# SIGIR2020-Controlling Fairness and Bias in Dynamic Learning-to-Rank

## the Solution for the Bias Problem

- To address the bias problem, minimize the loss function as follows
$$L(w)= \sum_t \sum_d R^2(d|x_t) + \frac{c_t(d)}{p_t(d)}[c_t(d)-2R(d|x_t)]$$
- where,
+ $t$ is a timestamp
+ $d$ is a document
+ $R$ is a function to fit
+ $c_t(d)$ is the click of document $d$ at timestamp $t$
+ $p_t(d)$ is the probablity of examing document $d$ at timestamp $t$
+ $x_t$ is all features that are available at timestamp $t$
- it can be proved that this f

## the Solution for the Unfairness Problem
$$argsort[R(d|x) + \lambda err_\tau (d)]$$
- where
+ $err_\tau (d)$ is defined as
$$err_\tau(d)=(\tau-1) \max_{G_i}[D_{\tau-1}(G_i, G)], d \in G$$
+ the larger $err_\tau (d)$ is,
+ $G$ is a group of documents (e.g., from the same source)
+ $D(G_i, G_j)$ is defined as
$$D(G_i, G_j)=\frac{\frac{1}{\tau}\sum_{t=1}^{\tau} E_t(G_i)}{M(G_i)} - \frac{\frac{1}{\tau}\sum_{t=1}^{\tau} E_t(G_j)}{M(G_j)}$$
+ if $D(G_i, G_j)$ is large, it means it's not fair for $G_j$ ($G_j$ get less exposure than it should have)
+ Exposure of group $G_i$: $E_t(G_i) = \frac{1}{|G_i|} \sum_{d \in G_i} p_t(d)$
+ Merit of group $G_i$: $M(G_i)=\frac{1}{|G_i|} \sum_{d \in G_i} R(d)$

## How are we going to use it?
- In our ranking model: 1) it only considers $c(d)$; 2) it considers $c(d)$ in the aggregated level; we should incorporate $p_t(d)$
- for diversification, it's hard to directly apply this because $R(d)$ is unknown for most cases;
+ we can approximite the ranking score as $R(d)$;
+ we can approximate the $E(G)$ and $M(G)$ from the sample buckets;