A flexible framework for evaluating user and item fairness in recommender systems

Abstract

One common characteristic of research works focused on fairness evaluation (in machine learning) is that they call for some form of parity (equality) either in treatment—meaning they ignore the information about users’ memberships in protected classes during training—or in impact—by enforcing proportional beneficial outcomes to users in different protected classes. In the recommender systems community, fairness has been studied with respect to both users’ and items’ memberships in protected classes defined by some sensitive attributes (e.g., gender or race for users, revenue in a multi-stakeholder setting for items). Again here, the concept has been commonly interpreted as some form of equality—i.e., the degree to which the system is meeting the information needs of all its users in an equal sense. In this work, we propose a probabilistic framework based on generalized cross entropy (GCE) to measure fairness of a given recommendation model. The framework comes with a suite of advantages: first, it allows the system designer to define and measure fairness for both users and items and can be applied to any classification task; second, it can incorporate various notions of fairness as it does not rely on specific and predefined probability distributions and they can be defined at design time; finally, in its design it uses a gain factor, which can be flexibly defined to contemplate different accuracy-related metrics to measure fairness upon decision-support metrics (e.g., precision, recall) or rank-based measures (e.g., NDCG, MAP). An experimental evaluation on four real-world datasets shows the nuances captured by our proposed metric regarding fairness on different user and item attributes, where nearest-neighbor recommenders tend to obtain good results under equality constraints. We observed that when the users are clustered based on both their interaction with the system and other sensitive attributes, such as age or gender, algorithms with similar performance values get different behaviors with respect to user fairness due to the different way they process data for each user cluster.

Appendices

Appendix A: Theoretical analysis of GCE properties

In appendix, we provide a theoretical analysis of the proposed probabilistic metric, i.e., GCE, for measuring unfairness. Previous work (Speicher et al. 2018) has explored four properties to be satisfied by inequality indices, including unfairness inequality. These properties are (1) anonymity, (2) population invariance, (3) transfer principle, and (4) zero normalization.

We claim that GCE satisfies these four properties. In appendix, for the sake of clarity, we prove that the mentioned properties are satisfied by a simplified version of the proposed probabilistic unfairness metric, i.e., GCE when $p_f$ is uniform. We follow our proofs based on the GCE formulation for discrete attributes, presented in Eq. (3). Assuming $p_f$ being uniform, the GCE formulation is:

$$\begin{aligned} I_{\text {uniform}}(m, a)&= \frac{1}{\beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{\left( \frac{1}{n}\right) ^\beta \cdot p_m^{(1-\beta )}(a_j)} - 1 \right] \nonumber \\&= \frac{1}{\beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{\left( \frac{1}{n}\right) ^\beta \cdot \left( \frac{v_j}{Z}\right) ^{(1-\beta )}} - 1 \right] \nonumber \\&= \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{ \left( \frac{v_j}{\mu }\right) ^{(1-\beta )}} - n \right] \end{aligned}$$

(9)

where $p_m(a_j)=v_j/Z$, i.e., $Z = \sum _{j=1}^n{v_j}$, and $\mu = Z / n$ denotes the average value. For brevity, we denote $I_{\text {uniform}}(m, a)$ as $I_{\text {uniform}}({\mathbf {v}})$ where ${\mathbf {v}} = [v_1, v_2, \cdots , v_n] \in {\mathbb {R}}^n$ is the vector of all values corresponding to the attribute a obtained by the model m.

Anonymity. According to the anonymity property, the inequality measure should not depend on the characteristics of attributes except for their values obtained by the model. As shown in Equation (9), the inequality measure only depends on the value of attributes, i.e., $v_j$s and the average value $\mu$ which again is computed based on the values as $\frac{\sum _{j=1}^n{v_j}}{n}$. Therefore, this property is satisfied by $I_{\text {uniform}}$.

Population invariance. This property indicates that the inequality measure is independent of the population size.

Proof

To prove that $I_{\text {uniform}}$ satisfies the population invariance property, assume ${\mathbf {v}}' = <{\mathbf {v}}, {\mathbf {v}}, \cdots , {\mathbf {v}}> \in {\mathbb {R}}^{nk}$ denotes a k-replication of the vector ${\mathbf {v}}$. Therefore, $I_{\text {uniform}}({\mathbf {v}}')$ is computed as:

$$\begin{aligned} I_{\text {uniform}}({\mathbf {v}}')&= \frac{1}{nk \beta \cdot (1-\beta )} \left[ \sum _{j=1}^{nk}{ \left( \frac{v'_j}{\mu } \right) ^{(1-\beta )}} - nk \right] \\&= \frac{1}{nk \beta \cdot (1-\beta )} \left[ \sum _{j=1}^{n}{ \left( k \frac{v_j}{\mu } \right) ^{(1-\beta )}} - nk \right] \\&= \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^{n}{ \left( \frac{v_j}{\mu } \right) ^{(1-\beta )}} - n \right] \\&= I_{\text {uniform}}({\mathbf {v}}) \end{aligned}$$

$\square$

The transfer principle. According to the transfer principle, also known as the Pigou–Dalton principle (Dalton 1920; Pigou 1912), transferring benefit from a high-benefit attribute value to a low-benefit value, if it does not reverse the relative position of values, must decrease the inequality.

Proof

Assume we transfer $\delta$ from $v_j$ to $v_{j'}$, such that $v_j > v_{j'}$ and $0< \delta < \frac{v_j - v_{j'}}{2}$ so this transfer does not reverse the relative position of these two attribute values. This results in ${\mathbf {v}}' = [v_1, v_2, \ldots , v_j-\delta , \ldots , v_{j'}+\delta , \ldots , v_n] \in {\mathbb {R}}^n$. Therefore,

$$\begin{aligned}&I_{\text {uniform}}({\mathbf {v}}') - I_{\text {uniform}}({\mathbf {v}}) \nonumber \\&\quad = \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{ \left( \frac{v'_j}{\mu }\right) ^{(1-\beta )}} - n \right] - \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{ \left( \frac{v_j}{\mu }\right) ^{(1-\beta )}} - n \right] \nonumber \\&\quad = \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{\left( \left( \frac{v'_j}{\mu }\right) ^{(1-\beta )} - \left( \frac{v_j}{\mu }\right) ^{(1-\beta )}\right) } \right] \nonumber \\&\quad = \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ (v_j - \delta )^{(1-\beta )} + (v_{j'} + \delta )^{(1-\beta )} - v_j^{(1-\beta )} - v_{j'}^{(1-\beta )} \right] \end{aligned}$$

(10)

To obtain the maximum value of this function, we compute its derivative with respect to $\delta$ and set it to zero, as follows:

$$\begin{aligned}&\frac{\partial (I_{\text {uniform}}({\mathbf {v}}') - I_{\text {uniform}}({\mathbf {v}}))}{\partial \delta } = 0 \\&\quad \Rightarrow \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ -(1-\beta )(v_j - \delta )^{-\beta } + (1-\beta )(v_{j'} + \delta )^{-\beta } \right] = 0 \\&\quad \Rightarrow -(v_j - \delta )^{-\beta } + (v_{j'} + \delta )^{-\beta } = 0 \\&\quad \Rightarrow \delta = \frac{v_j-v_{j'}}{2} \end{aligned}$$

Since $\frac{\partial ^2(I_{\text {uniform}}({\mathbf {v}}') - I_{\text {uniform}}({\mathbf {v}}))}{\partial \delta ^2} < 0$, the computed $\delta$ gives us the maximum value for the given function. Therefore, according to Eq. (10), since $0< \delta < \frac{v_j-v_{j'}}{2}$, we have:

$$\begin{aligned}&I_{\text {uniform}}({\mathbf {v}}') - I_{\text {uniform}}({\mathbf {v}})\\&\quad< \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ \left( v_j - \frac{v_j-v_{j'}}{2} \right) ^{(1-\beta )} + \left( v_{j'} + \frac{v_j-v_{j'}}{2} \right) ^{(1-\beta )} - v_j^{(1-\beta )} - v_{j'}^{(1-\beta )} \right] \\&\quad = \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ \left( \frac{v_j+v_{j'}}{2} \right) ^{(1-\beta )} + \left( \frac{v_j+v_{j'}}{2} \right) ^{(1-\beta )} - v_j^{(1-\beta )} - v_{j'}^{(1-\beta )} \right] \\&\quad = \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ 2^\beta (v_j+v_{j'})^{(1-\beta )} - v_j^{(1-\beta )} - v_{j'}^{(1-\beta )} \right] \\&\quad< \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ 2^\beta (2v_{j'})^{(1-\beta )} - v_j^{(1-\beta )} - v_{j'}^{(1-\beta )} \right] \\&\quad = \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ 2(v_{j'})^{(1-\beta )} - v_j^{(1-\beta )} - v_{j'}^{(1-\beta )} \right] \\&\quad = \frac{1}{n \beta \cdot (1-\beta ) \cdot \mu ^{(1-\beta )}} \left[ (v_{j'})^{(1-\beta )} - v_j^{(1-\beta )} \right] \\&\quad < 0 \end{aligned}$$

Therefore, $I_{\text {uniform}}({\mathbf {v}}') < I_{\text {uniform}}({\mathbf {v}})$, and thus, $I_{\text {uniform}}$ satisfies the transfer principle. $\square$

Zero normalization. According to this property, the inequality measure should be minimized when all attribute values are equal (i.e., the uniform distribution). The minimum value for the fairness metric should be zero.

Proof

To prove this property, we use the Lagrange multiplier approach. The Lagrange function is defined as:

$$\begin{aligned} {\mathcal {L}}({\mathbf {v}}, \lambda ) = \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{ \left( \frac{v_j}{\mu }\right) ^{(1-\beta )}} - n \right] - \lambda \left( \sum _{j=1}^n{\frac{v_j}{n}} - \mu \right) \end{aligned}$$

(11)

where $\lambda$ is the Lagrange multiplier. Therefore, we have:

$$\begin{aligned} {\left\{ \begin{array}{ll} \frac{\partial {\mathcal {L}}({\mathbf {v}}, \lambda )}{\partial v_j} = \frac{1}{n \beta \mu } \cdot \left( \frac{v_j}{\mu }\right) ^{-\beta } - \frac{\lambda }{n} \\ \frac{\partial {\mathcal {L}}({\mathbf {v}}, \lambda )}{\partial \lambda } = \sum _{j=1}^n{\frac{v_j}{n}} - \mu \end{array}\right. } \end{aligned}$$

(12)

Setting the above partial derivatives to zero results in $v_1 = v_2 = \cdots = v_n = \mu$. Therefore, we have:

$$\begin{aligned}&\min _{{\mathbf {v}}}{\frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{ \left( \frac{v_j}{\mu }\right) ^{(1-\beta )}} - n \right] } \\&\quad = \frac{1}{n \beta \cdot (1-\beta )} \left[ \sum _{j=1}^n{ \left( \frac{\mu }{\mu }\right) ^{(1-\beta )}} - n \right] \\&\quad = \frac{1}{n \beta \cdot (1-\beta )} \left[ n - n \right] \\&\quad = 0 \end{aligned}$$

Therefore, $I_{\text {uniform}}$ satisfies the zero normalization property. $\square$

Summary. In appendix, we theoretically study GCE and the provided proofs show that GCE satisfies the anonymity, population invariance, transfer principle, and zero normalization properties, under the uniformity assumption for the fair distribution. The proofs can be extended to the general case by relaxing the uniformity assumption, since we do not use any property of the uniform distribution in the proofs and just use its simple form to improve the readability and clarity.

Appendix B: Full results

In this section, we present the results for all the datasets and item and user attributes that were not included in the paper because of space constraints. First, we show in Table 14 the item GCE based on the price attribute for the datasets (instead of only limited to toys, as in Sect. 5.1).

Second, our results on user attributes—that is, interactions, helpfulness, and happiness for Amazon datasets, and age and gender for MovieLens—are presented for the datasets (together with the analysis already shown in Sect. 5.2 for Amazon Toys & Games): Amazon Electronics is described in Table 15, Amazon Video Games in Table 16, and MovieLens-1M in Table 17.

Table 14 ItemGCE using price as feature on the tested datasets

Table 15 UserGCE for Amazon Electronics dataset using the three user features considered

Table 16 UserGCE for Amazon Video Games dataset using the three user features considered

Table 17 UserGCE for MovieLens-1M dataset using the two user features considered for the rest of the datasets, plus age and gender