# 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).

# Fairness and discrimination, PhD Course, #4 Wasserstein Distances and Optimal Transport

For the fourth course, we will discuss Wasserstein distance and Optimal Transport. Last week, we mentioned distances, dissimilarity and divergences. But before talking about Wasserstein, we should mention Cramer distance.

#### Cramer and Wasserstein distances

The definition of Cramér distance, for $k\geq1$, is

while Wasserstein will be (also for $k\geq1$)

If we consider cumulative distribution functions, for the first one (Cramer), we consider some sort of “vertical” distance, while for the second one (Wasserstein), we consider some “horizontal” one,

Obviously, when $k=1$, the two distances are identical

c1 = function(x) abs(pnorm(x,0,1)-pnorm(x,1,2)) w1 = function(x) abs(qnorm(x,0,1)-qnorm(x,1,2)) integrate(c1,-Inf,Inf)$value [1] 1.166631 integrate(w1,0,1)$value [1] 1.166636

But when $k>1$, it is no longer the case.

c2 = function(x) (pnorm(x,0,1)-pnorm(x,1,2))^2 w2 = function(u) (qnorm(u,0,1)-qnorm(u,1,2))^2 sqrt(integrate(c2,-Inf,Inf)$value) [1] 0.5167714 sqrt(integrate(w2,0,1)$value) [1] 1.414214

For instance, we can illustrate with a simple multinomial distribution, and the distance with some Binomial one, with some parametric inference based on distance minimization $$\theta^\star=\text{argmin}\{d(p,q_{\theta})\}$$(where here a multinomial distribution with parameters $\boldsymbol{p}=(.5,.1,.4)$, taking values respectively in $\{0,1,10\}$, while the binomial distribution has probabilities $\boldsymbol{q}_{\theta}=(1-\theta,\theta)$, taking values in $\{0,10\}$)

One can prove that

while

When $k=1$, observe that the distance is easy to compute when distributions are ordered

When $k=2$, the two distances are not equal

In the Gaussian (and the Bernoulli) case, we can get an expression for the distance (and much more as we will see later on)

There are several representations for $W_2$

And finally, we can also discuss $W_{\infty}$

#### Wasserstein distances, and optimal transport

Wasserstein distance can also we written using some sort of expected values, when considering random variables instead of distributions, and some best-case scenario, or cheapest transportation cost,

which lead to the so call Kantorovich problem

An alternative way to look at this problem is to consider a transport map, and a push-forward measure

This is simply

Of course such mapping exist

We can then consider Monge problem

And interestingly, those two problems are (somehow) equivalent

#### Discrete case

If $\boldsymbol{a}_{{A}}\in\mathbb{R}_+^{\color{red}{n_{{A}}}}$ and $\boldsymbol{a}_{{B}}\in\mathbb{R}_+^{\color{blue}{n_{{B}}}}$, define$$U(\boldsymbol{a}_{{A}},\boldsymbol{a}_{{B}})=\big\lbrace M\in\mathbb{R}_+^{\color{red}{n_{{A}}}\times\color{blue}{n_{{B}}}}:M\boldsymbol{1}_{\color{blue}{n_{{B}}}}=\boldsymbol{a}_{A}\text{ and }{M}^\top\boldsymbol{1}_{\color{red}{n_{{A}}}}=\boldsymbol{a}_{B}\big\rbrace$$For convenience, let $U_{\color{red}{n_{{A}}},\color{blue}{n_{{B}}}}$ denote $$\displaystyle{U\left(\boldsymbol{1}_{n_{{A}}},\frac{\color{red}{n_{{A}}}}{\color{blue}{n_{{B}}}}\boldsymbol{1}_{n_{{B}}}\right)}$$ (so that $U_{\color{red}{n},\color{blue}{n}}$ is the set of permutation matrices associated with $\mathcal{S}_n$). Let $C_{i,j}=d(x_i,y_{j})^k$so that $$W_k^k(\boldsymbol{x},\boldsymbol{y}) = \underset{P\in U_{\color{red}{n_{{A}}},\color{blue}{n_{{B}}}}}{\text{argmin}} \Big\lbrace \langle P,C\rangle \Big\rbrace$$where$$\langle P,C\rangle = \sum_{i=1}^{\color{red}{n_{{A}}}} \sum_{j=1}^{\color{blue}{n_{{B}}}} P_{i,j}C_{i,j}$$ then consider $$P^* \in \underset{P\in U_{\color{red}{n_A},\color{blue}{n_B}}}{\text{argmin}} \Big\lbrace \langle P,C\rangle \Big\rbrace$$For the slides, if we have the same sample sizes in the two groups, we have

we can illustrate below (with costs, or distances)

And with different group sizes,

i.e., if we consider real datasets

And as usual, we can consider some penalized version. Define $$\mathcal{E}(P) = -\sum_{i=1}^{\color{red}{n_{{A}}}} \sum_{j=1}^{\color{blue}{n_{{B}}}} P_{i,j}\log P_{i,j}$$or$$\mathcal{E}'(P) = -\sum_{i=1}^{\color{red}{n_{{A}}}} \sum_{j=1}^{\color{blue}{n_{{B}}}} P_{i,j}\big[\log P_{i,j}-1\big]$$ or $$\mathcal{E}'(P) = -\sum_{i=1}^{\color{red}{n_{{A}}}} \sum_{j=1}^{\color{blue}{n_{{B}}}} P_{i,j}\big[\log P_{i,j}-1\big]$$ Define $$P^*_\gamma = \underset{P\in U_{\color{red}{n_{{A}}},\color{blue}{n_{{B}}}}}{\text{argmin}} \Big\lbrace \langle P,C\rangle -\gamma \mathcal{E}(P) \Big\rbrace$$ The problem is strictly convex.

#### Sinkhorn relaxation

This idea is related to the following theorem

Consider a simple optimal transportation problem between 6 points to 6 other points,

set.seed(123) x = (1:6)/7 y = runif(9) x [1] 0.14 0.29 0.43 0.57 0.71 0.86 y[1:6] [1] 0.29 0.79 0.41 0.88 0.94 0.05 library(T4transport) Wxy = wasserstein(x,y[1:6]) Wxy$plan that we can visualize below (the first observation of $\boldsymbol{x}$ is matched with the last one of $\boldsymbol{y}$, the second observation of $\boldsymbol{x}$ is matched with the first one of $\boldsymbol{y}$, etc) We observe that we simply match according to ranks. But we can also use a penalized version Sxy = sinkhorn(x, y[1:6], p = 2, lambda = 0.001) Sxy$plan

here with a very small pernalty

or a larger one

Sxy = sinkhorn(x, y[1:6], p = 2, lambda = 0.05) Sxy\$plan

In the discrete case, optimal transport can be related to Hardy-Littlewood-Polya inequality, that is related to the idea of matching based on ranks (corresponding to a monotone mapping function)

We have then

In the bivariate dicrete case, we have

#### Optimal mapping

We have mentioned that, in the univariate setting

and clearly, $\mathcal{T}^\star$ is increasing. In the Gaussian case, for example$$x_{{B}}=\mathcal{T}^\star(x_{{A}})= \mu_{{B}}+\sigma_{{B}}\sigma_{{A}}^{-1} (x_A-\mu_{{A}}).$$In the multivariate case, we need a more general concept of increasingness to define an “increasing” mapping $\mathcal{T}^\star:\mathbb{R}^k\to\mathbb{R}^k$.

In the Gaussian case, for example, we have a linear mapping,$$\boldsymbol{x}_{{B}} = \mathcal{T}^\star(\boldsymbol{x}_{{A}})=\boldsymbol{\mu}_{{B}} + \boldsymbol{A}(\boldsymbol{x}_{{A}}-\boldsymbol{\mu}_{{A}})$$where $\boldsymbol{A}$ is a symmetric positive matrix that satisfies $$\boldsymbol{A}\boldsymbol{\Sigma}_{{A}}\boldsymbol{A}=\boldsymbol{\Sigma}_{{B}},$$ which has a unique solution given by $$\boldsymbol{A}=\boldsymbol{\Sigma}_{{A}}^{-1/2}\big(\boldsymbol{\Sigma}_{{A}}^{1/2}\boldsymbol{\Sigma}_{{B}}\boldsymbol{\Sigma}_{{A}}^{1/2}\big)^{1/2}\boldsymbol{\Sigma}_{{A}}^{-1/2},$$ where $\boldsymbol{M}^{1/2}$ is the square root of the square (symmetric) positive matrix $\boldsymbol{M}$ based on the Schur decomposition ($\boldsymbol{M}^{1/2}$ is a positive symmetric matrix). In R, for example, use the expm package,

M = expm::sqrtm(matrix(c(1,1.2,1.2,2),2,2)) M [,1] [,2] [1,] 0.8244771 0.5658953 [2,] 0.5658953 1.2960565 M %*% M [,1] [,2] [1,] 1.0 1.2 [2,] 1.2 2.0

#### Optimal mapping, on real data

To illustrate, it is possible to consider the optimal matching, between the height of $n$ men and $n$ women,

Another example (discussed in Optimal Transport for Counterfactual Estimation: A Method for Causal Inference – with a nice R notebook created by Ewen), consider Black and non-Black mothers in the U.S.

or the joint mapping, in dimension 2

We will spend more time on those functions (and the related concept) in a few weeks, when discussing barycenters and geodesics… More details in the slides (online) and in the forthcoming textbook,

# Fairness and ethics for Insurance Pricing, at TD Insurance Seminar

This Thursday, I will give a talk on fairness and ethics at the TD Insurance IA/ML annuel conference. Slides are available (it is an overview of Insurance, biaises, discrimination and fairness v2, emphasizing the use of optimal transport techniques).

# Talk at the ESSEC Risk Seminar

Thursday, I will be at La Défense, in Paris (France), to give a talk at the ESSEC Risk Seminar, entitled Causal Inference and Counterfactuals with Optimal Transport, with Applications in Fairness and Discrimination. Slides are now available, and the talk will be based on same recent papers, starting with Mitigating Discrimination in Insurance with Wasserstein Barycenters (presented last week-end at BIAS 2023) but also Fairness in Multi-Task Learning via Wasserstein Barycenters, and A Sequentially Fair Mechanism for Multiple Sensitive Attributes. There is also the textbook that should appear before the winter.

# 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.