1. Introduction
This paper introduces new measures of unfairness in algorithmic recommendation and demonstrates how to optimize these metrics to reduce different forms of unfairness. Since recommender systems make predictions based on observed data, they can easily inherit bias that may already exist. To address this issue, we first describe a process that leads to unfairness in recommender systems and identify the insufficiency of demographic parity for this setting. We then propose four new unfairness metrics that address different forms of unfairness. To improve model fairness, we provide five fairness objectives that can be optimized as regularizers.
We focus on a frequently practiced approach for recommendation called collaborative filtering. With this approach, predictions are made based on cooccurrence statistics, and most methods assume that the missing ratings are missing at random. Unfortunately, research has shown that sampled ratings have markedly different properties from the users’ true preferences (marlin2012collaborative, ; marlin:recsys09, ), which is a potential source of unfairness.
We consider a running example of unfair recommendation in education (sacin2009recommendation, ; thai2010recommender, ; dascalu2016educational, ), in which the underrepresentation of women in science, technology, engineering, and mathematics (STEM) topics (beede2011women, ; smith2011women, ; griffith2010persistence, ) causes the learned model to underestimate women’s preferences and be biased towards men. We find this setting a serious motivation to advance understanding of unfairness—and methods to reduce unfairness—in recommendation.
Related Work
Various studies have considered algorithmic fairness in problems such as classification (pedreshi2008discrimination, ; lum2016statistical, ; zafar2017fairness, ). Removing sensitive features (e.g., gender, race, or age) is often insufficient for fairness. Features are often correlated, so other unprotected attributes can be related to the sensitive features (kamishima2011fairness, ; zemel2013learning, ). Moreover, in problems such as collaborative filtering, algorithms do not directly consider measured features and instead infer latent user attributes from their behavior.
Another frequently practiced strategy for encouraging fairness is to enforce demographic parity, which is to ensure that the overall proportion of members in the protected group receiving positive (negative) classifications are identical to the proportion of the population as a whole (zemel2013learning, ). Based on this nonparity unfairness concept, Kamishima et al. (kamishima2011fairness, ; kamishima2012enhancement, ; kamishima2013efficiency, ) try to solve the unfairness issue in recommender systems by adding a regularization term that enforces demographic parity. However, demographic parity is only appropriate when preferences are unrelated to the sensitive features. In recommendation, user preferences are indeed influenced by sensitive features such as gender, race and age (chausson2010watches, ; daymont1984job, ).
To address the issues of demographic parity, Hardt et al. (hardt2016equality, ) measure unfairness with the true positive rate and true negative rate. They propose that, in a binary setting, given a decision , a protected attribute and the true label , the constraints are (hardt2016equality, ) . This idea encourages equal opportunity and no longer relies on the assumption of demographic parity, that the target variable is independent of sensitive features. Similarly, Calders et al. (calders2013controlling, )
propose to impose constrains on the residuals of linear regression models, which requires not only the mean prediction but also the mean residuals to be the same across groups. These ideas form the basis of the unfairness metrics we propose for recommendation.
2. Fairness Objectives for Collaborative Filtering
This section introduces fairness objectives for collaborative filtering. We begin by reviewing the matrix factorization method. We then describe the various fairness objectives we consider, providing formal definitions and discussion of their motivations.
2.1. Matrix Factorization
We consider the task of collaborative filtering using matrix factorization (koren2009matrix, ). We have a set of users indexed from 1 to and a set of items indexed from 1 to . For the th user, let be a variable indicating which group the th user belongs to. For the th item, let indicate the item group that it belongs to. Let be the preference score of the th user for the th item.
The matrixfactorization formulation assumes that each rating can be represented as , where is a
dimensional vector representing the
th user, is a dimensional vector representing the th item, and and are scalar bias terms for the user and item, respectively. The matrixfactorization learning algorithm seeks to learn these parameters from observed ratings , typically by minimizing a regularized, squared reconstruction error:(1) 
where and are the vectors of bias terms, and represents the Frobenius norm.
2.2. Fairness Metrics
We consider a binary group feature distinguishing disadvantaged and advantaged groups. In the STEM example, the disadvantaged group may be women and nonbinary gender identities, and the advantaged group may be men.
The first metric is value unfairness
, which measures inconsistency in signed estimation error across the user types, computed as
(2) 
where is the average predicted score for the th item from disadvantaged users, is the average predicted score for advantaged users, and and are the average ratings for the disadvantaged and advantaged users, respectively. Value unfairness occurs when one class of user is consistently given higher or lower predictions than their true preferences.
The second metric is absolute unfairness, which measures inconsistency in absolute estimation error across user types, computed as
(3) 
Absolute unfairness is unsigned, so it captures the quality of prediction for each user type.
The third metric is underestimation unfairness, which measures inconsistency in how much the predictions underestimate the true ratings:
(4) 
where is the hinge function, i.e., if and 0 otherwise. Underestimation unfairness is important in settings where missing recommendations are more critical than extra recommendations.
Conversely, the fourth new metric is overestimation unfairness, which measures inconsistency in how much the predictions overestimate the true ratings:
(5) 
Finally, a nonparity unfairness measure based on the regularization term introduced by Kamishima et al. (kamishima2011fairness, ) can be computed as the absolute difference between the overall average ratings of disadvantaged users and that of advantaged users .
To optimize the metric(s), we solve for a local minimum of .
3. Experiments
We run experiments on simulated courserecommendation data and real movie rating data (harper2016movielens, ).
3.1. Synthetic data
In our synthetic experiments, we consider four user groups and three item groups . The user groups represent women who do not enjoy STEM topics (W), women who do enjoy STEM topics (WS), men who do not enjoy STEM topics (M), and men who do (MS). The item groups represent courses that tend to appeal to women (Fem), STEM courses, and courses that tend to appeal to men (Masc). We generate simulated courserecommendation data with two stochastic block models (holland1976local, )
. Our rating block model determines the probability that a user in a user group likes an item in an item group
(6) 
We use two observation block models that determine the probability a user in a user group rates an item in an item group: one with uniform observation probability for all groups and one with unbalanced observation probabilities inspired by realworld biases
(7) 
We define two different user group distributions: one in which each of the four groups is exactly a quarter of the population, and an imbalanced setting where 0.4 of the population is in W, 0.1 in WS, 0.4 in MS, and 0.1 in M. This heavy imbalance is inspired by some of the severe gender imbalance in certain STEM areas today.
Unfairness from different types of underrepresentation
Using standard matrix factorization, we measure the various unfairness metrics under the different sampling settings. We average over five random trials and plot the average score in Fig. 1. In each trial, we generated ratings by 400 users and 300 items with the block models. We label the settings as follows: uniform user groups and uniform observation probabilities (U), uniform groups and biased observation probabilities (O), biased user group populations and uniform observations (P), and biased populations and observations (P+O).
Unfairness  Error  Value  Absolute  Underestimation  Overestimation  NonParity 

None  0.317 1.3e02  0.649 1.8e02  0.443 2.2e02  0.107 6.5e03  0.544 2.0e02  0.362 1.6e02 
Value  0.130 1.0e02  0.245 1.4e02  0.177 1.5e02  0.063 4.1e03  0.199 1.5e02  0.324 1.2e02 
Absolute  0.205 8.8e03  0.535 1.6e02  0.267 1.3e02  0.135 6.2e03  0.400 1.4e02  0.294 1.0e02 
Under  0.269 1.6e02  0.512 2.3e02  0.401 2.4e02  0.060 3.5e03  0.456 2.3e02  0.357 1.6e02 
Over  0.130 6.5e03  0.296 1.2e02  0.172 1.3e02  0.074 6.0e03  0.228 1.1e02  0.321 1.2e02 
NonParity  0.324 1.3e02  0.697 1.8e02  0.453 2.2e02  0.124 6.9e03  0.573 1.9e02  0.251 1.0e02 
Unfairness  Error  Value  Absolute  Underestimation  Overestimation  NonParity 

None  0.887 1.9e03  0.234 6.3e03  0.126 1.7e03  0.107 1.6e03  0.153 3.9e03  0.036 1.3e03 
Value  0.886 2.2e03  0.223 6.9e03  0.128 2.2e03  0.102 1.9e03  0.148 4.9e03  0.041 1.6e03 
Absolute  0.887 2.0e03  0.235 6.2e03  0.124 1.7e03  0.110 1.8e03  0.151 4.2e03  0.023 2.7e03 
Under  0.888 2.2e03  0.233 6.8e03  0.128 1.8e03  0.102 1.7e03  0.156 4.2e03  0.058 9.3e04 
Over  0.885 1.9e03  0.234 5.8e03  0.125 1.6e03  0.112 1.9e03  0.148 4.1e03  0.015 2.0e03 
NonParity  0.887 1.9e03  0.236 6.0e03  0.126 1.6e03  0.110 1.7e03  0.152 3.9e03  0.010 1.5e03 
The statistics suggest that each underrepresentation type contributes to various forms of unfairness. For all metrics except parity, there is a strict order of unfairness, where uniform data is the most fair and biasing the populations and observations causes the most unfairness. Because of the observation bias, there is actually nonparity in the labeled ratings, so a high nonparity score does not necessarily indicate an unfair situation. These tests verify that unfairness can occur with imbalanced populations or observations even when the measured ratings accurately represent user preferences.
Optimization of unfairness metrics
We optimize fairness objectives under the most imbalanced setting: the user populations are imbalanced, and the sampling rate is skewed. We optimize for 500 iterations of Adam
(kingma2014adam, ).The results are listed in Table 1. The learning algorithm successfully minimizes the unfairness penalties, generalizing to unseen, heldout useritem pairs. And reducing any unfairness metric does not lead to a significant increase in reconstruction error. The combined objective “Over+Under” leads to scores that are close to the minimum of each metric except parity.
3.2. Real data
We use the Movielens Million Dataset (harper2016movielens, ), which contains ratings in [1,5] by 6,040 users and 3,883 movies. We manually selected five genres (action, crime, musical, romance, and scifi) that each have different forms of gender imbalance and only consider movies that list these genres. Then we filtered the users to only consider those who rated at least 50 of the selected movies. After filtering by genre and rating frequency, we have 2,953 users and 1,006 movies in the dataset.
We run three trials in which we randomly split the ratings into training and testing sets, the average scores are listed in Table 2. As in the synthetic setting, the results show that optimizing each unfairness metric leads to the best performance on that metric without a significant change in the reconstruction error.
4. Conclusion
In this paper, we discussed various types of unfairness that can occur in collaborative filtering. We demonstrate that these forms of unfairness can occur even when the observed rating data accurately reflects the users’ preferences. We propose four fairness metrics and demonstrate that augmenting matrix factorization objectives with these metrics as penalty functions enables their minimization. Our experiments on synthetic and real data show that minimization of these unfairness metrics is possible with no significant increase in reconstruction error. However, no single objective was the best for all unfairness metrics, so it remains necessary for practitioners to consider precisely which form of unfairness is most important in their application and optimize that specific objective.
Future Work
While our work here focused on improving fairness among user groups, we did not address fair treatment of different item groups. The model could be biased towards certain items, e.g., performing better for some items than others. Achieving fairness for both user and items may be important when considering that the items may also suffer from discrimination or bias, e.g., when courses are taught by instructors with different demographics.
Moreover, our fairness metrics assume that users rate items according to their true preferences. This assumption is likely violated in real data, since ratings can also be influenced by environmental factors. E.g., in education, a student’s rating for a course also depends on whether the course has an inclusive and welcoming learning environment. However, addressing this type of bias may require additional information or external interventions beyond the provided rating data.
Finally, we are investigating methods to reduce unfairness by directly modeling the twostage sampling process we used in Section 3.1. Explicitly modeling the rating and observation probabilities as separate variables may enable a principled, probabilistic approach to address these forms of data imbalance.
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