1 Introduction
As machine learning algorithms increasingly affect decision making in society, many have raised concerns about the fairness and biases of these algorithms, especially in applications to healthcare or criminal justice, where human lives are at stake
(Angwin et al., 2016; Barocas & Selbst, 2016). It is often hoped that the use of automatic decision support systems trained on observational data will remove human bias and improve accuracy. However, factors such as data quality and model choice may encode unintentional discrimination, resulting in systematic disparate impact.We study fairness in prediction of outcomes such as recidivism, annual income, or patient mortality. Fairness is evaluated with respect to protected groups of individuals defined by attributes such as gender or ethnicity (Ruggieri et al., 2010). Following previous work, we measure discrimination in terms of differences in prediction cost across protected groups (Calders & Verwer, 2010; Dwork et al., 2012; Feldman et al., 2015). Correcting for issues of data provenance and historical bias in labels is outside of the scope of this work. Much research has been devoted to constraining models to satisfy costbased fairness in prediction, as we expand on below. The impact of data collection on discrimination has received comparatively little attention.
Fairness in prediction has been encouraged by adjusting models through regularization (Bechavod & Ligett, 2017; Kamishima et al., 2011), constraints (Kamiran et al., 2010; Zafar et al., 2017), and representation learning (Zemel et al., 2013). These attempts can be broadly categorized as modelbased approaches to fairness. Others have applied data preprocessing to reduce discrimination (Hajian & DomingoFerrer, 2013; Feldman et al., 2015; Calmon et al., 2017). For an empirical comparison, see for example Friedler et al. (2018). Inevitably, however, restricting the model class or perturbing training data to improve fairness may harm predictive accuracy (CorbettDavies et al., 2017).
A tradeoff of predictive accuracy for fairness is sometimes difficult to motivate when predictions influence highstakes decisions. In particular, posthoc correction methods based on randomizing predictions (Hardt et al., 2016; Pleiss et al., 2017) are unjustifiable for ethical reasons in clinical tasks such as severity scoring. Moreover, as pointed out by Woodworth et al. (2017), posthoc correction may lead to suboptimal predictive accuracy compared to other equally fair classifiers.
Disparate predictive accuracy can often be explained by insufficient or skewed sample sizes or inherent unpredictability of the outcome given the available set of variables. With this in mind, we propose that fairness of predictive models should be analyzed in terms of model bias, model variance, and outcome noise
before they are constrained to satisfy fairness criteria. This exposes and separates the adverse impact of inadequate data collection and the choice of the model on fairness. The cost of fairness need not always be one of predictive accuracy, but one of investment in data collection and model development. In highstakes applications, the benefits often outweigh the costs.In this work, we use the term “discrimination" to refer to specific kinds of differences in the predictive power of models when applied to different protected groups. In some domains, such differences may not be considered discriminatory, and it is critical that decisions made based on this information are sensitive to this fact. For example, in prior work, researchers showed that causal inference may help uncover which sources of differences in predictive accuracy introduce unfairness (Kusner et al., 2017). In this work, we assume that observed differences are considered discriminatory and discuss various means of explaining and reducing them.
Main contributions
We give a procedure for analyzing discrimination in predictive models with respect to costbased definitions of group fairness, emphasizing the impact of data collection. First, we propose the use of biasvariancenoise decompositions for separating sources of discrimination. Second, we suggest procedures for estimating the value of collecting additional training samples. Finally, we propose the use of clustering for identifying subpopulations that are discriminated against to guide additional variable collection. We use these tools to analyze the fairness of common learning algorithms in three tasks: predicting income based on census data, predicting mortality of patients in critical care, and predicting book review ratings from text. We find that the accuracy in predictions of the mortality of cancer patients vary by as much as between protected groups. In addition, our experiments confirm that discrimination level is sensitive to the quality of the training data.
2 Background
We study fairness in prediction of an outcome . Predictions are based on a set of covariates and a protected attribute . In mortality prediction, represents the medical history of a patient in critical care, the selfreported ethnicity, and mortality. A model is considered fair if its errors are distributed similarly across protected groups, as measured by a cost function . Predictions learned from a training set are denoted for some from a class . The protected attribute is assumed to be binary, , but our results generalize to the nonbinary case. A dataset consists of samples distributed according to . When clear from context, we drop the subscript from .
A popular costbased definition of fairness is the
equalized odds
criterion, which states that a binary classifier is fair if its false negative rates (FNR) and false positive rates (FPR) are equal across groups (Hardt et al., 2016). We define FPR and FNR with respect to protected group byExact equality, , is often hard to verify or enforce in practice. Instead, we study the degree to which such constraints are violated. More generally, we use differences in cost functions between protected groups to define the level of discrimination ,
(1) 
In this work we study cost functions in binary classification tasks, with the zeroone loss. In regression problems, we use the groupspecific meansquared error . According to (1), predictions satisfy equalized odds on if and .
Calibration and impossibility
A scorebased classifier is calibrated if the prediction score assigned to a unit equals the fraction of positive outcomes for all units assigned similar scores. It is impossible for a classifier to be calibrated in every protected group and satisfy multiple costbased fairness criteria at once, unless accuracy is perfect or base rates of outcomes are equal across groups (Chouldechova, 2017). A relaxed version of this result (Kleinberg et al., 2016) applies to the discrimination level . Inevitably, both constraintbased methods and our approach are faced with a choice between which fairness criteria to satisfy, and at what cost.
3 Sources of perceived discrimination
There are many potential sources of discrimination in predictive models. In particular, the choice of hypothesis class and learning objective has received a lot of attention (Calders & Verwer, 2010; Zemel et al., 2013; Fish et al., 2016). However, data collection—the chosen set of predictive variables , the sampling distribution , and the training set size —is an equally integral part of deploying fair machine learning systems in practice, and it should be guided to promote fairness. Below, we tease apart sources of discrimination through biasvariancenoise decompositions of costbased fairness criteria. In general, we may think of noise in the outcome as the effect of a set of unobserved variables , potentially interacting with . Even the optimal achievable error for predictions based on may be reduced further by observing parts of . In Figure 1, we illustrate three common learning scenarios and study their fairness properties through bias, variance, and noise.
To account for randomness in the sampling of training sets, we redefine discrimination level (1) in terms of the expected cost over draws of a random training set .
Definition 1.
The expected discrimination level of a predictive model learned from a random training set , is
is not observed in practice when only a single training set is available. If is small, it is recommended to estimate through resampling methods such as bootstrapping (Efron, 1992).
3.1 Biasvariancenoise decompositions of discrimination level
An algorithm that learns models from datasets is given, and the covariates and size of the training data are fixed. We assume that is a deterministic function given the training set , e.g. a thresholded scoring function. Following Domingos (2000)
, we base our analysis on decompositions of loss functions
evaluated at points . For decompositions of costs we let this be the zeroone loss, , and for , the squared loss, . We define the main prediction as the average prediction over draws of training sets for the squared loss, and the majority vote for the zeroone loss. The (Bayes) optimal prediction achieves the smallest expected error with respect to the random outcome .Definition 2 (Bias, variance and noise).
Following Domingos (2000), we define bias , variance and noise at a point below.
(2) 
Here, and , are all deterministic functions of , while
is a random variable.
In words, the bias is the loss incurred by the main prediction relative to the optimal prediction. The variance is the average loss incurred by the predictions learned from different datasets relative to the main prediction. The noise is the remaining loss independent of the learning algorithm, often known as the Bayes error. We use these definitions to decompose under various definitions of .
Theorem 1.
With the groupspecific zeroone loss or classconditional versions (e.g. FNR, FPR), or the mean squared error, and the discrimination level admit decompositions of the form
where we leave out in the decomposition of for brevity. With defined as in (2), we have
For the zeroone loss, if , otherwise . For the squared loss . The noise term for population losses is
and for classconditional losses w.r.t class ,
For the zeroone loss, and classconditional variants, and for the squared loss, .

Conditioning and exchanging order of expectation, the cases of mean squared error and zeroone losses follow from Domingos (2000). Classconditional losses follow from a casebycase analysis of possible errors. See the supplementary material for a full proof. ∎
Theorem 1 points to distinct sources of perceived discrimination. Significant differences in bias indicate that the chosen model class is not flexible enough to fit both protected groups well (see Figure 0(c)). This is typical of (misspecified) linear models which approximate nonlinear functions well only in small regions of the input space. Regularization or posthoc correction of models effectively increase the bias of one of the groups, and should be considered only if there is reason to believe that the original bias is already minimal.
Differences in variance, , could be caused by differences in sample sizes or groupconditional feature variance , combined with a high capacity model. Targeted collection of training samples may help resolve this issue. Our decomposition does not apply to posthoc randomization methods (Hardt et al., 2016) but we may treat these in the same way as we do random training sets and interpret them as increasing the variance of one group to improve fairness.
When noise is significantly different between protected groups, discrimination is partially unrelated to model choice and training set size and may only be reduced by measuring additional variables.
Proposition 1.
If , no model can be 0discriminatory in expectation without access to additional information or increasing bias or variance w.r.t. to the Bayes optimal classifier.
Proof.
By definition, . As the Bayes optimal classifier has neither bias nor variance, the result follows immediately. ∎
In line with Proposition 1, most methods for ensuring algorithmic fairness reduce discrimination by trading off a difference in noise for one in bias or variance. However, this tradeoff is only motivated if the considered predictive model is close to Bayes optimal and no additional predictive variables may be measured. Moreover, if noise is homoskedastic in regression settings, posthoc randomization is illadvised, as the difference in Bayes error is zero, and discrimination is caused only by model bias or variance (see the supplementary material for a proof).
Estimating bias, variance and noise
Groupspecific variance may be estimated through sample splitting or bootstrapping (Efron, 1992). In contrast, the noise and bias are difficult to estimate when is highdimensional or continuous. In fact, no convergence results of noise estimates may be obtained without further assumptions on the data distribution (Antos et al., 1999). Under some such assumptions, noise may be approximately estimated using distancebased methods (Devijver & Kittler, 1982), nearestneighbor methods (Fukunaga & Hummels, 1987; Cover & Hart, 1967), or classifier ensembles (Tumer & Ghosh, 1996). When comparing the discrimination level of two different models, noise terms cancel, as they are independent of the model. As a result, differences in bias may be estimated even when the noise is not known (see the supplementary material).
Testing for significant discrimination
When sample sizes are small, perceived discrimination may not be statistically significant. In the supplementary material, we give statistical tests both for the discrimination level and the difference in discrimination level between two models .
4 Reducing discrimination through data collection
In light of the decomposition of Theorem 1, we explore avenues for reducing group differences in bias, variance, and noise without sacrificing predictive accuracy. In practice, predictive accuracy is often artificially limited when data is expensive or impractical to collect. With an investment in training samples or measurement of predictive variables, both accuracy and fairness may be improved.
4.1 Increasing training set size
Standard regularization used to avoid overfitting is not guaranteed to improve or preserve fairness. An alternative route is to collect more training samples and reduce the impact of the biasvariance tradeoff. When supplementary data is collected from the same distribution as the existing set, covariate shift may be avoided (QuioneroCandela et al., 2009). This is often achievable; labeled data may be expensive, such as when paying experts to label observations, but given the means to acquire additional labels, they would be drawn from the original distribution. To estimate the value of increasing sample size, we predict the discrimination level as increases in size.
The curve measuring generalization performance of predictive models as a function of training set size is called a Type II learning curve (Domhan et al., 2015). We call , as a function of , the learning curve with respect to protected group . We define the discrimination learning curve (see Figure 1(a) for an example). Empirically, learning curves behave asymptotically as inverse powerlaw
curves for diverse algorithms such as deep neural networks, support vector machines, and nearestneighbor classifiers, even when model capacity is allowed to grow with
(Hestness et al., 2017; Mukherjee et al., 2003). This observation is also supported by theoretical results (Amari, 1993).Assumption 1 (Learning curves).
The population prediction loss , and groupspecific losses , for a fixed learning algorithm , behave asymptotically as inverse powerlaw curves with parameters . That is, such that for ,
(3) 
Intercepts, in (3) represent the asymptotic bias and the Bayes error , with the former vanishing for consistent estimators. Accurately estimating from finite samples is often challenging as the first term tends to dominate the learning curve for practical sample sizes.
In experiments, we find that the inverse powerlaws model fit group conditional () and classconditional (FPR, FNR) errors well, and use these to extrapolate based on estimates from subsampled data.
4.2 Measuring additional variables
When discrimination is dominated by a difference in noise, , fairness may not be improved through model selection alone without sacrificing accuracy (see Proposition 1). Such a scenario is likely when available covariates are not equally predictive of the outcome in both groups. We propose identification of clusters of individuals in which discrimination is high as a means to guide further variable collection—if the variance in outcomes within a cluster is not explained by the available feature set, additional variables may be used to further distinguish its members.
Let a random variable represent a (possibly stochastic) clustering such that indicates membership in cluster . Then let denote the expected prediction cost for units in cluster with protected attribute . As an example, for the zeroone loss we let
and define analogously for false positives or false negatives. Clusters for which is large identify groups of individuals for which discrimination is worse than average, and can guide targeted collection of additional variables or samples. In our experiments on income prediction, we consider particularly simple clusterings of data defined by subjects with measurements above or below the average value of a single feature with . In mortality prediction, we cluster patients using topic modeling. As measuring additional variables is expensive, the utility of a candidate set should be estimated before collecting a large sample (Koepke & Bilenko, 2012).
5 Experiments
We analyze the fairness properties of standard machine learning algorithms in three tasks: prediction of income based on national census data, prediction of patient mortality based on clinical notes, and prediction of book review ratings based on review text.^{1}^{1}1A synthetic experiment validating groupspecific learning curves is left to the supplementary material. We disentangle sources of discrimination by assessing the level of discrimination for the full data,estimating the value of increasing training set size by fitting Type II learning curves, and using clustering to identify subgroups where discrimination is high. In addition, we estimate the Bayes error through nonparametric techniques.
In our experiments, we omit the sensitive attribute from our classifiers to allow for closer comparison to previous works, e.g. Hardt et al. (2016); Zafar et al. (2017). In preliminary results, we found that fitting separate classifiers for each group increased the error rates of both groups due to the resulting smaller sample size, as classifiers could not learn from other groups. As our model objective is to maximize accuracy over all data points, our analysis uses a single classifier trained on the entire population.
5.1 Income prediction

Predictions of a person’s salary may be used to help determine an individual’s market worth, but systematic underestimation of the salary of protected groups could harm their competitiveness on the job market. The Adult dataset in the UCI Machine Learning Repository (Lichman, 2013) contains 32,561 observations of yearly income (represented as a binary outcome: over or under $50,000) and twelve categorical or continuous features including education, age, and marital status. Categorical attributes are dichotomized, resulting in a total of 105 features.
We follow Pleiss et al. (2017) and strive to ensure fairness across genders, which is excluded as a feature from the predictive models. Using an 80/20 traintest split, we learn a random forest predictor, which is is wellcalibrated for both groups (Brier (1950) scores of 0.13 and 0.06 for men and women). We find the difference in zeroone loss
has a 95%confidence interval
^{2}^{2}2Details for computing statistically significant discrimination can be found in the supplementary material. with decision thresholds at 0.5. At this threshold, the false negative rates are and for men and women respectively, and the false positive rates and. We focus on random forest classifiers, although we found similar results for logistic regression and decision trees.
We examine the effect of varying training set size on discrimination. We fit inverse powerlaw curves to estimates of and using repeated sample splitting where at least 20% of the full data is held out for evaluating generalization error at every value of
. We tune hyperparameters for each training set size for decision tree classifiers and logistic regression but tuned over the entire dataset for random forest. We include full training details in the supplementary material. Metrics are averaged over 50 trials. See Figure
1(a) for the results for random forests. Both FPR and FNR decrease with additional training samples. The discrimination level for false negatives decreases by a striking 40% when increasing the training set size from 1000 to 10,000. This suggests that trading off accuracy for fairness at small sample sizes may be illadvised. Based on fitted powerlaw curves, we estimate that for unlimited training data drawn from the same distribution, we would have and .In Figure 1(b), we compare estimated upper and lower bounds on noise ( and ) for men and women using the Mahalanobis and Bhattacharyya distances (Devijver & Kittler, 1982), and a nearest neighbor method (Cover & Hart, 1967) with and 5fold cross validation. Men have consistently higher noise estimates than women, which is consistent with the differences in zeroone loss found using all models. For nearest neighbors estimates, intervals for men and women are nonoverlapping, which suggests that noise may contribute substantially to discrimination.
To guide attempts at reducing discrimination further, we identify clusters of individuals for whom false negative predictions are made at different rates between protected groups, with the method described in Section 4.2. We find that for individuals in executive or managerial occupations (12% of the sample), false negatives are more than twice as frequent for women (0.412) as for men (0.157). For individuals in all other occupations, the difference is significantly smaller, 0.543 for women and 0.461 for men, despite the fact that the disparity in outcome base rates in this cluster is large (0.26 for men versus 0.09 for women). A possible reason is that in managerial occupations the available variable set explains a larger portion of the variance in salary for men than for women. If so, further subcategorization of managerial occupations could help reduce discrimination in prediction.
5.2 Intensive care unit mortality prediction
Unstructured medical data such as clinical notes can reveal insights for questions like mortality prediction; however, disparities in predictive accuracy may result in discrimination of protected groups. Using the MIMICIII dataset of all clinical notes from 25,879 adult patients from Beth Israel Deaconess Medical Center (Johnson et al., 2016), we predict hospital mortality of patients in critical care. Fairness is studied with respect to five selfreported ethnic groups of the following proportions: Asian (2.2%), Black (8.8%), Hispanic (3.4%), White (70.8%), and Other (14.8%). Notes were collected in the first 48 hours of an intensive care unit (ICU) stay; discharge notes were excluded. We only included patients that stayed in the ICU for more than 48 hours. We use the tfidf statistics of the 10,000 most frequent words as features. Training a model on 50% of the data, selecting hyperparameters on 25%, and testing on 25%, we find that logistic regression with L1regularization achieves an AUC of 0.81. The logistic regression is wellcalibrated with Brier scores ranging from 0.060.11 across the five groups; we note better calibration is correlated with lower prediction error.
We report cost and discrimination level in terms of generalized zeroone loss (Pleiss et al., 2017). Using an ANOVA test (Fisher, 1925) with
, we reject the null hypothesis that loss is the same among all five groups. To map the 95% confidence intervals, we perform pairwise comparisons of means using Tukey’s range test
(Tukey, 1949) across 5fold crossvalidation. As seen in Figure 2(a), patients in the Other and Hispanic groups have the highest and lowest generalized zeroone loss, respectively, with relatively few overlapping intervals. Notably, the largest ethnic group (White) does not have the best accuracy, whereas smaller ethnic groups tend towards extremes. While racial groups differ in hospital mortality base rates (Table 1 in the Supplementary material), Hispanic (10.3%) and Black (10.9%) patients have very different error rates despite similar base rates.To better understand the discrimination induced by our model, we explore the effect of changing training set size. To this end, we repeatedly subsample and split the data, holding out at least 20% of the full data for testing. In Figure 2(b), we show loss averaged over 50 trials of training a logistic regression on increasingly larger training sets; estimated inverse powerlaw curves show good fits. We see that some pairwise differences in loss decrease with additional training data.
Next, we identify clusters for which the difference in prediction errors between protected groups is large. We learn a topic model with topics generated using Latent Dirichlet Allocation (Blei et al., 2003). Topics are concatenated into an matrix where designates the proportion of topic in note . Following prior work on enrichment of topics in clinical notes (Marlin et al., 2012; Ghassemi et al., 2014)
, we estimate the probability of patient mortality
given a topic as where is the hospital mortality of patient . We compare relative error rates given protected group and topic using binary predicted mortality , actual mortality , and group for patient throughwhich follows using substitution and conditioning on . These error rates were computed using a logistic regression with L1 regularization using an 80/20 traintest split over 50 trials. While many topics have consistent error rates across groups, some topics (e.g. cardiac patients or cancer patients as shown in Figure 2(c)) have large differences in error rates across groups. We include more detailed topic descriptions in the supplementary material. Once we have identified a subpopulation with particularly high error, for example cancer patients, we can consider collecting more features or collecting more data from the same data distribution. We find that error rates differ between and across protected groups of cancer patients, and between and for cardiac patients.
5.3 Book review ratings
In the supplementary material, we study prediction of book review ratings from review texts (Gnanesh, 2017). The protected attribute was chosen to be the gender of the author as determined from Wikipedia. In the dataset, the difference in meansquared error has 95%confidence interval with for reviews for male authors and . Strikingly, our findings suggest that may be completely eliminated by additional targeted sampling of the less represented gender.
6 Discussion
We identify that existing approaches for reducing discrimination induced by prediction errors may be unethical or impractical to apply in settings where predictive accuracy is critical, such as in healthcare or criminal justice. As an alternative, we propose a procedure for analyzing the different sources contributing to discrimination. Decomposing wellknown definitions of costbased fairness criteria in terms of differences in bias, variance, and noise, we suggest methods for reducing each term through model choice or additional training data collection. Case studies on three realworld datasets confirm that collection of additional samples is often sufficient to improve fairness, and that existing posthoc methods for reducing discrimination may unnecessarily sacrifice predictive accuracy when other solutions are available.
Looking forward, we can see several avenues for future research. In this work, we argue that identifying clusters or subpopulations with high predictive disparity would allow for more targeted ways to reduce discrimination. We encourage future research to dig deeper into the question of local or contextspecific unfairness in general, and into algorithms for addressing it. Additionally, extending our analysis to intersectional fairness (Buolamwini & Gebru, 2018; HébertJohnson et al., 2017), e.g. looking at both gender and race or all subdivisions, would provide more nuanced grappling with unfairness. Finally, additional data collection to improve the model may cause unexpected delayed impacts (Liu et al., 2018) and negative feedback loops (Ensign et al., 2017) as a result of distributional shifts in the data. More broadly, we believe that the study of fairness in nonstationary populations is an interesting direction to pursue.
Acknowledgements
The authors would like to thank Yoni Halpern and Hunter Lang for helpful comments, and Zeshan Hussain for clinical guidance. This work was partially supported by Office of Naval Research Award No. N000141712791 and NSF CAREER award #1350965.
References
 Amari (1993) Amari, ShunIchi. A universal theorem on learning curves. Neural networks, 6(2):161–166, 1993.
 Angwin et al. (2016) Angwin, Julia, Larson, Jeff, Mattu, Surya, and Kirchner, Lauren. Machine bias. ProPublica, May, 23, 2016.
 Antos et al. (1999) Antos, András, Devroye, Luc, and Gyorfi, Laszlo. Lower bounds for bayes error estimation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 21(7):643–645, 1999.
 Barocas & Selbst (2016) Barocas, Solon and Selbst, Andrew D. Big data’s disparate impact. Cal. L. Rev., 104:671, 2016.
 Bechavod & Ligett (2017) Bechavod, Yahav and Ligett, Katrina. Learning fair classifiers: A regularizationinspired approach. arXiv preprint arXiv:1707.00044, 2017.

Bhattacharyya (1943)
Bhattacharyya, Anil.
On a measure of divergence between two statistical populations defined by their probability distributions.
Bull. Calcutta Math. Soc., 35:99–109, 1943.  Blei et al. (2003) Blei, David M, Ng, Andrew Y, and Jordan, Michael I. Latent dirichlet allocation. Journal of machine Learning research, 3(Jan):993–1022, 2003.
 Brier (1950) Brier, Glenn W. Verification of forecasts expressed in terms of probability. Monthey Weather Review, 78(1):1–3, 1950.
 Brown et al. (2001) Brown, Lawrence D, Cai, T Tony, and DasGupta, Anirban. Interval estimation for a binomial proportion. Statistical science, pp. 101–117, 2001.
 Buolamwini & Gebru (2018) Buolamwini, Joy and Gebru, Timnit. Gender shades: Intersectional accuracy disparities in commercial gender classification. In Conference on Fairness, Accountability and Transparency, pp. 77–91, 2018.

Calders & Verwer (2010)
Calders, Toon and Verwer, Sicco.
Three naive bayes approaches for discriminationfree classification.
Data Mining and Knowledge Discovery, 21(2):277–292, 2010.  Calmon et al. (2017) Calmon, Flavio, Wei, Dennis, Vinzamuri, Bhanukiran, Ramamurthy, Karthikeyan Natesan, and Varshney, Kush R. Optimized preprocessing for discrimination prevention. In Advances in Neural Information Processing Systems, pp. 3995–4004, 2017.
 Chouldechova (2017) Chouldechova, Alexandra. Fair prediction with disparate impact: A study of bias in recidivism prediction instruments. arXiv preprint arXiv:1703.00056, 2017.
 CorbettDavies et al. (2017) CorbettDavies, Sam, Pierson, Emma, Feller, Avi, Goel, Sharad, and Huq, Aziz. Algorithmic decision making and the cost of fairness. In Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 797–806. ACM, 2017.
 Cover & Hart (1967) Cover, Thomas and Hart, Peter. Nearest neighbor pattern classification. IEEE transactions on information theory, 13(1):21–27, 1967.
 Devijver & Kittler (1982) Devijver, Pierre A. and Kittler, Josef. Pattern recognition: a statistical approach. Sung Kang, 1982.

Domhan et al. (2015)
Domhan, Tobias, Springenberg, Jost Tobias, and Hutter, Frank.
Speeding up automatic hyperparameter optimization of deep neural
networks by extrapolation of learning curves.
In
TwentyFourth International Joint Conference on Artificial Intelligence
, 2015.  Domingos (2000) Domingos, Pedro. A unified biasvariance decomposition. In Proceedings of 17th International Conference on Machine Learning, pp. 231–238, 2000.
 Dwork et al. (2012) Dwork, Cynthia, Hardt, Moritz, Pitassi, Toniann, Reingold, Omer, and Zemel, Richard. Fairness through awareness. In Proceedings of the 3rd Innovations in Theoretical Computer Science Conference, pp. 214–226. ACM, 2012.
 Efron (1992) Efron, Bradley. Bootstrap methods: another look at the jackknife. In Breakthroughs in statistics, pp. 569–593. Springer, 1992.
 Ensign et al. (2017) Ensign, Danielle, Friedler, Sorelle A., Neville, Scott, Scheidegger, Carlos Eduardo, and Venkatasubramanian, Suresh. Runaway feedback loops in predictive policing. CoRR, abs/1706.09847, 2017. URL http://arxiv.org/abs/1706.09847.
 Feldman et al. (2015) Feldman, Michael, Friedler, Sorelle A, Moeller, John, Scheidegger, Carlos, and Venkatasubramanian, Suresh. Certifying and removing disparate impact. In Proceedings of the 21th ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, pp. 259–268. ACM, 2015.
 Fish et al. (2016) Fish, Benjamin, Kun, Jeremy, and Lelkes, Ádám D. A confidencebased approach for balancing fairness and accuracy. In Proceedings of the 2016 SIAM International Conference on Data Mining, pp. 144–152. SIAM, 2016.
 Fisher (1925) Fisher, R.A. Statistical methods for research workers. Edinburgh Oliver & Boyd, 1925.
 Friedler et al. (2018) Friedler, Sorelle A, Scheidegger, Carlos, Venkatasubramanian, Suresh, Choudhary, Sonam, Hamilton, Evan P, and Roth, Derek. A comparative study of fairnessenhancing interventions in machine learning. arXiv preprint arXiv:1802.04422, 2018.
 Fukunaga & Hummels (1987) Fukunaga, Keinosuke and Hummels, Donald M. Bayes error estimation using parzen and knn procedures. IEEE Transactions on Pattern Analysis and Machine Intelligence, 9(5):634–643, 1987.
 Ghassemi et al. (2014) Ghassemi, Marzyeh, Naumann, Tristan, DoshiVelez, Finale, Brimmer, Nicole, Joshi, Rohit, Rumshisky, Anna, and Szolovits, Peter. Unfolding physiological state: Mortality modelling in intensive care units. In Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining, pp. 75–84. ACM, 2014.
 Gnanesh (2017) Gnanesh. Goodreads book reviews, 2017. URL https://www.kaggle.com/gnanesh/goodreadsbookreviews.
 Hajian & DomingoFerrer (2013) Hajian, Sara and DomingoFerrer, Josep. A methodology for direct and indirect discrimination prevention in data mining. IEEE transactions on knowledge and data engineering, 25(7):1445–1459, 2013.

Hardt et al. (2016)
Hardt, Moritz, Price, Eric, Srebro, Nati, et al.
Equality of opportunity in supervised learning.
In Advances in Neural Information Processing Systems, pp. 3315–3323, 2016.  HébertJohnson et al. (2017) HébertJohnson, Ursula, Kim, Michael P, Reingold, Omer, and Rothblum, Guy N. Calibration for the (computationallyidentifiable) masses. arXiv preprint arXiv:1711.08513, 2017.
 Hestness et al. (2017) Hestness, Joel, Narang, Sharan, Ardalani, Newsha, Diamos, Gregory, Jun, Heewoo, Kianinejad, Hassan, Patwary, Md, Ali, Mostofa, Yang, Yang, and Zhou, Yanqi. Deep learning scaling is predictable, empirically. arXiv preprint arXiv:1712.00409, 2017.
 Johnson et al. (2016) Johnson, Alistair EW, Pollard, Tom J, Shen, Lu, Lehman, Liwei H, Feng, Mengling, Ghassemi, Mohammad, Moody, Benjamin, Szolovits, Peter, Celi, Leo Anthony, and Mark, Roger G. Mimiciii, a freely accessible critical care database. Scientific data, 3, 2016.
 Kamiran et al. (2010) Kamiran, Faisal, Calders, Toon, and Pechenizkiy, Mykola. Discrimination aware decision tree learning. In Data Mining (ICDM), 2010 IEEE 10th International Conference on, pp. 869–874. IEEE, 2010.
 Kamishima et al. (2011) Kamishima, Toshihiro, Akaho, Shotaro, and Sakuma, Jun. Fairnessaware learning through regularization approach. In Data Mining Workshops (ICDMW), 2011 IEEE 11th International Conference on, pp. 643–650. IEEE, 2011.
 Kleinberg et al. (2016) Kleinberg, Jon, Mullainathan, Sendhil, and Raghavan, Manish. Inherent tradeoffs in the fair determination of risk scores. arXiv preprint arXiv:1609.05807, 2016.
 Koepke & Bilenko (2012) Koepke, Hoyt and Bilenko, Mikhail. Fast prediction of new feature utility. arXiv preprint arXiv:1206.4680, 2012.
 Kusner et al. (2017) Kusner, Matt J, Loftus, Joshua, Russell, Chris, and Silva, Ricardo. Counterfactual fairness. In Advances in Neural Information Processing Systems, pp. 4069–4079, 2017.
 Lichman (2013) Lichman, M. UCI machine learning repository, 2013. URL http://archive.ics.uci.edu/ml.
 Liu et al. (2018) Liu, Lydia T, Dean, Sarah, Rolf, Esther, Simchowitz, Max, and Hardt, Moritz. Delayed impact of fair machine learning. arXiv preprint arXiv:1803.04383, 2018.
 Mahalanobis (1936) Mahalanobis, Prasanta Chandra. On the generalized distance in statistics. National Institute of Science of India, 1936.
 Marlin et al. (2012) Marlin, Benjamin M, Kale, David C, Khemani, Robinder G, and Wetzel, Randall C. Unsupervised pattern discovery in electronic health care data using probabilistic clustering models. In Proceedings of the 2nd ACM SIGHIT International Health Informatics Symposium, pp. 389–398. ACM, 2012.
 Mukherjee et al. (2003) Mukherjee, Sayan, Tamayo, Pablo, Rogers, Simon, Rifkin, Ryan, Engle, Anna, Campbell, Colin, Golub, Todd R, and Mesirov, Jill P. Estimating dataset size requirements for classifying dna microarray data. Journal of computational biology, 10(2):119–142, 2003.
 Pedregosa et al. (2011) Pedregosa, F., Varoquaux, G., Gramfort, A., Michel, V., Thirion, B., Grisel, O., Blondel, M., Prettenhofer, P., Weiss, R., Dubourg, V., Vanderplas, J., Passos, A., Cournapeau, D., Brucher, M., Perrot, M., and Duchesnay, E. Scikitlearn: Machine learning in Python. Journal of Machine Learning Research, 12:2825–2830, 2011.
 Pleiss et al. (2017) Pleiss, Geoff, Raghavan, Manish, Wu, Felix, Kleinberg, Jon, and Weinberger, Kilian Q. On fairness and calibration. In Advances in Neural Information Processing Systems, pp. 5684–5693, 2017.
 QuioneroCandela et al. (2009) QuioneroCandela, Joaquin, Sugiyama, Masashi, Schwaighofer, Anton, and Lawrence, Neil D. Dataset shift in machine learning. The MIT Press, 2009.
 Ruggieri et al. (2010) Ruggieri, Salvatore, Pedreschi, Dino, and Turini, Franco. Data mining for discrimination discovery. ACM Transactions on Knowledge Discovery from Data (TKDD), 4(2):9, 2010.
 Tukey (1949) Tukey, John W. Comparing individual means in the analysis of variance. Biometrics, pp. 99–114, 1949.
 Tumer & Ghosh (1996) Tumer, Kagan and Ghosh, Joydeep. Estimating the bayes error rate through classifier combining. In Pattern Recognition, 1996., Proceedings of the 13th International Conference on, volume 2, pp. 695–699. IEEE, 1996.
 Woodworth et al. (2017) Woodworth, Blake, Gunasekar, Suriya, Ohannessian, Mesrob I, and Srebro, Nathan. Learning nondiscriminatory predictors. Conference On Learning Theory, 2017.
 Zafar et al. (2017) Zafar, Muhammad Bilal, Valera, Isabel, Gomez Rodriguez, Manuel, and Gummadi, Krishna P. Fairness constraints: Mechanisms for fair classification. arXiv preprint arXiv:1507.05259, 2017.
 Zemel et al. (2013) Zemel, Richard S, Wu, Yu, Swersky, Kevin, Pitassi, Toniann, and Dwork, Cynthia. Learning fair representations. ICML (3), 28:325–333, 2013.
Appendix A Testing for significant discrimination
In general, neither nor can be computed exactly, as the expectations and , for are known only approximately through a set of samples drawn from the (possibly classconditional) population . The Monte Carlo estimate,
with , may be used to form an estimate
. By the central limit theorem, for sufficiently large
, and . As a result, the significance of can be tested with a twotailed ztest or using the test of Woodworth et al. (2017). If sample sizes are small and the target binary, more appropriate tests are available (Brown et al., 2001). In addition, we will often want to compare the discrimination levels of predictors , resulting from different learning algorithms, models, or sets of observed variables. The random variableis not Normal distributed, but is an absolute difference of foldednormal variables. However, for any
, is Normal distributed. Further, by enumerating the signs of and , we can show that . As a result, to reject the null hypothesis , we require that the observed values of both and are unlikely under at given significance.Appendix B Additional experimental details
b.1 Datasets

Adult Income Dataset (Lichman, 2013). The dataset has 32,561 instances. The target variable indicates whether or not income is larger than 50K dollars, and the sensitive feature is Gender. Each data object is described by 14 attributes which include 8 categorical and 6 numerical attributes. We quantize the categorical attributes into binary features and keep the continuous attributes, which results in 105 features for prediction. We note the label imbalance as 30% of male adults have income over 50K whereas only 10% of female adults have income over 50K. Additionally 24% of all adults have salary over 50K, and the dataset has 33% women and 67% men.

Goodreads reviews Gnanesh (2017), only included in the supplemental materials. The dataset was collected from Oct 12, 2017 to Oct 21, 2017 and has 13,244 reviews. The target variable is the rating of the review, and the sensitive feature is the gender of the author. Genders were gathered by querying Wikipedia and using pronoun inference, and the dataset is a subset of the original Goodreads dataset because it only includes reviews about the top 100 most popular authors. Each datum consists of the review text, vectorized using TfIdf. The review scores occurred with counts 578, 2606, 4544, 5516 for scores 1,3,4, and 5 respectively. Books by women authors and men authors had average scores of 4.088 and 4.092 respectively.

MIMICIII dataset (Johnson et al., 2016). The dataset includes 25,879 adult patients admitted to the intensive care unit of the Beth Israel Deaconess Medical Center in downtown Boston. Clinical notes from the first 48 hours are used to predict hospital mortality after 48 hours. Of all adult patients, 13.8% patients died in the hospital. We are interested in the difference in performance between the five selfreported ethnic groups and following data sizes and hospital mortality rates.
Race # patients % total Hospital Mortality Asian 583 2.3 14.2 Black 2,327 9.0 10.9 Hispanic 832 3.2 10.3 Other 3,761 14.5 18.4 White 18,377 71.0 13.4 Table 1: Summary statistics of clinical notes dataset
b.2 Synthetic experiments
To illustrate the effect of training set size and model choice, and the validity of the powerlaw learning curve assumption, we conduct a small synthetic experiment in which and with . The outcome is a quadratic function with heteroskedastic noise, , with . We fit decision tree, random forest and ridge regressors of the outcome to using default parameters in the implementation in scikitlearn (Pedregosa et al., 2011), but limiting the decision tree to depth . The size of the training set is varied exponentially between and samples, and at each size, trees are fit 200 times. In Figure 4, we show the resulting learning curves and as well as fits of Pow3 curves to them. Shown in dotted lines are extrapolations of learning curves from different sample sizes, illustrating the difficulty of estimating the intercepts and the Bayes error with high accuracy.
b.3 Book review ratings
Sentiment and rating prediction from text reveal quantitative insights from unstructured data; however deficiencies in algorithmic prediction may incorrectly represent populations. Using a dataset of 13,244 reviews collected from Goodreads (Gnanesh, 2017) with inferred author sex scraped from Wikipedia, we seek to predict the review rating based on the review text. We use as features the TfIdf statistics of the 5000 most frequent words. Our protected attribute is gender of the author of the book, and the target attribute is the rating (15) of the review. The data is heavily imbalanced, with 18% reviews about female authors versus 82% reviews about male authors.
We observe statistically significant levels of discrimination with respect to mean squared error (MSE
) with linear regression, decision trees and random forests. Using a random forest and training on 80% of the dataset and testing on 20%, we find that our
has 95%confidence interval with for reviews for male authors and for reviews for female authors using a difference in means statistical test. Results were found after hyperparameter turning for each training set size and taking an average over 50 trials. We observe similar patterns with linear regression and decision trees.To estimate the impact of additional training data, we evaluate the effect of varying training set size on predictive performance and discrimination. Through repeated sample spitting, we train a random forest on increasing training set sizes, reserving at least 20% of the dataset for testing. In Figure 4(a), additional training data lowers and , fitting an inverse powerlaw. Based on the intercept terms of the extrapolated powerlaws ( for reviews with male authors and for reviews with female authors), we may expect that can be explained more by differences in bias and variance than by noise since our estimated difference in noise .
In order to further measure the effect of collecting more samples, we analyze a onesized increase in training data. Because of the initial skew of author genders in the dataset, we vary the number of reviews for female authors, creating a shift in populations in the training data. We fix the training set size of reviews for male authors at , which represents the size of the full data for female authors , reserving 20% of the dataset as test data. We then vary the training data size for female authors such that the ratio varies evenly between 0.1 to 1.0. Using a linear regression in Figure 4(b), we see that as the ratio increases, decreases far below and far below our best reported MSE of the random forest on the full dataset. This suggests that shifting the data ratio and collecting more data for the underrepresented group can adapt our model to reduce discrimination.
b.4 Clinical notes
Here we include additional details about topic modeling. Topics were sampled using Markov Chain Monte Carlo after 2,500 iterations. We present the topics with highest and lowest variance in error rates among groups in Table
2. Error rates were computed using a logistic regression with L1 regularization over 10,000 TFIDF features using 80/20 training and testing data split over 50 trials. Based on the most representative words for each topic, we can infer topic descriptions, for example cancer patients for topic 48 and cardiac patients for topic 45.Topic  Top words  Asian  Black  Hispanic  Other  White 
31  no(t pain present normal edema tube history pulse absent left respiratory monitor  5.9  8.4  17.6  30.8  11.1 
17  hospital lymphoma continue s/p unit bmt thrombocytopenia line rash  34.3  13.6  34.9  30.2  26.0 
43  bowel abdominal abd abdomen surgery s/p small pain obstruction fluid ngt  16.6  11.8  5.7  26.8  13.2 
45  artery carotid aneurysm left identifier numeric vertebral internal clip  5.4  5.3  3.8  20.4  10.0 
48  mass cancer metastatic lung tumor patient cell left malignant breast hospital  21.6  25.4  12.3  30.2  18.5 
1  neo gtt pain resp neuro wean clear plan insulin good  3.3  1.8  1.6  3.6  2.7 
2  assessment insulin mg/dl plan pain meq/l mmhg chest cabg action  0.3  0.6  0.9  3.6  2.2 
0  chest reason tube clip left artery s/p pneumothorax cabg pulmonary  3.2  5.5  2.5  5.6  4.0 
25  c/o pain clear denies oriented sats plan alert stable monitor  7.3  3.9  5.9  8.2  6.5 
47  pacer pacemaker icd s/p paced rhythm ccu amiodarone cardiac  8.2  9.1  8.3  13.8  10.1 
We identified patients with notes corresponding to topic 48, corresponding to cancer, as a subpopulation with large differences in errors between groups. By varying the training size while saving 20% of the data for testing, we estimate that more data would not be beneficial for decreasing error (see Figure 5(c)). The mean over 50 trials is reported with hyperparameters chosen for each training size. Instead, we recommend collecting more features (e.g. structured data from lab results, more detailed patient history) as a way of improving error for this subpopulation.
Furthermore, we compute the 95% confidence intervals for false positive and false negative rates for a logistic regression with L1 regularization in Figure 5(a) and Figure 5(b).
Appendix C Exploring model choice
If a difference in bias is the dominating source of discrimination between groups, changing the class of models under consideration could have a large impact on discrimination.Consider for example Figure 1c in which the true outcome has higher complexity in regions where one protected group is more densely distributed than the other. Increasing model capacity in such cases, or exploring other model classes of similar capacity, may reduce as long as the biasvariance tradeoff is beneficial. Bias is not identifiable in general, as this requires estimation or bounding of noise components , or an assumption that they are equal, , or negligible, . However, as noise is independent of model choice, a difference in bias of different models is identifiable even if the noise is not known, provided that the variance is estimated. With , and , and , two predictors for comparison, we may test the hypothesis .
Appendix D Regression with homoskedastic noise
By definition of , we can state the following result.
Proposition 2.
Homoskedastic noise, i.e. , does not contribute to discrimination level under the squared loss .
Proof.
Under the squared loss, , as . ∎
In contrast, for the zeroone loss and classspecific variants, the expected noise terms do not cancel, as they depend on the factor .
Appendix E Biasvariance decomposition. Proof of Theorem 1.
Lemma A1 (Squared loss and zeroone loss).
The following claim holds for both:
a) the zeroone loss with and ,
b) a) the squared loss with .
Proof.
See Domingos (2000).∎
Lemma A2 (Classspecific zeroone loss).
With the zeroone loss, it holds with and
Proof.
We begin by showing that with .
As the above should be zero for all options, this implies that .
We now show that,
We have that if ,
A similar calculation for the case where yields the claim.
Finally, We have that
which gives us our result. ∎
Appendix F Difference between power law curves
Let and . Then has at most 2 local minima. We see this by rewriting
and so
Comments
There are no comments yet.