# Fairness and discrimination, PhD Course, #10 Mitigation, Post-processing

For the last part, in our graduate course, we will discuss further mitigation, and after pre-processing and in-processing techniques, we will present post-processing ones. It simply means that we created a model, that could be seen as discriminatory. But besides that, it was a “good” model, so still want to use the predictions we obtained. Quite heuristically, in the context of binary sensitive attributes, we could agree that we got two sets of predictions, for the two sensitive attribute, and the fair model should probability “in between”. And that is a natural idea when dealing with convex objects, and it is related to averages, centroids or barycenters,

As mentioned above, there are several ways to defined such a quantity

but the most interesting one will be related to the one based on optimization.

Interestingly, this idea can be extended to more complexe objects than points, in some metric space, but more generally on distributions. And since we have seen several distances on the set of distributions, we can consider for instance the Wasserstein (2) barycenter, as in Agueh and Carlier (2011),

An interesting point, it that in the univariate setting, there is a simple connection with optimal transport, where averages of push-forward mapping are considered,

(even if it remains computationnaly difficult to get)

In our context, if we have a model, consider two scores, $m(\boldsymbol{x},s=A)$ and $m(\boldsymbol{x},s=B)$, on the two sensistive groups, and consider quite naturally the fair barycenter.

The heuristic interpretation is simple, and interstingly, using that new model $m^\star(\boldsymbol{x})$, individuals we be ordered the same way within each group. In the Gaussian case, it is also possible to compute those barycenters.

To do so, we need to define $\boldsymbol{\Sigma}^t$, for $t\in[0,1]$, including some square root of $\boldsymbol{\Sigma}$, i.e. $\boldsymbol{\Sigma}^{1/2}$, that we expect to be symmetric. To do so, we need simply the exponential of matrices

and the logarithm

Then define the square root, for instance,

Here, we can prove that the barycenter of Gaussian vectors is Gaussian. The mean is the average of means, and a slightly more complex formula is used for the variance

based on matrix equations,

In dimension 2, we can write it more simply

Interestingly, the average is variances is larger than the variance of the barycenter. And a natural property can be obtained whem variances are diagonalizables in the same basis

Here we can illustrate iso-probabilities curves

More generally, use histograms

But it becomes hard to compute

An easier approach is to use numerical simulations and a kernel estimate to obtain a smooth density,

We can use it on pictures. For instance, we have several observations of a “3”, handwriten

We can compute there barycenter, and generate based on it

Quite naturally, once we have barycenters, we can consider geodesics

and apply it again to Gaussian vectors

On our dataset, with scores, we have

We can then compute the “barycenter score”, that will be, for people in group A

and people in group B

Consider now the score of three models, on the motor dataset

The score $m^\star$ for people in group A is

while the score $m^\star$ for people in group B is

We can now compare predictions for people in the two groups

Numerically, we obtain

So we are now able to mitigate unfair scores.

# Online Seminar Finance & Modeling, Centre d’Économie de la Sorbonne

In a week,  I will give a talk at the Modélisation Financière seminar (“Online Seminar Finance & Modeling” according to the invitation) on Using optimal transport to mitigate unfair predictions. Slides are now on line.

The insurance industry is heavily reliant on predictions of risks based on characteristics of potential customers. Although the use of said models is common, researchers have long pointed out that such practices perpetuate discrimination based on sensitive features such as gender or race. Given that such discrimination can often be attributed to historical data biases, an elimination or at least mitigation is desirable. With the shift from more traditional models to machine-learning based predictions, calls for greater mitigation have grown anew, as simply excluding sensitive variables in the pricing process can be shown to be ineffective. In this talk, we first investigate why predictions are a necessity within the industry and why correcting biases is not as straightforward as simply identifying a sensitive variable. We then propose to ease the biases through the use of Wasserstein barycenters instead of simple scaling. To demonstrate the effects and effectiveness of the approach we employ it on real data and discuss its implications. The talk will be based on recent work with François Hu and Philipp Ratz (2310.20508, 2309.06627, 2306.12912 and 2306.10155).

# Parametric Fairness with Statistical Guarantees

Our paper Parametric Fairness with Statistical Guarantees is now available on ArXiv.

Algorithmic fairness has gained prominence due to societal and regulatory concerns about biases in Machine Learning models. Common group fairness metrics like Equalized Odds for classification or Demographic Parity for both classification and regression are widely used and a host of computationally advantageous post-processing methods have been developed around them. However, these metrics often limit users from incorporating domain knowledge. Despite meeting traditional fairness criteria, they can obscure issues related to intersectional fairness and even replicate unwanted intra-group biases in the resulting fair solution. To avoid this narrow perspective, we extend the concept of Demographic Parity to incorporate distributional properties in the predictions, allowing expert knowledge to be used in the fair solution. We illustrate the use of this new metric through a practical example of wages, and develop a parametric method that efficiently addresses practical challenges like limited training data and constraints on total spending, offering a robust solution for real-life applications.

# Mitigating Discrimination in Insurance with Wasserstein Barycenters

Our new paper, with François Hu and Philipp Ratz, Mitigating Discrimination in Insurance with Wasserstein Barycenters is now available on ArXiv.

The insurance industry is heavily reliant on predictions of risks based on characteristics of potential customers. Although the use of said models is common, researchers have long pointed out that such practices perpetuate discrimination based on sensitive features such as gender or race. Given that such discrimination can often be attributed to historical data biases, an elimination or at least mitigation is desirable. With the shift from more traditional models to machine-learning based predictions, calls for greater mitigation have grown anew, as simply excluding sensitive variables in the pricing process can be shown to be ineffective. In this article, we first investigate why predictions are a necessity within the industry and why correcting biases is not as straightforward as simply identifying a sensitive variable. We then propose to ease the biases through the use of Wasserstein barycenters instead of simple scaling. To demonstrate the effects and effectiveness of the approach we employ it on real data and discuss its implications.

(fictitious maps used in the article)

# Fairness in Multi-Task Learning via Wasserstein Barycenters

Our new paper, with François Hu and Philipp Ratz, Fairness in Multi-Task Learning via Wasserstein Barycenters, is now available.

Algorithmic Fairness is an established field in machine learning that aims to reduce biases in data. Recent advances have proposed various methods to ensure fairness in a univariate environment, where the goal is to de-bias a single task. However, extending fairness to a multi-task setting, where more than one objective is optimised using a shared representation, remains underexplored. To bridge this gap, we develop a method that extends the definition of Strong Demographic Parity to multi-task learning using multi-marginal Wasserstein barycenters. Our approach provides a closed form solution for the optimal fair multi-task predictor including both regression and binary classification tasks. We develop a data-driven estimation procedure for the solution and run numerical experiments on both synthetic and real datasets. The empirical results highlight the practical value of our post-processing methodology in promoting fair decision-making.

It will be presented in September, at the European Conference on Machine Learning and Principles and Practice of Knowledge Discovery in Databases (ECML PKDD 2023), in Torino.