Tag Archives: Wasserstein

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}}}}, defineU(\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\rbraceFor 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})^kso 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\rbracewhere\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\rbraceFor 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 examplex_{{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,

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.

Fairness in Multi-Task Learning via Wasserstein Barycenters, at ECML PKDD 2023

Toady, presentation of our paper Fairness in Multi-Task Learning via Wasserstein Barycenters, ECML PKDD, in Torino, by François. Slides are available online (and a poster can be found below)

The paper was actually published in Machine Learning and Knowledge Discovery in Databases: Research Track (295–312), available here.

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.