SlowGuess/NeurIPS_2023_OCR huggingface dataset:
zhoua@usc.edu
In consequential domains, it is often impossible to compel individuals to take treatment, so that optimal treatment assignments are merely suggestions when humans make the final treatment decisions. On the other hand, there can be different heterogeneity in both the actual response to treatment and final treatment decisions given recommendations. For example, in social services, a persistent puzzle is the gap in take-up of beneficial services among those who may benefit from them the most. When decision-makers have equity- for fairness-minded preferences over both access and average outcomes, the optimal decision rule changes due to these differing heterogeneity patterns. We study identification and improved/robust estimation under potential violations of positivity. We consider fairness constraints such as demographic parity in treatment take-up, and other constraints, via constrained optimization. We develop a two-stage, online learning-based algorithm for solving over parametrized policy classes under general constraints to obtain variance-sensitive regret bounds. Our framework can be extended to handle algorithmic recommendations under an often-reasonable covariate-conditional exclusion restriction, using our robustness checks for lack of positivity in the recommendation.
The intersection of causal inference and machine learning for heterogeneous treatment effect estimation can improve public health, increase revenue, and improve outcomes by personalizing treatment decisions, such as medications, e-commerce platform interactions, and social interventions, to those who benefit from it the most [4, 30, 35, 49]. But, in many important settings, we do not have direct control over treatment, and can only optimize over encouragements, or recommendations for treatment. For example, in e-commerce, companies can rarely compel users to sign up for certain services, rather nudge or encourage users to sign up via promotions and offers. When we are interested in optimizing the effects of signing up – or other voluntary actions beyond a platform's control – on important final outcomes such as revenue, we therefore need to consider fairness-constrained optimal encouragement designs. Often human expert oversight is required in the loop in important settings where ensuring fairness in machine learning [6] is also of interest: doctors prescribe treatment from recommendations [31], managers and workers combine their expertise to act based on decision support [8], and in the social sector, caseworkers assign to beneficial programs based on recommendations from risk scores that support triage [14, 19, 48].
The human in the loop requires new methodology for optimal encouragement designs because
when the human in the loop makes the final prescription, algorithmic recommendations do not have direct causal effects on outcomes; they change the probability of treatment assignment.
On the other hand, this is analogous to the well-understood notion of non-compliance/non-adherence in randomized controlled trials in the real world [22, 21]. For example, patients who are prescribed treatment may not actually take medication. A common strategy is to conduct an intention-to-treat analysis: under assumptions of no unobserved confounders affecting treatment take-up and outcome, we may simply view encouragement as treatment. But, in the case of prediction-informed decisions in social settings, if we are concerned about access to the intervention in addition to utility of the policy over the population, finer-grained analysis is warranted. If an outcome-optimal policy results in wide disparities in access, for example in marginalized populations not taking up incentives for healthy food due to lack of access in food deserts, or administrative burden that screens out individuals applying for social services that could benefit the most, this could be a serious concern for decision-makers. We ultimately may seek optimal decision rules that improve disparities in treatment access. In contrast, previous work in algorithmic accountability primarily focuses on auditing recommendations, but not both the access and efficacy achieved under the final decision rule. Therefore, previous methods can fall short in mitigating potential disparities.
Our contributions are as follows: we characterize optimal and resource fairness-constrained optimal decision rules, develop statistically improved estimators and robustness checks for the setting of algorithmic recommendations with sufficiently randomized decisions. We also develop methodology for optimizing over a constrained policy class with less conservative out-of-sample fairness constraint satisfaction by a two-stage procedure, and we provide sample complexity bounds. We assess improved recommendation rules in a stylized case study of optimizing recommendation of supervised release in the PSA-DMF pretrial risk-assessment tool while reducing surveillance disparities.
In the main text, we briefly highlight the most relevant methodological and substantive work and defer additional discussion to the appendix.
Optimal encouragement designs/policy learning with constraints. There is extensive literature on off-policy evaluation and learning, empirical welfare maximization, and optimal treatment regimes [5, 49, 35, 30]. [39] studies an optimal individualized encouragement design, though their focus is on optimal individualized treatment regimes with instrumental variables. [27] study fairness in pricing, and some of the desiderata in that setting on revenue (here, marginal welfare) and demand (take-up) are again relevant here, but in a more general setting beyond pricing. The most closely related work in terms of problem setup is the formulation of "optimal encouragement designs" in [39]. However, they focus on knapsack resource constraints, which have a different solution structure than fairness constraints. Their outcome models in regression adjustment are conditional on the recommended/not recommended partitions which would not allow our fairness constraints that introduce treatment- and group-dependent costs. [44] has studied uniform feasibility in constrained resource allocation, but without encouragement or fairness. [9] studies robust extrapolation in policy learning from algorithmic recommendation, but not fairness. Our later case study is on supervised release, where there is a lot of randomness in final treatment decisions, rather than pretrial detention.
Fair off-policy learning We highlight some most closely related works in off-policy learning (omitting works in the sequential setting). [37] studies high-probability fairness constraint satisfaction. [29] studies doubly-robust causal fair classification, while others have imposed deterministic resource constraints on the optimal policy formulation [13]. [26] studies (robust) bounds for treatment responders in binary outcome settings; this desiderata is coupled to classification notions of direct treatment. Again, our focus is on modeling the fairness implications of non-adherence. Indeed, in order to provide general algorithms and methods, we do build on prior fair classification literature. A different line of work studies "counterfactual" risk assessments which models a different concern.
Other causal methodology for intention-to-treat We focus on deriving estimators for intention-to-treat analyses in view of fairness constraints (which result in group-specific welfare weights). Our interest is in imposing separate desiderata on treatment realizations under non-compliance; but we don't conduct instrumental variable inference and we assume unconfoundedness holds. We argue ITT is policy-relevant whereas complier-strata specific analysis is less policy-relevant since the compliers are unknown. Since our primary interest is in characterizing fair optimal decision rules, we don't model this as a mediation analysis problem (isolating the impact of recommendation even under the same ultimate treatment in a nested counterfactual), which may be more relevant for descriptive
characterization. [27] studies multi-objective desiderata for pricing and notes intention-to-treat structure in pricing, but not fairness considerations in more general problems. A related literature studies principal stratification [24], which has similar policy-relevance disadvantages regarding interpretability as complier analysis does.
We briefly describe the problem setup. We work in the Neyman-Rubin potential outcomes framework for causal inference [40]. We define the following:
- recommendation flag  $R \in \{0,1\}$ , where  $R = 1$  means encouraged/recommended. (We will use the terms encouragement/recommendation interchangeably).
- treatment  $T(R) \in \mathcal{T}$ , where  $T(r) = 1$  indicates the treatment decision was 1 when the recommendation reported  $r$ .
- outcome  $Y\left( {t\left( r\right) }\right)$  is the potential outcome under encouragement  $r$  and treatment  $t$  .
Regarding fairness, we will be concerned about disparities in utility and treatment benefits (resources or burdens) across different groups, denoted  $A \in \{a, b\}$ . (For notational brevity, we may generically discuss identification/estimation without additionally conditioning on the protected attribute). For example, recommendations arise from binary high-risk/low-risk labels of classifiers. In practice, in consequential domains, classifier decisions are rarely automated, rather used to inform humans in the loop. The human expert in the loop decides whether or not to assign treatment. For binary outcomes, we will interpret  $Y(t(r)) = 1$  as the positive outcome, and when treatments are also binary, we may further develop analogues of fair classification criteria. We let  $c(r, t, y) \colon \{0, 1\}^3 \mapsto \mathbb{R}$  denote the cost function for  $r \in \{0, 1\}, t \in \mathcal{T}, y \in \{0, 1\}$ , which may sometimes be abbreviated  $c_{rt}(y)$ . We discuss identification and estimation based on the following recommendation, treatment propensity, and outcome models:
$$
e _ {r} (X, A) := P (R = r \mid X, A), \quad p _ {t \mid r} (X, A) := P (T = t \mid R = r, X, A),
$$
$$
\mu_ {r, t} (X, A) := \mathbb {E} \left[ c _ {r t} (Y) \mid R = r, T = t, X, A \right] = \mathbb {E} \left[ c _ {r t} (Y) \mid T = t, X, A \right] := \mu_ {t} (X, A) (\text {a s n . 2})
$$
We are generally instead interested in personalized recommendation rules  $\pi(r \mid X) = \pi_r(X)$  which describes the probability of assigning the recommendation  $r$  to covariates  $X$ . The average encouragement effect is the difference in average outcomes if we refer everyone vs. no one, while the encouragement policy value  $V(\pi)$  is the population expectation induced by the potential outcomes and treatment assignments realized under a recommendation policy  $\pi$ .
$$
A E E = \mathbb {E} [ Y (T (1)) - Y (T (0)) ], \quad V (\pi) = \mathbb {E} [ c (\pi , T (\pi), Y (\pi)) ].
$$
Because algorithmic decision makers may be differentially responsive to recommendation, and treatment effects may be heterogeneous, the optimal recommendation rule may differ from the (infeasible) optimal treatment rule when taking constraints into account or for simpler policy classes.
Assumption 1 (Consistency and SUTVA).  $Y_{i} = Y_{i}(T_{i}(R_{i}))$
Assumption 2 (Conditional exclusion restriction).  $Y(T(R)) \perp R \mid T, X, A$
Assumption 3 (Unconfoundedness).  $Y(T(r)) \perp T(r) \mid X, A$ .
Assumption 4 (Stable responsivities under new recommendations).  $P(T = t \mid R = r, X)$  remains fixed from the observational to the future dataset.
Assumption 5 (Decomposable costs).  $c(r,t,y) = c_r(r) + c_t(t) + c_y(y)$
Assumption 6 (Overlap).  $\nu_{r} \leq e_{r}(X, A) \leq 1 - \nu_{r}$ ;  $\nu_{t} \leq p_{t|r}(X, A) \leq 1 - \nu_{t}$ ;  $\nu_{r}, \nu_{t} \leq 0$
Our key assumption beyond standard causal inference assumptions is the conditional exclusion restriction assumption 2, i.e. that conditional on observable information  $X$ , the recommendation has no causal effect on the outcome beyond its effect on increasing treatment probability. This assumes that all of the covariate information that is informative of downstream outcomes is measured. Although this may not exactly hold in all applications, stating this assumption is also a starting point for sensitivity analysis under violations of it [25]. Assuming assumption 6 is like assuming
we consider a randomized controlled trial with nonadherence. But later we give arguments using robustness to go beyond this, leveraging our finer-grained characterization.
Assumption 4 is a structural assumption that limits our method to most appropriately re-optimize over small changes to existing algorithmic recommendations. This is also required for the validity of intention-to-treat analyses. For example,  $p_{0|1}(x)$  (disagreement with algorithmic recommendation) could be a baseline algorithmic aversion. Not all settings are appropriate for this assumption. We don't assume micro-foundations on how or why human decision-makers were deviating from algorithmic recommendations, but take these patterns as given. One possibility for relaxing this assumption is via conducting sensitivity analysis, i.e. optimizing over unknown responsivity probabilities near known ones.
Later on, we will be particularly interested in constrained formulations on the intention-to-treat effect that impose separate desiderata on outcomes under treatment, as well as treatment.
We consider two settings: in the first,  $R$  is (as-if) randomized and satisfies overlap. Then  $R$  can be interpreted as intention to treat or prescription, whereas  $T$  is the actual realization thereof. We study identification of optimal encouragement designs with potential constraints on treatment or outcome utility patterns by group membership. We characterize optimal unconstrained/constrained decisions under resource parity. In the second,  $R$  is an algorithmic recommendation that does not satisfy overlap in recommendation (but there is sufficient randomness in human decisions to satisfy overlap in treatment): we derive robustness checks in this setting by being robust. First we discuss causal identification in optimal encouragement designs.
Proposition 1 (Regression adjustment identification).
$$
\mathbb {E} [ c (\pi , T (\pi), Y (\pi)) ] = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) \mu_ {t} (X) p _ {t | r} (X) ]
$$
$$
\begin{array}{l} \mathbb {E} [ c (\pi , T (\pi), Y (\pi)) ] = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) \mathbb {E} [ \mathbb {I} \left[ T (r) = t \right] c _ {r t} (Y (r, t)) \mid R = r, X) ] \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi_ {r} (X) P (T = t \mid R = r, X) \mathbb {E} [ c _ {r t} (Y (r, t)) \mid R = r, X ] ] \\ = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) P (T = t \mid R = r, X) \mathbb {E} \left[ c _ {r t} (Y) \mid T = t, X \right] \right] \\ \end{array}
$$
where the last line follows by the conditional exclusion restriction (Assumption 2) and consistency (Assumption 1).
Resource-parity constrained optimal decision rules We consider a resource/burden parity fairness constraint:
$$
V _ {\epsilon} ^ {*} = \max  _ {\pi} \left\{\mathbb {E} [ c (\pi , T (\pi), Y (\pi)) ]: \mathbb {E} [ T (\pi) \mid A = a ] - \mathbb {E} [ T (\pi) \mid A = b ] \leq \epsilon \right\} \tag {1}
$$
Enforcing absolute values, etc. follows in the standard way. Not all values of  $\epsilon$  may be feasible; in the appendix we give an auxiliary program to compute feasible ranges of  $\epsilon$ . We first characterize a threshold solution when the policy class is unconstrained.
Proposition 2 (Threshold solutions). Define
$$
\begin{array}{l} L (\lambda , X, A) = (p _ {1 | 1} (X, A) - p _ {1 | 0} (X, A)) \left\{\tau (X, A) + \frac {\lambda}{p (A)} (\mathbb {I} [ A = a ] - \mathbb {I} [ A = b ]) \right\} + \lambda (p _ {1 | 0} (X, a) - p _ {1 | 0} (X, b)) \\ \lambda^ {*} \in \arg \min  _ {\lambda} \mathbb {E} [ L (\lambda , X, A) _ {+} ], \pi^ {*} (x, u) = \mathbb {I} \{L (\lambda^ {*}, X, u) > 0 \} \\ \end{array}
$$
If instead  $d(x)$  is a function of covariates  $x$  only,
$$
\lambda^ {*} \in \arg \min  _ {\lambda} \mathbb {E} [ \mathbb {E} [ L (\lambda , X, A) \mid X ] _ {+} ], \pi^ {*} (x) = \mathbb {I} \{\mathbb {E} [ L (\lambda^ {*}, X, A) \mid X ] > 0 \}
$$
Establishing this threshold structure (follows by duality of infinite-dimensional linear programming) allows us to provide a generalization bound argument.
Proposition 3 (Policy value generalization). Assume the nuisance models  $\eta = [p_{1|0}, p_{1|1}, \mu_1, \mu_0]^\top$ ,  $\eta \in H$  are consistent and well-specified with finite VC-dimension  $V_\eta$  over the product function class  $H$ . Let  $\Pi = \{\mathbb{I}\{\mathbb{E}[L(\lambda, X, A; \eta) \mid X] > 0 : \lambda \in \mathbb{R}; \eta \in \mathcal{F}\}$ .
$$
\sup  _ {\pi \in \Pi , \lambda \in \mathbb {R}} | (\mathbb {E} _ {n} [ \pi L (\lambda , X, A) ] - \mathbb {E} [ \pi L (\lambda , X, A) ]) | = O _ {p} (n ^ {- \frac {1}{2}})
$$
This bound is stated for known nuisance functions: verifying stability under estimated nuisance functions further requires rate conditions.
Doubly-robust estimation We may improve statistical properties of estimation by developing doubly robust estimators which can achieve faster statistical convergence when both the probability of recommendation assignment (when it is random), and the probability of outcome are consistently estimated; or otherwise protect against misspecification of either model. We first consider the ideal setting when algorithmic recommendations are randomized so that  $e_r(X) = P(R = r \mid X)$ .
Proposition 4 (Variance-reduced estimation).
$$
V (\pi) = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) \left\{\frac {\mathbb {I} [ R = r ]}{e _ {r} (X)} (\mathbb {I} [ T = t ] c _ {r 1} (Y) - \mu_ {1} (X) p _ {t | r} (X)) + \mu_ {1} (X) p _ {t | r} (X) \right\} \right]
$$
$$
\mathbb {E} [ T (\pi) ] = \sum_ {r \in \{0, 1 \}} \mathbb {E} \left[ \pi_ {r} (X) \left\{\frac {\mathbb {I} [ R = r ]}{e _ {r} (X)} (T (r) - p _ {1 | r} (x)) + p _ {1 | r} (x) \right\} \right]
$$
Although similar characterization appears in [39] for the doubly-robust policy value alone, note that doubly-robust versions of the constraints we study would result in differences in the Lagrangian so we retain the full expression rather than simplifying. For example, for regression adjustment, Proposition 9 provides interpretability on how constraints affect the optimal decision rule. In the appendix we provide additional results describing extensions of Proposition 8 with improved estimation.
4.2 Robust estimation with treatment overlap but not recommendation overlap
When recommendations are e.g. the high-risk/low-risk labels from binary classifiers, we may not satisfy the overlap assumption, since algorithmic recommendations are deterministic functions of covariates. However, note that identification in Proposition 1 requires only SUTVA and consistency, and the exclusion restriction assumption. Additional assumptions may be required to extrapolate  $p_{t|r}(X)$  beyond regions of common support. On the other hand, supposing that positivity held with respect to  $T$  given covariates  $X$ , given unconfoundedness, our finer-grained approach can be beneficial because we only require robust extrapolation of  $p_{t|r}(X)$ , response to recommendations, rather than the outcome models  $\mu_t(X)$ .
We first describe what can be done if we allow ourselves parametric extrapolation on  $p_{1|1}(X)$ , treatment responsivity. In the case study later on, the support of  $X \mid R = 1$  is a superset of the support of  $X \mid R = 0$  in the observational data. Given this, we derive the following alternative identification based on marginal control variates (where  $p_t = P(T = t \mid X)$  marginalizes over the distribution of  $R$  in the observational data):
Proposition 5 (Control variate for alternative identification). Assume that  $Y(T(r)) \perp T(r) \mid R = r, X$ .
$$
V (\pi) = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} \left[ \left\{c _ {r t} (Y (t)) \frac {\mathbb {I} [ T = t ]}{p _ {t} (X)} + \left(1 - \frac {\mathbb {I} [ T = t ]}{p _ {t} (X)}\right) \mu_ {t} (X) \right\} p _ {t | r} (X) \right]
$$
Robust extrapolation of  $p_{t|r}(X)$  Let  $\mathcal{X}^{\mathrm{no}} = \{x : P(R = 1 \mid X) = 0\}$  denote the no overlap region; on this region there are no joint observations of  $(t,r,x)$ . We consider uncertainty sets for ambiguous treatment recommendation probabilities. For example, one plausible structural assumption is monotonicity, that is, making an algorithmic recommendation can only increase the probability of being treated.
We define the following uncertainty set:
$$
\mathcal {U} _ {q _ {t | r}} := \left\{q _ {1 | r} (x ^ {\prime}): q _ {1 | r} (x) \geq p _ {1 | r} (x), \forall x \in \mathcal {X} ^ {\mathrm {n o}}, r \sum_ {t \in \mathcal {T}} q _ {t | r} (x) = 1, \forall x, r \right\}
$$
We could assume uniform bounds on unknown probabilities, or more refined bounds, such as Lipschitz-smoothness with respect to some distance metric  $d$ , or boundedness.
$$
\mathcal {U} _ {\mathrm {l i p}} := \left\{q _ {1 | r} (x ^ {\prime}) \colon d (q _ {1 | r} (x ^ {\prime}), p _ {1 | r} (x)) \leq L d (x ^ {\prime}, x), (x ^ {\prime}, x) \in (\mathcal {X} ^ {\mathrm {n o}} \times \mathcal {X} ^ {\mathrm {n o}}) \right\}, \mathcal {U} _ {\mathrm {b n d}} := \left\{q _ {1 | r} (x ^ {\prime}) \colon \underline {{b}} (x) \leq q _ {1 | r} (x ^ {\prime}) \leq \overline {{b}} (x) \right\}
$$
Define  $V_{ov}(\pi) \coloneqq \sum_{t \in \mathcal{T}, r \in \{0,1\}} \mathbb{E}[\pi(r \mid X)p_{t|r}(X)\mu_t(X)\mathbb{I}_{ov}]$ . Let  $\mathcal{U}$  denote the uncertainty set including any custom constraints, e.g.  $\mathcal{U} = \mathcal{U}_{q_{t|r}} \cap \mathcal{U}_{\mathrm{lip}}$ . For brevity we use  $\mathbb{I}_{no}$  to denote  $\mathbb{I}[X \in \mathcal{X}^{\mathrm{no}}]$ ,  $\mathbb{I}_{ov}$  to denote  $\mathbb{I}[X \in \mathcal{X}^{\mathrm{ov}}]$ . Then we may obtain robust bounds by optimizing over regions of no overlap:
$$
\overline {{V}} (\pi) := V _ {o v} (\pi) + \overline {{V}} _ {n o} (\pi), \quad \overline {{V}} _ {n o} (\pi) := \max  _ {q _ {t r} (X) \in \mathcal {U}} \left\{\sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ \pi (r \mid X) \mu_ {t} (X) q _ {t r} (X) \mathbb {I} _ {n o} ] ] \right\}
$$
In the specialized, but practically relevant case of binary outcomes/treatments/recommendations, we obtain the following simplifications for bounds on the policy value, and the minimax robust policy that optimizes the worst-case overlap extrapolation function. In the special case of constant uniform bounds, it is equivalent (in the case of binary outcomes) to consider marginalizations:
Lemma 1 (Binary outcomes, constant bound). Let  $\mathcal{U}_{cbnd} \coloneqq \left\{q_{t|r}(x') : \underline{B} \leq q_{1|r}(x') \leq \overline{B}\right\}$  and  $\mathcal{U} = \mathcal{U}_{q_{t|r}} \cap \mathcal{U}_{cbnd}$ . Define  $\beta_{t|r}(a) \coloneqq \mathbb{E}[q_{t|r}(X,A) \mid T = t, A = a]$ . If  $T \in \{0,1\}$ ,
$$
\overline {{V}} _ {n o} (\pi) = \sum_ {t \in \mathcal {T}, r \in \{0, 1 \}} \mathbb {E} [ c _ {r t} ^ {*} \beta_ {t | r} \mathbb {E} [ Y \pi (r \mid X) \mid T = t ] \mathbb {I} _ {n o} ],
$$
$$
w h e r e c _ {r t} ^ {*} = \left\{ \begin{array}{l l} \overline {{B}} \mathbb {I} [ t = 1 ] + \underline {{B}} \mathbb {I} [ t = 0 ] & i f \mathbb {E} [ Y \pi (r \mid X) \mid T = t ] \geq 0 \\ \overline {{B}} \mathbb {I} [ t = 0 ] + \underline {{B}} \mathbb {I} [ t = 1 ] & i f \mathbb {E} [ Y \pi (r \mid X) \mid T = t ] <   0 \end{array} \right.
$$
We state the next result for simple uncertainty sets, like intervals, to deduce insights about the robust policy. In the appendix we include a more general reformulation for polytopic uncertainty sets.
Proposition 6 (Robust linear program). Suppose  $R, T \in \{0,1\}$ , and  $q_{r1}(\cdot ,u) \in \mathcal{U}_{bnd}, \forall r,u$ . Define
$$
\tau (x, a) = \mu_ {1} (x, a) - \mu_ {0} (x, a), \Delta B _ {r} (x, u) = (\overline {{B}} _ {r} (x, u) - \underline {{B}} _ {r} (x, u)), B _ {r} ^ {\mathrm {m i d}} (x, u) = \underline {{B}} _ {r} (x, u) + \frac {1}{2} \Delta B _ {r} (x, u),
$$