RISE: Robust Individualized Decision Learning with Sensitive Variables
Anonymous Author(s)
AffiliationAddress email
Abstract
This paper introduces RISE, a robust individualized decision learning frameworkwith sensitive variables, where sensitive variables are collectible data and importantto the intervention decision, but their inclusion in decision making is prohibiteddue to reasons such as delayed availability or fairness concerns.
The convention isto ignore these sensitive variables in learning decision rules, leading to significantuncertainty and bias.
To address this, we propose a decision learning frameworkto incorporate sensitive variables during offline training but do not include themin the input of the learned decision rule during deployment.
Specifically, froma causal perspective, the proposed framework intends to improve the worst-caseoutcomes of individuals caused by sensitive variables that are unavailable at thetime of decision.
Unlike most existing literature that uses mean-optimal objectives,the proposed learning framework robustifies sensitive variables via finding a newlydefined quantile- or infimum-optimal decision rule for improving the worst-offgroup among all sensitive variable realizations.
The reliable performance of theproposed method is demonstrated through synthetic experiments and three real-dataapplications.
Introduction
Recently, there has been a widespread interest in developing methodology for individualized decisionrules (IDRs) based on observational data.
When deriving IDRs, some collectible data are importantto the intervention decision, while their inclusion in decision making is prohibited due to reasonssuch as delayed availability or fairness concerns.
For example, sensitive characteristics of subjectsregarding their income, sex, race and ethnicity may not be appropriate to be used directly fordecision making due to fairness concerns.
In the medical field especially for patients in severelife-threatening conditions such as sepsis, timely bedside intervention decisions have to be madebefore lab measurements are ordered, assayed and returned to the attending physicians.
However, dueto the delayed availability of lab results, most of the decisions are made with great uncertainty andbias due to partial information at hand.
We define sensitive variables as variables whose inclusion intodecision rules is prohibited.
The formal definition of sensitive variables will be given in Section 3.
In this work we propose RISE ( R obust I ndividualized decision learning with SE nsitive variables), arobust IDR framework to improve the outcome of individuals when there are informative yet sensitivevariables that are either not available or prohibited from using during IDR deployment.
The mainquestion of interest is whether the learned IDR could yield similar outcomes across all realizationsof the sensitive variables.
To achieve this, we propose to estimate the optimal IDR by optimizinga quantile- or infimum-based objective, respectively, for continuous or discrete sensitive variables.
Our idea falls along the lines of work that considers algorithmic fairness [1] while extending it tothe setting of causal inference [2] in the sense that decisions are driven by causality rather than ageneral utility function.
We show in our empirical analyses that this leads to fairer and safer real-life decisions with little sacrifice of the overall performance.
This optimization problem is then shownto be equivalent to a weighted classification problem where most existing statistical and machinelearning classifiers can be readily applied .
Assuming that a large outcome value is preferable, optimal IDRs are traditionally derived throughmaximizing the mean outcome of the sample population.
In this paper, we are interested in a specificyet broadly applicable setting of learning that involves sensitive variables.
We consider offlinelearning where sensitive variables are collected and can be used in training the IDRs , but they cannotbe used as input in the resulting IDRs .
This is a setting commonly considered in the fairness andprivacy-related literature for classification (e.g., [3]), but not from a causal standpoint.
When thereexist important variables that are simply left out from training, the estimated IDR will be biased.
This bias can be removed if all important variables are used during training, which we will show in
Section 3.1 a mean-optimal approach.
The optimal action maximizes the mean outcome where themean is taken over the sensitive variables, conditioning on other variables.
This method, however,has no control of the disparity in sensitive variables.
Subjects with different sensitive values mayreport large outcome differences, hence unfairly or unsafely treated.
Therefore, objective functionswith robustness guarantees for sensitive variables are preferred, since they offer protection to subjectsin the lower tail of the outcome distribution with regards to the sensitive values.
For illustration, we consider a toy example with binary actions, A ∈ {− 1 , 1 } .
We note that thedecision can only be made based on the variable X whereas S is a sensitive variable.
The setup isshown in Table 1 and partial simulation results are shown in Table 2.
The details can be found in Section 4.1 under Example 1.
We consider vulnerable subjects as those with low outcome values,as highlighted in red in Table 1 (A full definition will be given in Section 3.2).
For X ≤ 0 .
5 , thetraditional mean-optimal rule assigns action A = 1 as it achieves the largest average reward across
S = 0 and S = 1 .
However, this action results in great harm for subjects with S = 1 as they couldget the worst expected outcome of 0.
On the contrary, the proposed RISE improves the worst-caseoutcome by assigning A = − 1 , protecting the vulnerable subjects.
Likewise, for X > 0 . 5 , the meanoptimal rule assigns A = − 1 while the proposed rule assigns A = 1 protecting those with S = {+
 thatcould+} have experienced an outcome of 5 .
Compared to the mean-optimal rule, the proposed RISEachieves a much larger reward among vulnerable subjects while maintaining a comparable overallexpected reward.
The worst-case outcomes of the rule by the proposed RISE are colored in blue.
[Table]
[Equation]
[Table]
[Equation]
[Table] [Equation] [Table] [Table]
Main contributions.
Methodology-wise, 1) we propose a novel framework, RISE, to handlesensitive variables in causality-driven decision making.
Robustness is introduced to improve theworst-case outcome caused by sensitive variables, and as a result, it reduces the outcome variationacross subjects.
The latter is directly associated with fairness and safety in decision making.
Tothe best of our knowledge, we are among the first to propose a robust-type fairness criterion undercausal inference.
We introduce a classification-based optimization framework that can easilyleverage most existing classification tools catered to different functional classes, including state-ofthe-art random forest, boosting, or neural network models.
Application-wise, 3) the consideration ofsensitive variables in decision learning is important to applications in policy, education, healthcare,etc.
Specifically, we illustrate the application of RISE using three real-world examples from fairnessand safety perspectives where robust decision rules are needed, across which we have observedrobust performance of the proposed approach.
From a fairness perspective, we consider a job trainingprogram where age is considered as a sensitive variable.
From a safety perspective, we consider twoapplications to healthcare where lab measurements are considered as sensitive variables.
Related Work
Our work focuses on individualized decision rules, which aim at assigning treatment decision basedon subject characteristics.
See [4, 5, 6, 7] and references therein for a comprehensive review. In
Appendix A.1, we provide additional literature review on general IDRs under causal settings.
Beyondthe field of causal inference, fairness and safety, and robustness are two areas of research that extendwell beyond the learning of IDRs.
In the following, we provide a review on both, with focus given towork related to causal inference and IDRs.
Fairness and safety in IDRs.
The consideration of fairness and safety in machine learning has seenan explosion of interest in the past few years. We refer to [1, 8, 9, 10, 11, 12, 13, 14] and referencestherein for a review of this topic in classification and regression problems.
In these work, sensitivevariables are also referred to as sensitive or protected attributes.
We extend the definition of sensitivevariables to include delayed information that is not available at deployment as it is also suitable forthis framework.
Among earlier work, preprocessing approaches [3, 15, 16, 17] and inprocess training approaches[18, 11, 19] consider disentangling the input X from a known or unknown sensitive variable S sothat the transformed X does not contain any information that can be used to trace back to S .
Dueto the causal nature of IDRs, effect of IDRs cannot be estimated consistently when an informative S is left out and the resulting rule is sub-optimal.
This follows from the classic argument that anyunmeasured confounding (i.e., S ), if not accounted for, would lead to bias.
Similar issues persist incontextual bandits [20, 21].
Inside the causal framework, [22, 23] extend fairness from prediction topolicy learning using causal graphical models by incorporating fairness constraints.
[24] considerscounterfactual fairness that seeks to achieve conditional independence of the decisions via datapreprocessing.
Despite earlier efforts in bringing fairness into the causal framework, most of theseapproaches only ensure mean zero disparity in S but do not have robustness guarantees in the sensethat the variance of the disparity in S is not controlled.
Besides, most examples consider a singlecategorical sensitive variable, but not multiple or continuous ones.
Robustness in IDRs.
Recently the statistical literature has witnessed a growing interest in developingrobust methods for estimating IDRs.
They introduce robustness into the objective function by usingquantile-optimal treatment regimes or mean-optimal treatment regimes under certain constraintsto improve the gain of individuals at the lower tail of the reward spectrum [25, 26, 27, 28, 29].
Robustness, in their sense, pertains to the outcome distribution subject to sampling error.
Whensensitive variables are present, we consider instead the robustness of the outcome distribution subjectto the uncertainty due to sensitive variables, providing a more targeted way of ensuring robustness,which is directly related to fairness and safety.
Compared to algorithms based on explicit fairnessconstraints (for example [30, 31] in classification and [22, 24] in causal inference) that seek to removethe disparity across different values of S , our method reduces the variance of disparity across S .
Inaddition, constraint-based approaches typically require specialized optimization procedures whereasour approach presents an elegant and systematic way for optimization.
To our knowledge, we are thefirst few to consider decision fairness via a robust objective under the causal framework.
Robust Decision Learning Framework with Sensitive Variables
3.1 Preliminaries
Notation.
We let random variables be represented by upper-case letters, and their realizations berepresented by lower-case letters.
Suppose there are n independent subjects sampled from a givenpopulation.
For subject i , let A i ∈ {− 1 , 1 } denote a binary treatment assignment and Y i denotethe corresponding outcome.
Without loss of generality, we assume a larger value of outcome isdesirable.
Under the potential outcomes framework [32, 33], let Y i ( − 1) be the potential outcomehad the subject been assigned to control and Y i (1) be the potential outcome had the subject beenassigned to treatment.
Let X i ∈
X be the feature vector and, for now, S i be a single sensitive variable.
Extension to multiple sensitive variables is presented in Section 3.4.
We consider S ∈ S where 130 S
= { 1 , . .
, K } if S is discrete and S
= R if S is continuous.
Definition of sensitive variables.
We define sensitive variables that are important to the interventiondecision, but their inclusion in decision making is prohibited.
Formally, consider variables X and
S that are both available during model training and are both determinants of conditional averagetreatment effect [34].
While X and S may be both involved in training, the derived decision rule d ( · )precludes the input of S due to sensitive concerns.
Hence, the derived IDR is only a function withthe form d ( X ) :
X → A .
Following the above definition, we consider an offline learning framework where sensitive variables are collected and can be used in obtaining the IDRs, but they cannot be usedin the resulting IDRs.
Assumption 1.
Assume the following conditions hold:
(1a) Consistency: Y
=
Y ( − 1) 1 ( A = − 1)
+ Y (1) 1 ( A = 1) .
(1b) Positivity: 0 <
Pr ( A = 1 | X, S ) < 1 .
(1c) Unconfoundedness: { Y ( − 1) , Y (1) }
⊥ A |{ X, S } and { Y ( − 1) , Y (1) } ̸⊥ A | X .
Assumption 1a is the standard consistency assumption in causal inference and Assumption 1b statesthat every subject has a nonzero probability of getting the treatment.
Assumption 1c states thatgiven X and S , the potential outcomes are independent of the treatment assignments.
Besides, theunconfoundedness does not hold when only X is conditional, signifying the importance of S .
Undercausal settings, Assumption 1c ensures that treatment effects cannot be non-parametrically identifiablewithout S (See [35, 34, 36, 37, 38] and references therein).
Approaches such as disentanglementof X from S under supervised learning settings mentioned in Section 2 will introduce bias towardsestimating the IDR.
Naive approaches that omit sensitive variables.
When S is not available for future deployment, anaive approach is to maximize E X { E ( Y | X, A = d ( X )) } over d using ( X, A, Y ) during the trainingprocedure.
This approach will introduce bias in the estimation of potential outcomes and lead to asuboptimal IDR due to the unmeasured confounder S .
Mean-optimal approaches that use the sensitive variables.
It is thus important that one includes Sinto the training procedure.
For example, if we consider the value function framework (i.e., expectedoutcome) used by most existing works such as [39, 40], we can show that
[Equation]
where the third equality in (1) holds by Assumption 1 and the last inequality also indicates thenaive approaches without using S will in general fail.
Then one valid approach is to maximize
E X (cid:2) E S
| X { E ( Y | X, S, A = d ( X )) } (cid:3) over d using ( X, S, A, Y ) .
The optimal IDR under this criterion is, for every X ∈ X ,
[Equation]
which guarantees to find the treatment that maximizes the conditional expected outcome given each
X by averaging out the effect of the sensitive variable S .
The mean-optimal approaches, however,fail to control the disparities across realizations of the sensitive variables due to the integration over S , which may lead to unsatisfactory decisions to certain subgroups, as illustrated in the toy examplein Section 1.
3.2 Robust Optimality with Sensitive Variables
Driven by the limitation of existing approaches, our goal is to derive a robust decision rule thatmaximizes the worst-case scenarios of subjects when some sensitive information is not available atthe time of deploying the decision rule.
Specifically, our robust decision learning framework drawsdecisions based on individuals’ available characteristics summarized in the vector X without thesensitive variable S , while improving the worst-case outcome of subjects in terms of the sensitivevariable in the population.
Formally, given a collection D of all treatment decision rules dependingonly on X , the proposed RISE approach estimates the following IDR, which is defined as
[Equation]
where G S | X ( · ) could be chosen as some risk measure for evaluating E ( Y | X, S, A = d ( X )) for each
S ∈ S .
Examples include variance, conditional value at risk, quantiles, etc.
In this paper, we consider
G S | X as the conditional quantiles (for a continuous S ) or the infimum (for a discrete S ) over S .
Specifically, for a discrete S , G S | X is consider as an infimum operator of E ( Y | X, S, A = d ( X )) over S .
We thus aim to find
[Equation]
where inf is the infimum taken with respect to E ( Y | X, s, A = d ( X )) over s ∈ S .
This implies that for a given X , d ∗ ( X ) assigns the treatment that yields the best worst-case scenario among all possible values of S for every X ∈ X , or equivalently,
[Equation]
For a continuous S , we consider G S | X { E ( Y | X, S, A = d ( X )) } as Q τS | X { E ( Y | X, S, A = d ( X )) } , which is the τ -th quantile of { E ( Y | X, S, A = d ( X )) } and τ ∈ (0 , 1) is the quantile level of interest.
Specifically, Q τS | X { E ( Y | X, S, A
= d ( X )) }
= inf { t : F ( t ) ≥ τ } with F denoting the conditional distribution function of E ( Y | X, S, A = d ( X )) given X and d .
Note the randomness behind E ( Y | X, S, A = d ( X )) given X and d is fully determined by the sensitive variable S .
Then optimal IDR under this criterion is defined as
[Equation]
This implies that for a given X , d ∗ ( X ) assigns a treatment that yields the largest τ -th quantile of the outcome over the distribution related to S , or equivalently,
[Equation]
We let τ = 0 .
25 throughout the paper and suppress τ for simplicity.
Results on varying the value ofτ is provided in Appendix; see Section 4.1 for details.
Identifying vulnerable subjects.
Our RISE framework provides a natural way to define vulnerablegroups .
Specifically, for a discrete S , if inf S { E ( Y | X, S, A = 1) } > inf S { E ( Y | X, S, A = 0) } , thenarg inf S { E ( Y | X, S, A = 0) } is vulnerable given X , otherwise is arg inf S { E ( Y | X, S, A = 1) } .
In other words, the vulnerable subjects are those in the worst-off group that needs protection.
Similarly, for a continuous S , if Q S { E ( Y | X, S, A = 1) } > Q S { E ( Y | X, S, A = 0) } , then theset { S : E ( Y | X, S, A = 0) ≤ Q S { E ( Y | X, S, A = 0) }} defines the vulnerable subjects given X ,otherwise this group is defined as { S : E
( Y | X, S, A = 1) ≤ Q S { E ( Y | X, S, A = 1) }} .
3.3 Estimation and Algorithm
Here we provide a transformation of the proposed RISE from an optimization problem to a weightedclassification problem.
There are several advantages to this conversion: 1) The optimization problemdefined in (2) involves a nonsmooth and nonconvex objective function that could lead to computationalchallenges.
2) With multiple powerful statistical and machine learning toolbox to choose from, aclassification problem can be more readily solved in practice.
Hyperparameter tuning and modelselection could be conducted to further boost performance.
Compared to a direct optimizationof (2), a classification-based optimizer allows the use of off-the-shelf software packages that can betailored to different functional classes or incorporate different properties such as model sparsity.
Proposition 1.
Maximizing the objective function in (2) is equivalent to maximizing
[Equation]
1) } ] (cid:9) .
With Proposition 1 and a proper estimator of the outcome model E ( Y | X, S, A ) using our trainingdata D n = { X i , S i , A
i , Y i } ni
=1 , we replace the expectation of Y i by its estimate ˆ Y i and solve thefollowing problem.
[Equation]
where g 1 ( x i ) = G s | x { ˆ Y i ( x i , s, a i = 1) } and g 2 ( x i ) =
G s | x { ˆ Y i ( x i , s, a i = − 1) } .
Note thatg 1 ( x i ) − g 2 ( x i ) may not be positive, which makes Problem (3) difficult to solve. We have thefollowing key proposition 2 to address this issue and transform it into a classification problem.
Proposition 2.
Maximizing the empirical objective in (3) is equivalent to a weighted classificationproblem of minimizing
[Equation]
with features x i , the true label sgn { g 1 ( x i ) − g 2 ( x i ) } , and the sample weight | g 1 ( x i ) − g 2 ( x i ) | , forsubject i , i = 1 , . . . , n .
With Proposition 2, we have transformed the optimization problem (2) into a weighted classificationproblem (4) where for subject i with features x i , the true label is sgn { g 1 ( x i ) − g 2 ( x i ) } and thesample weight is | g 1 ( x i ) − g 2 ( x i ) | .
The estimated optimal decision rule by (4) is then given byˆ d ( x ) =
sgn { ˆ f ( x ) } .
The proof of Proposition 1 and Proposition 2 is presented in Appendix [-A.2.
-]A.2.
Algorithm 1 provides an algorithmic overview.
The inner expectation E ( Y | X, S, A ) can be modeledas ˆ Y ( X, S, A ) using a twin model separated by the treatment and control groups.
For a continuous S ,we propose to estimate G ( X, A ) =
Q S | X,A { E ( Y | X, S, A ) } via a quantile regression of ˆ Y on X butwithout S .
For a discrete S , we propose to obtain an estimate of G ( X, A ) =
inf S { E ( Y | X, S, A ) } byfinding the minimum among { E ( Y | X, S = 1 , A ) , . .
, E ( Y | X, S = K, A ) } .
The estimated decisionrule can then be obtained from the weighted classification.
In our implementation, neural networksare used to fit models in the training data sets.
The details on modeling and hyperparameter tuningvia cross-validations are given in Appendix A.3.
A Python package rise based on neural networksis built.
Note that the model choices are flexible.
[Table]
3.4 Extension to multiple discrete sensitive variables
For multiple discrete sensitive variables, similar estimation procedure can be conducted as outlinedin Section 3.3.
Suppose there are L discrete sensitive variables,
i.e., S =
{ S 1 , S 2 , . . .
, S L } .
Theinner expectation E ( Y | X, S 1 , . . .
, S L , A ) can be obtained with a twin model of Y on X and all Sfor each treatment level.
The infimum over S is obtained by finding the minimum iterating spaceof possible parameter values for each sensitive variable.
See Section 4.2 for an example of usingmultiple discrete sensitive variables.
We will discuss in Section 5 the challenges and future workrelated to scenarios with multiple continuous sensitive variables or that with a mixture of continuousand discrete sensitive variables.
Numerical Studies
In this section, we perform extensive numerical experiments to investigate the merit of robustness ofthe proposed framework via simulations and three real-data applications.
The results demonstratethat the proposed rules achieve a robust objective with sensitive variables unavailable at the time ofdecision while maintaining comparable mean outcomes.
Compared approaches.
For comparison, we consider the naive and mean-optimal approachesdescribed in Section 3.1, which correspond to different choices of G ( · ) functions.
The naive decisionrule that simply disregard information of S , denoted as Base , can be formulated in our optimizationframework of (2) by letting G ( X, A ) =
E ( Y | X, A ) .
The IDR can be estimated directly by fittinga model of Y on X in each treatment arm.
The resulting IDR is not sensitive variables-aware andis biased due to confounding, as discussed.
Another IDR that resembles traditional mean-optimaldecision rules, denoted as Exp , can be formulated as G ( X, S, A ) =
E ( Y | X, S, A ) .
This can beobtained by training a classification model without S , i.e., only using X , after obtaining an outcomemodel for the inner expectation E ( Y | X, S, A ) .
Note that this approach is not robust to extreme behaviors in S .
The modeling approaches described in Appendix A.3 applies to here.
We also remarkthat there is limited work in the IDR-related literature that can be immediately compared with ours.
Evaluation metrics.
Objective: the quantile objective is estimated and reported for a continuous
S and the infimum objective is for a discrete S .
The objective, when τ < 0 .
5 , (here τ = 0 .
25 )represents the value of the “low performers” among all possible value of S under a given d . 2)
Value: the value function, or expected reward used by the most existing methods, such as [39, 40],is defined as V ( d ) =
E { Y ( d ) } .
It represents the “average performers”.
For randomized trials, anunbiased estimator of V ( d ) is given by ˆ V ( d ) = { (cid:80)
Ti =1 Y
i 1( A i = ˆ d ) /π
( A i , X i ) } / { (cid:80)
Ti =1 1( A i =ˆ d ) /π
( A i , X i ) } [41], where T is the sample size of the test data.
For observational studies, the valueis estimated with ˆ V ( d ) =
T − 1 (cid:80) Ti
=1 ˆ Y
i
( x i , s i , a i = ˆ d ) .
We report the metrics among all subjectsand among the potential vulnerable subgroup, respectively.
For simulation, we consider trainingdata and the testing data, respectively, with sample sizes of 5,000.
For real-data applications, weconsider a 80-20 split of the dataset into a training data and a testing data.
Continuous covariatesare standardized before the estimation.
All results are based on 100 replications.
Experiments areperformed on a 6-core Intel Xeon CPU E5-2620 v3 2.40GHz equipped with 64GB RAM.
Pythoncode is provided as part of Supplementary Materials.
4.1 Simulation Studies
Example 1.
Here we provide the detail for the simulation of the motivating example introduced in
Section 1.
The outcome is generated using the following model:
Y i = 1 ( X i > 0 . 5) { 5 + 10 [-
-]1 ( A i =1) + 22 S
i − 24
( A i = 1) S i } + 1 ( X i ≤ 0 . 5) { 11 + 19 1 ( A i = 1) + 2 S
i − 32
( A i = 1) S i } + ϵ i ,where the covariate X i ∼ Unif [0 , 1] , treatment assignment A
i ∼ Bernoulli (0 . 5) , and the noiseϵ i ∼ N (0 , 1) .
For a discrete type S , the sensitive variable S i ∼ Bernoulli (0 . 5) .
For a continuoustype S , S i is generated from a mixture of beta distributions, Beta (4 , 1) and Beta (1 , 4) , with equalmixing proportions.
Example 2.
We generate the outcome Y using the following model: Y i = { 0 .
5 + 1 ( A i = 1)+exp( S i ) − 2 .
5 S
i 1 ( A i = 1) }{ 1+ X i 1 − X i 2 + X 2 i 3 +exp( X i 4 ) } + { 1+2 1 ( A i = 1)+0 .
2 exp( S i ) −3 .
5 S i 1 ( A i = 1) }{ 1+5 X
i 1 − 2 X i 2 +3 X i 3 +2 exp( X i 4 ) } + ϵ , where X ij ∼ U (0 , 1) , j = 1 , . . .
A satisfies log { P ( A i = 1 | X i ) /P
( A = 0 | X i ) }
= − 0 .
6( S i + X
i 1 − X i
+ X
i 3 − X i
+ X
i 5 − X i 6 ) ,and ϵ i ∼ N (0 , 1) .
For a continuous type S , S i is generated from a mixture of beta distributions,
Beta (4 , 1) and Beta (1 , 4) , with equal mixing proportions; for a discrete type S , we consider a binary
S i that satisfies log { P ( S i =
| X i ) /P
( S i =
| X i ) }
= − 2 .
5 + 0 .
8( X i
+ X
i
+ X
i
+ X
i
+ X i 5 + X i 6 ) .
Table 3 summarizes the performance of the proposed IDRs compared to the mean criterion for
Example 1 and Example 2.
The proposed RISE achieves the largest objectives and improves thevalue among vulnerable subjects, while maintaining comparative overall values.
As demonstrated inthe toy example introduced in Section 1, we expect that RISE helps improve the value among thevulnerable subjects while maintaining a comparable overall value.
As for the objective, intuitively,the proposed rule is expected to have a larger objective.
We also point out that there is no directrelationship between the objective among all subjects versus the objective among vulnerable subjects.
For example, using the toy example with setup in Table 1, and limiting to subjects with X ≤ 0 .
5 only, S = 1 is vulnerable and is assigned A = − 1 by the proposed RISE.
The objective among S = 1 is13 but the objective among both S
= 0 and S = 1 is 12 =
(11 + 13) / 2 , which is smaller than thatamong the vulnerable group.
In other words, by protecting the vulnerable subjects, the proposed rulemay lead to an increase in the outcome of the vulnerable group, and the gain may result in a higheroutcome than the overall mean outcome.
In the appendix, we consider for a continuous S different quantile criteria τ = 0 .
1 and 0 . 5 , respectively, to test the robustness of the proposed RISE.
Results show that when τ is small, there is morestrength in the proposed method, as the algorithm aims to improve the worst-outcome scenarios.
The proposed RISE has the largest gain in objective and value among vulnerable subjects when τis 0.1, and has similar performance as the compared approaches when τ is 0.5.
Also, we considera scenario where S is not involved in the data generation of Y , i.e., Assumption 1c is simplified as{ Y ( − 1) , Y (1) }
⊥ A | X .
The estimated objective and value function are similar across all comparedapproaches, which indicates the robustness of RISE.
The details can be found in Appendix A.4.
[Table]
4.2 Real-data Applications
We present three real-data examples to showcase the robust performance of RISE.
These applicationsconsider either fairness or safety in the context of policy [42] and healthcare [43, 44] where sensitivevariables exist.
The information of data access is provided in Appendix A.5.
Fairness in a job training program.
To illustrate the implication of the proposed method from afairness perspective, we consider the National Supported Work (NSW) program [42] for improvingpersonalized recommendations of a job training program on increasing incomes.
This programintended to provide a 6 to 18-month training for individuals in face of economic and social problemssuch as former drug addicts and juvenile delinquents.
The original experimental dataset consistsof 185 individuals who received the job training program ( A = 1 ) and 260 individuals who did not( A = − 1 ).
The baseline covariates are age, years of schooling, race (1 = African Americans or
Hispanics, 0 = others), married (1 = yes, 0 = no), high school diploma (1 = yes, 0 = no), earning in1974, and earning in 1975.
The outcome variable is the earning in 1978.
In the exploratory analysisusing causal forest [45], we observe that age may play an important role in the causal effect of thejob training program on the long-term post-market earning.
In the following data example we use ageas the sensitive variable S and other baseline covariates as X .
The earnings in years 1974, 1975, and1978 are transformed by taking the logarithm of the earning plus one.
Improvement of HIV treatment.
To illustrate the implication of the proposed method from asafety perspective when there is delayed information, we consider the ACTG175 dataset among HIVpositive patients [43].
The original study considers a total of 2,139 patients who were randomlyassigned into four treatment groups.
In this data application, we focus on finding the optimal IDRsbetween two treatments: zidovudine combined with didanosine ( A = − 1 ) and zidovudine combinedwith zalcitabine ( A = 1 ).
The total number of patients receiving these two treatments is 1,046.
Thebaseline covariates we consider are age, weight, CD4 T-cell amount at baseline, hemophilia (1 = yes,
= no), homosexual activity (1 = yes, 0 = no), Karnofsky score, history of intravenous drug use (1 =yes, 0 = no), gender (1 = male, 0 = female), CD8 T-cell amount at baseline, race (1 = non-Caucasian,0 = Caucasian), number of days of previously received antiretroviral therapy, use of zidovudine in the30 days prior to treatment initiation (1 = yes, 0 = no), and symptomatic indicator (1 = symptomatic, 0= asymptomatic).
The outcome variable is the CD4 T-cell amount at 96 ± 5 weeks from the baseline.
We consider CD8 T-cell amount at baseline as the sensitive variable.
The response of CD8 T-cellamong HIV positive patients has not been fully understood [46].
Clinically, it is plausible that only
CD4 is measured in clinical visits where treatments are based on, hence CD8 might not be measuredand not used in decision making.
As our exploratory analysis using causal forest shows, CD8 T-cellamount may play an important part in the treatment effect of the outcome.
Safe resuscitation for patients with sepsis.
For this application, we apply the proposed method totreating sepsis, a life-threatening disease.
This application intends to provide an example to applyour method with multiple categorical sensitive variables in the scenario where there is missing yetimportant information at the time of decision making.
We apply the proposed method to a sepsis studyfrom the University of Pittsburgh Medical Center (UPMC).
The original study cohort includes 30,687 patients with Sepsis-3 [44] within 6 hours of hospital arrival from 14 UPMC hospitals between 2013and 2017.
For our data analysis, we consider X to be baseline patient characteristics 4 hours beforesepsis onset, which includes patient demographics of age, gender (1 = male, 0 = female), race (1 =
Caucasian, 0 = others), and weight, and vital signs of usage of mechanical ventilation (1 = yes, 0 =no), respiratory rate, temperature, intravenous fluids (1 = yes, 0 = no), Glasgow Coma Scale score,platelets, blood urea nitrogen, white blood cell counts, glucose, creatinine.
We consider two sensitivevariables, lactate and Sequential Organ Failure Assessment (SOFA) score 4 hours before sepsis onset.
Note that their measurements are obtained retrospectively after treatment decisions have been madeand are not available at times of decision.
According to the new definition of Sepsis-3 [47], a serumlactate level >2 mmol/L is considered to be in critical conditions and is highly likely to indicate aseptic shock. Also, a SOFA score greater than 6 has been associated with a higher mortality [48, 49].
The treatment option is whether the patient took any vasopressors during the first 24 hours after sepsisonset.
The outcome is patient survival at day 90.
The analysis cohort contains 6,539 patients in total.
We are interested in making decision about whether to treat patients with vasopressors in the first 24hours after sepsis onset given the measurements of lactate and SOFA are not available at the time ofdecision making.
Additional rational and background on this example is provided in Appendix A.5.
Results.
Table 4 presents the performance of various IDRs on the three applications.
As expected,
RISE has the largest objective as well as value among vulnerable subjects.
The patterns are similar tothat in the synthetic experiments in Section 4.1.
In applications to the job training data and the sepsisstudy, results show that RISE has a larger value among all subjects than other IDRs.
This is possiblewhen there are more gains in the vulnerable subjects than other subjects, which further demonstratethe superiority of the proposed approach in improving worst-case outcomes caused by sensitivevariables.
We provide visualizations by Shapley additive explanations (SHAP) [50] for RISE and Exp, respectively, in Appendix A.5 about feature importance in the final classification models tohelp interpret important covariates in making the decisions.
The SHAP approach provides unitedvalues to describe the correlation between each feature and the predicted decision rule, respectively[50].
Overall, the direction of correlations is similar for RISE and Exp, but their ranking of featureimportance may be different.
[Table]
Discussion
[Table] We have proposed RISE, a robust decision learning framework with a novel quantile-, or infimumoptimal treatment objective intended to improve the worst-case scenarios of individuals when decisions with uncertainty are needed to be made with sensitive yet important information missing.
Ourapproach can be applied to a board of applications, including but not limited to politics, education,healthcare, etc.
For multiple continuous sensitive variables, the estimated decision rule by RISE canbe easily obtained by fitting similar quantile regression as described in Section 3.3.
For a mixture ofcontinuous and discrete sensitive variables, the estimated rule can be obtained by taking infimumover the discrete ones, then obtaining quantile over the continuous ones.
However, challenges remainin find vulnerable subjects described in Section 3.2 under these settings as it may be computationallydifficult to find a vulnerable set of S when they are multi-dimensional.
Another future work includesextension of the current binary treatment option to a multi-treatment option.
Checklist
The checklist follows the references.
Please read the checklist guidelines carefully for information onhow to answer these questions.
For each question, change the default [TODO] to [Yes] [No] or[N/A] .
You are strongly encouraged to include a justification to your answer , either by referencingthe appropriate section of your paper or providing a brief inline description.
For example:
• Did you include the license to the code and datasets?
[Yes] See Section 2.
• Did you include the license to the code and datasets?
[No] The code and the data are
proprietary.
• Did you include the license to the code and datasets?
[N/A]
Please do not modify the questions and only use the provided macros for your answers.
Note that the
Checklist section does not count towards the page limit.
In your paper, please delete this instructionsblock and only keep the Checklist section heading above along with the questions/answers below.
For all authors...
(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope?
[Yes] (b) Did you describe the limitations of your work?
[Yes] See Section 3.4 and Section 5.
(c) Did you discuss any potential negative societal impacts of your work?
[N/A]
(d) Have you read the ethics review guidelines and ensured that your paper conforms to them?
[Yes]
If you are including theoretical results...
(a) Did you state the full set of assumptions of all theoretical results?
[Yes] See Section 3.1.
(b) Did you include complete proofs of all theoretical results?
[Yes] See Section A.2.
If you ran experiments...
(a) Did you include the code, data, and instructions needed to reproduce the main experi-
mental results (either in the supplemental material or as a URL)?
[Yes] See Section 4.
Code is provided as part of Supplementary Materials.
(b) Did you specify all the training details (e g., data splits, hyperparameters,
how they
were chosen)?
[Yes] See Section 4 and Section A.3.
(c) Did you report error bars (e g., with respect to the random seed after running experi- ments multiple times)?
[Yes] See Section 4 and Section A.4.
(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)?
[Yes] See Section 4.
Experiments are performed on a 6-core Intel Xeon CPU E5-2620 v3 2.40GHz equipped with 64GB RAM
If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
(a) If your work uses existing assets, did you cite the creators?
[Yes] See Section 4.2 andAppendix A.5.
(b) Did you mention the license of the assets?
[Yes] The first two datasets are publiclyavailable data, and the third dataset is proprietary.
We have cited the published papersthat originated the three data applications.
(c) Did you include any new assets either in the supplemental material or as a URL?
[Yes]Code is provided as part of Supplementary Materials.
(d) Did you discuss whether and how consent was obtained from people whose data you’re
using/curating?
[Yes] See Appendix A.5.
(e) Did you discuss whether the data you are using/curating contains personally identifiable
information or offensive content?
[Yes] See Appendix A.5.
The first two real-data applications are publicly available data, and the third application uses deidentified data
that all personally identifiable information has been removed.
If you used crowdsourcing or conducted research with human subjects...
(a) Did you include the full text of instructions given to participants and screenshots, ifapplicable?
[N/A]
(b) Did you describe any potential participant risks, with links to Institutional Review
Board (IRB) approvals, if applicable?
[N/A]
(c) Did you include the estimated hourly wage paid to participants and the total amountspent on participant compensation?
[N/A]
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