Fair Predictions
by
Thomas Kehrenberg
and
Oliver Thomas
Predictive Analytics Lab, University of Sussex
Mon, 15
th
April 2019
## Summary - How does our data become unfair? - Define "Fair" - Methods to enforce fairness - In the algorithm - Adjusting the data itself - Conclusion
## How bias gets into the dataset ### Sampling Bias
## Sampling Bias <img src="images/sampling_1.svg" alt="Data Generation Pipeline" style="width: 55%; border: none; box-shadow: none;">
## Sampling Bias <img src="images/sampling_2.svg" alt="Data Generation Pipeline" style="width: 55%; border: none; box-shadow: none;">
## Sampling Bias <img src="images/sampling_3.svg" alt="Data Generation Pipeline" style="width: 55%; border: none; box-shadow: none;">
## How bias gets into the dataset ### Unfairness <br/> <img src="images/data_generation.svg" alt="Data Generation Pipeline" style="width: 50%; border: none;">
## Confusion matrix | | $y=1$ | $y=0$ | |-----------------|:-------------------:|:-------------------:| | **$\hat{y}=1$** | true positive (TP) | false positive (FP) | | **$\hat{y}=0$** | false negative (FN) | true negative (TN) | <br/> **acceptance rate**: $AR = P(\hat{y}=1) = \frac{TP + FP}{TP + FP + FN + TN}$ **true positive rate**: $TPR = P(\hat{y}=1|y=1) = \frac{TP}{TP + FN}$
## Fairness definitions: ### Group Fairness ### Individual Fairness
## Definitions – Group Fairness ### Demographic Parity Make (positive) predictions at the same rate across different subgroups. $$ P(\hat{Y} = 1 | S=0) = P(\hat{Y} = 1 | S=1) $$ → Equalise acceptance rate accross groups.
## Definitions – Group Fairness ### Equal Odds Make (positive) predictions at the same rate across different subgroups, conditioned on the known outcome. $$ P(\hat{Y}=y | S=0, Y=y) = P(\hat{Y}=y | S=1, Y=y) $$ with $Y \in \\{0,1\\}$ → Equalise the TPR and the TNR (or FPR) across groups.
## Definitions – Group Fairness ### Equal Opportunity Make (positive) predictions at the same rate across different subgroups, conditioned on if the known outcome is positive. $$ P(\hat{Y} = 1 | S=0, Y=1) = P(\hat{Y} = 1 | S=1, Y=1) $$ → Equalise the TPR across groups.
## Fairness definitions: ### Individual Fairness > Similar individuals should be treated similarly. Given some distance measure between individuals (e.g. the Euclidean distance in the feature space), the distance between individuals and the distance of their associated outcome should be proportional.
## Fairness-accuracy trade-off If the class label is *biased*, or the features themselves are biased, then it cannot be possible to be both fair and accurate. However, if bias solely occurs due to sampling bias, then fairness should improve generalisation. <!-- Not sure about the above statement, will have a think -->
## How to remove bias? - During training an algortithm - Adjusting the data
## Notation - $Y \in \\{0,...,n\\}$: Class label - $S \in \\{0,...,k\\}$: Sensitive label - $X = (X_1, ... , X_m)$: Features of a datapoint (Covariates) - $\theta$: Model parameters that we learn.
## Notation $\hat{Y}=$ Model's prediction of $Y$ $\hat{S}=$ Model's prediction of $S$ $\hat{X}=$ Model's prediction of $X$
## Example Dataset: UCI Adult U.S. census data from 1994 $Y=$ Does the individual earn > $\$50,000$? $S=$ Is the individual white? $X=$ Features of the individual: relationship status, age, education level, hours worked per week etc.
## In-process methods - Enforce fairness *during training* - **Example**: Logistic Regression * model: $\; f(x) = Wx + b$, $\; \theta = \\{W, b\\}$ * probability score: $\; P(y=1|x, \theta)=\sigma(f(x))$ * with dependency on $s$: $\; P(y=1|x, \theta, s)$
## In-process: Likelihood - Manipulate the likelihood $P(y=1|x, \theta, s)$ to enforce fairness - Introduce target labels $\bar{y}$ - $P(y=1|x, \theta, s) = \sum\limits_{\bar{y}} P(y=1|\bar{y},s)P(\bar{y}|x, \theta, s)$ - Intuition: change the learning goal Paper: *Kehrenberg et al.*, `arXiv:1810.05598`
## In-process: Likelihood <!-- - conceptually works best with *Demographic Parity* * but was also extended to Equal Odds/Opportunity --> transparent and easy to control  <!-- .element: style="height: 7em; border: none;"-->  <!-- .element: style="height: 7em; border: none;"-->
## In-process: Constraint - Different approach: add constraint to model parameters so that model will be fair - Measure of bias in output $\; f$: $\; Cov(s, f) \approx \frac{1}{N} \sum\limits_{i=1}^{N} (s_i - \bar{s}) \cdot f_i$ * Assumption: $E[f] =0$ which is fulfilled in the Logistic Regression example - $Cov(s, f)=0$ → $s$ and $f$ independent → Demographic Parity
## In-process: Constraint - Goal: $\; f$ is predictive of $y$ while keeping $Cov(s, f)$ small - Classification loss: $\; \mathcal{L}_c(y, \sigma(f(x))\;$ (e.g. cross entropy) - Total loss: $\;\mathcal{L} = \mathcal{L}_c + \gamma \cdot \| Cov(s, f) \|$ * $\gamma$ is a trade-off parameter - A variation on this can also enforce Equal Odds/Opportunity
## Pre-process ### Change of problem set-up We are no longer a decision making company and are now a data vendor. Our concern is that our clients will be unwittingly making *unfair* decisions based on the data that we provide.
## Pre-process ### Possible solutions - generate some new, fairer examples - transform the data to a new *fair* space, where everyone is neither male nor female, but somewhere in between.
## Pre-process ### Generate Fairer Samples <img src="images/data_generation.svg" alt="Data Generation Pipeline" style="width: 50%; border: none;">
## Pre-process ### Generate Fairer Samples By going back to $\tilde{X}$, and $\tilde{Y}$, we can imagine the likely $X$ and $Y$ realised from a different sensitive attibute value. Model: Auto-Encoding Variational Bayes to obtain $\tilde{X}$ and $\tilde{Y}$. Use these `imagined` examples alongside actual training data. Paper: *Thomas et al.*, `Under review at IJCAI'19`
## Pre-process ### Transform the dataset learn some function $g: X \rightarrow Z$ s.t. $Z \perp S$ Then a classifier $f: Z \rightarrow Y$ will be fair.
## Pre-process ### Transform the dataset <img src="images/Latent.svg" alt="Mapping" style="width: 50%; border: none;">
## Pre-process ### Transform the dataset Paper: *Quadrianto et al.*, `arXiv:1810.06755` IEEE Conference on Computer Vision and Pattern Recognition (CVPR) 2019 ** Current state-of-the-art approach** Constrains $Z$ to be in the same space as $X$, so **meaning** retained. Enforces independent *distributions* so that `$S \in \{0,1,...,n\}$`
## Pre-process ### Transform the dataset <table style="width:100%"> <tr> <th>X</th> <th>Fair X</th> </tr> <tr> <td><img src="images/x_rel_giv_y1.png" alt="X" style="width: 70%; border: none;"></td> <td><img src="images/x_tilde_rel_giv_y1.png" alt="X tilde" style="width: 70%; border: none;"></td> </tr> </table>
## Take-aways <p class="fragment">Data and algorithms are **not** inherently fair</p> <p class="fragment">Fairness is **application specific**</p> <p class="fragment">Strategies to be fairer exist</p>