completely eliminate it. A second approach is to use _post-processing_ de-biasing methods. These methods are model-agnostic and, therefore, not specific to NLP since they modify the results of previously trained classifiers in order to achieve fairer results. References [11; 24] investigated this for binary classification, and Reference [25] proposed a method for multiclass classification. The last approach to mitigate biases in AI is to use fairness-aware algorithms, which are specifically designed to take into account the potential for bias and to learn from the data in a way that reduces the risk of making biased decisions. These are the _in-processing_ methods, which generally do not depend on the type of data input either. The method we propose in this paper is one of them. To achieve this, we can use adversarial learning by adjusting the discriminator. Adversarial learning involves training a model to make predictions while also training a second model to identify and correct any biases in the first model's predictions. By incorporating this technique into the training process, References [26; 27] demonstrated that it is possible to reduce the amount of bias present in machine learning models. Another technique is to constrain the predictions with a regularisation technique, such as [28], but this technique was only used on a logistic regression classifier. On the other hand, Reference [29] mitigated fairness specifically in neural networks. Finally, References [30; 31] used fairness metrics constraints and solved the training problem subject to those constraints. All these _in-processing_ methods apply in the case of binary classification. There is indeed an _in-processing_ paper that proposes a method for multiclass classification for a computer vision task [32], but this paper focused on the regularisation of the mean bias amplification and, therefore, did not deal with the classic fairness metrics. **Research implications:** We want to emphasise that the _pre-processing_ and _post-processing_ methods are complementary to _in-processing_ methods. Especially when using neural network models, which simultaneously project the input data into an optimal representation (the so-called _latent space_ or _feature space_) and use this optimal data representation for their predictions, we believe that _in-processing_ methods are those that should be the most-efficient ones. They can indeed both constrain the neural networks to learn fair data representations and fair decision rules based on these data representations. Our paper hence focused on an _in-processing_ method. In this context, our methodology tackles an issue that was still not addressed in the fair learning literature, as far as the authors know: we tackled algorithmic biases on multi-class neural network classifiers and not on binary classifiers or on non-neural network classifiers. We believe that the potential of such a strategy is high for the future certification of commercial AI systems. The key methodological contribution of our work is to show how to extend [7] to multi-class classification for regularised mini-batch training of neural networks. As described in Section 3.3, extending this optimal transport regularisation strategy to multi-class classification requires tackling an important technical lock related to the algorithmic cost of the procedure, which is the heart of this methodological contribution. Note that this regularisation strategy applies to any type of multi-class classification neural network model. Thus, the method is particularly flexible and can be applied in various industrial classification problems.