SlowGuess/NeurIPS_2023_OCR huggingface dataset:
$$
\mathbb {E} [ \Delta_ {o v} T (\pi) ] = \mathbb {E} [ T (\pi) \mathbb {I} _ {o v} \mid A = a ] - \mathbb {E} [ T (\pi) \mathbb {I} _ {o v} \mid A = b ], c _ {1} (\pi) = \sum_ {r} \mathbb {E} [ \tau_ {r} \pi_ {r} B ^ {\mathrm {m i d}} ]
$$
Then the robust linear program is:
$$
\max  V _ {o v} (\pi) + \mathbb {E} [ \mu_ {0} ] + c _ {1} (\pi) - \frac {1}{2} \sum_ {r} \mathbb {E} [ | \tau | \pi (r \mid X) \Delta B _ {r} (X, A) \mathbb {I} _ {n o} ]
$$
$$
\text {s . t .} \sum_ {r} \left\{\mathbb {E} \left[ \pi (r \mid X) \bar {B} _ {r} (X, A) \mathbb {I} _ {n o} \mid A = a \right] - \mathbb {E} \left[ \pi_ {r} \underline {{B}} _ {r} (X, A) \mathbb {I} _ {n o} \mid A = b \right] \right\} + \Delta_ {o v} ^ {T} (\pi) \leq \epsilon
$$
5 Additional fairness constraints and policy optimization
We previously discussed policy optimization, over unrestricted decision rules, given estimates. We now introduce general methodology to handle 1) optimization over a policy class of restricted functional form and 2) more general fairness constraints. We first introduce the fair-classification algorithm of [1], describe our extensions to obtain variance-sensitive regret bounds and less conservative policy optimization (inspired by a regularized ERM argument given in [12]), and then provide sample complexity analysis.
Algorithm and setup In the following, to be consistent with standard form for linear programs, note that we consider outcomes  $Y$  to be costs so that we can phrase the saddle-point as minimization-maximization. Consider  $|\mathcal{K}|$  linear constraints and  $J$  groups (values of protected attribute  $A$ ), with coefficient matrix  $M \in \mathbb{R}^{K \times J}$ , the constraint moment function  $h_{j}(\pi), j \in [J]$  (with  $J$  the number of groups),  $O = (X, A, R, T, Y)$  denoting our data observations, and  $d$  the constraint constant vector:
$$
h _ {j} (\pi) = \mathbb {E} \left[ g _ {j} (O, \pi (X)) \mid \mathcal {E} _ {j} \right] \quad \text {f o r} j \in J, M h (\pi) \leq d
$$
Importantly,  $g_{j}$  depends on  $\pi$  while the conditioning event  $\mathcal{E}_j$  cannot depend on  $\pi$ . Many important fairness constraints can nonetheless be written in this framework, such as burden/resource parity,
Algorithm 1 MW2REDFAIR(D,g,E,M,d)
1: Input:  $\mathcal{D} = \{(X_i,R_i,T_i,Y_i,A_i)\}_{i = 1}^n$ $g,\mathcal{E},M,\widehat{d},\mathrm{B},\nu ,\alpha ,\theta_1 = 0\in \mathbb{R}^{|\mathcal{K}|}$    
2: for  $t = 1,2,\ldots$  do   
3: Set  $\lambda_{t,k} = B\frac{\exp\{\theta_k\}}{1 + \sum_{k'\in\mathcal{X}}\exp\{\theta_{k'}\}}$  for all  $k\in \mathcal{K},\beta_t\gets \mathrm{BEST}_\beta (\lambda_t),\widehat{Q}_t\gets \frac{1}{t}\sum_{t' = 1}^t\beta_{t'}$ $\bar{L}\gets L(\hat{Q}_t,\mathrm{BEST}_\lambda (\hat{Q}_t)),\hat{\lambda}_t\gets \frac{1}{t}\sum_{t' = 1}^t\lambda_{t'},\underline{L}\gets (\mathrm{BEST}_\beta (\hat{\lambda}_t),\hat{\lambda}_t),$    
5:  $\nu_{t}\gets \max \{L(\hat{Q}_{t},\hat{\lambda}_{t}) - \underline{L},\quad \bar{L} -L(\hat{Q}_{t},\hat{\lambda}_{t})\}$  If  $\nu_{t}\leq \nu$  then return  $(\hat{Q}_t,\hat{\lambda}_t)$    
6:  $\theta_{t + 1,i} = \theta_t + \log (1 - \eta (M\hat{\mu} (h_t) - \hat{c})_j),\forall i$    
7: end for
parity in true positive rates, but not measures such as calibration whose conditioning event does depend on  $\pi$ . (See Appendix B.2 for examples omitted for brevity). We further consider a convexification of  $\Pi$  via randomized policies  $Q \in \Delta(\Pi)$ , where  $\Delta(\Pi)$  is the set of distributions over  $\Pi$ , i.e., a randomized classifier that samples a policy  $\pi \sim Q$ . Therefore we solve
$$
\min  _ {Q \in \Delta (\Pi)} \left\{V (\pi): M h (\pi) \leq d \right\}
$$
On the other hand, the optimization is solved using sampled moments, so that we ought to interpret the constraint values  $\hat{d}_k = d_k + \epsilon_k$ , for all  $k$ . The overall algorithmic scheme is similar: we seek an approximate saddle point so that the constrained solution is equivalent to the Lagrangian,
$$
L (Q, \lambda) = \hat {V} (Q) + \lambda^ {\top} (M \hat {h} (Q) - \hat {d}), \qquad \min  _ {Q \in \Delta (\Pi)} \{V (\pi) \colon M h (\pi) \leq d \} = \min  _ {Q \in \Delta (\Pi)} \max  _ {\lambda \in \mathbb {R} _ {+} ^ {K}} L (Q, \lambda).
$$
We simultaneously solve for an approximate saddle point and bound the domain of  $\Lambda$  by  $B$ :
$$
\min  _ {Q \in \Delta} \max  _ {\lambda \in \mathbb {R} _ {+} ^ {| \mathcal {X} |}, \| \lambda \| _ {1} \leq B} L (Q, \lambda), \quad \max  _ {\lambda \in \mathbb {R} _ {+} ^ {| \mathcal {X} |}, \| \lambda \| _ {1} \leq B} \min  _ {Q \in \Delta} L (Q, \lambda)
$$
We play a no-regret (second-order multiplicative weights [11, 43], a slight variant of Hedge/exponentiated gradient [18]) algorithm for the  $\lambda$ -player, while using best-response oracles for the  $Q$ -player. Full details are in Algorithm 1. Given  $\lambda_{t}$ ,  $\mathrm{BEST}_{\beta}(\lambda_t)$  computes a best response over  $Q$ ; since the worst-case distribution will place all its weight on one classifier, this step can be implemented by a reduction to cost-sensitive/weighted classification [10, 49], which we describe in further detail below. Computing the best response over  $\mathrm{BEST}_{\lambda}(\hat{Q}_t)$  selects the most violated constraint. We include further details in Appendix B.2.
Weighted classification reduction. There is a well-known reduction of optimizing the zero-one loss for policy learning to weighted classification. Taking the Lagrangian will introduce datapoint-dependent additional weights. This reduction requires  $\pi \in \{-1, + 1\}$ ,  $T\in \{-1, + 1\}$  (for notational convenience alone). We consider parameterized policy classes so that  $\pi (x) = \mathrm{sign}(g_{\theta}(x))$  for some index function  $g$  depending on a parameter  $\beta \in \mathbb{R}^d$ . Consider the centered regret  $J(\pi) = \mathbb{E}[Y(\pi)] - \frac{1}{2}\mathbb{E}[\mathbb{E}[Y\mid R = 1,X] + \mathbb{E}[Y\mid R = 0,X]]$ . Then  $J(\beta) = J(\mathrm{sgn}(g_{\beta}(\cdot))) = \mathbb{E}[\mathrm{sgn}(g_{\beta}(X))\{\psi \} ]$  where  $\psi$  can be one of, where  $\mu_r^R (X) = \mathbb{E}[Y\mid R = r,X]$ ,
$$
\psi_ {D M} = (p _ {1 | 1} (X) - p _ {1 | 0} (X)) (\mu_ {1} (X) - \mu_ {0} (X)), \psi_ {I P W} = \frac {R Y}{e _ {R} (X)}, \psi_ {D R} = \psi_ {D M} + \psi_ {I P W} + \frac {R \mu^ {R} (X)}{e _ {R} (X)}
$$
We can apply the standard reduction to cost-sensitive classification since  $\psi_i \operatorname{sgn}(g_\beta(X_i)) = |\psi_i|$  ( $1 - 2\mathbb{I}[\operatorname{sgn}(g_\beta(X_i)) \neq \operatorname{sgn}(\psi_i)]$ ). Then we can use surrogate losses for the zero-one loss. Although many functional forms for  $\ell(\cdot)$  are Fisher-consistent, one such choice of  $\ell$  is the logistic (cross-entropy) loss given below:
$$
L (\beta) = \mathbb {E} [ | \psi | \ell (g _ {\beta} (X), \operatorname {s g n} (\psi)) ], \quad l (g, s) = 2 \log (1 + \exp (g)) - (s + 1) \tag {2}
$$
Two-stage variance-constrained algorithm. We seek to improve upon this procedure so that we may obtain regret bounds on policy value and fairness constraint violation that exhibit more favorable dependence on the maximal variance over small-variance slices near the optimal policy, rather than worst-case constants over all policies, [12, 5]. Further, note that the algorithm sets the constraint feasibility slacks via generalization bounds that previously depended on these worst-case constants.
Algorithm 2 Two-stage localized fair classification via reductions
1: Randomly split the data into  $\mathcal{D}_1, \mathcal{D}_2$
2: Obtain  $\hat{\pi}_1^*$  and the index set of binding constraints  $\hat{\mathcal{L}}_1$  by learning nuisances  $\eta_{1}$  and running Algorithm 1 on  $\mathcal{D}_1$  with MW2REDFAIR  $(\mathcal{D}_1,h,\mathcal{E},M,d;\eta_1)$
3:  $\hat{\sigma}_j^2\gets \mathrm{Var}(g_j(O,\hat{\pi}_1^*)\mathbb{I}[\mathcal{E}_j] / p_j),\forall j;\hat{d}\gets d + 2\hat{\sigma}^2 n^{-\alpha}$
4: Augment additional constraints with  $\epsilon_{n}$ -policy-value and constraint slices relative to  $\hat{\pi}_{1}$ : define an augmented system (where subscripting by  $\hat{\mathcal{I}}_1$  indexes the corresponding matrix or vector):
$$
\tilde {h} _ {j ^ {\prime}} = \mathbb {E} _ {n _ {1}} \left[ \left\{g _ {j ^ {\prime}} ^ {\hat {\mathcal {I}} _ {1}} (O; \hat {\pi} _ {1} ^ {*}) - g _ {j ^ {\prime}} ^ {\hat {\mathcal {I}} _ {1}} (O; \pi) \right\} \mid \mathcal {E} _ {j} \right], \forall j ^ {\prime} \in \hat {\mathcal {I}} _ {1}, \tilde {h} ^ {v} = \mathbb {E} _ {n _ {1}} \left[ v _ {D R} (O; \pi , \eta_ {1}) - v _ {D R} (O; \pi , \eta_ {1}) \right]
$$
$$
\tilde {M} = \left[ M; M _ {\hat {\mathcal {L}} _ {1}}, \vec {1} \right], \tilde {h} = \left[ h (\cdot ; \pi), \tilde {h}, \tilde {h} ^ {v} \right] ^ {\top}, \tilde {d} = \left[ \hat {d}, \epsilon_ {n}, \epsilon_ {n} \right] ^ {\top}, \tilde {\mathcal {E}} = \left[ \mathcal {E}, \mathcal {E} _ {\hat {\mathcal {L}} _ {1}}, \emptyset \right] ^ {\top}
$$
5: Obtain  $\hat{\pi}_2^*$  by running Algorithm 1 on  $\mathcal{D}_2$  with MW2REDFAIR  $(\mathcal{D},\tilde{g},\tilde{\mathcal{E}},\tilde{M},\tilde{d},\eta_2)$
These challenges motivate the two-stage procedure, described formally in Algorithm 2 and verbally here. We adapt an out-of-sample regularization scheme developed in [12], which recovers variance-sensitive regret bounds via a small modification to ERM. We split the data, learn nuisance estimators  $\eta_{1}$  for use in our policy value and constraint estimates, and run Algorithm 1 (MW2REDFAIR  $(\mathcal{D}_1,h,\mathcal{E},M,d;\eta_1)$ ) to obtain an estimate of the optimal policy  $\hat{\pi}_{1}$ , and the constraint variances at  $\hat{\pi}_{1}$ . Next, we augment the constraint matrix with additional constraints that require the second-stage  $\pi$  to achieve  $\epsilon_{n}$  close policy value and constraint moment values relative to  $\hat{\pi}_{1}$ . Since errors concentrate fast, this can be viewed as variance regularization. And, we set the constraint slacks  $\tilde{d}$  in the second stage using estimated variance constants from  $\hat{\pi}_{1}$ , which will be less conservative than using worst-case bounds. Define
$$
v ^ {(\cdot)} (Q) = \mathbb {E} _ {\pi \sim Q} [ \pi \psi_ {(\cdot)} \mid O ], \text {f o r} (\cdot) \in \{\emptyset , \mathrm {D R} \}, g _ {j} (O; Q) = \mathbb {E} _ {\pi \sim Q} [ g _ {j} (O; \pi) \mid O, \mathcal {E} _ {j} ],
$$
so that  $V^{(\cdot)}(Q) = \mathbb{E}[v^{(\cdot)}(Q)]$  and  $h_j(Q) = \mathbb{E}[g_j(O;Q)\mid \mathcal{E}_j]$ . Define the function classes
$$
\mathcal {F} _ {\Pi} = \left\{v _ {D R} (\cdot , \pi ; \eta): \pi \in \Pi , \eta \in H \right\}, \mathcal {F} _ {j} = \left\{g (\cdot , \pi ; \eta): \pi \in \Pi , \eta \in H \right\}
$$
and the empirical entropy integral  $\kappa(r, \mathcal{F}) = \inf_{\alpha \geq 0} \{4\alpha + 10\int_{\alpha}^{r}\sqrt{\frac{\mathcal{H}_2(\epsilon, \mathcal{F}, n)}{n}} d\epsilon\}$  where  $H_2(\epsilon, \mathcal{F}, n)$  is the  $L_2$  empirical entropy, i.e. log of the  $\|\cdot\|_2$ $\epsilon$ -covering number. We make a mild assumption of a learnable function class (bounded entropy integral) [46].
Assumption 7. The function classes  $\mathcal{F}_{\Pi}$ ,  $\{\mathcal{F}_j\}_{j\in \mathcal{I}}$  satisfy that for any constant  $r,\kappa (r,\mathcal{F})\to 0$  as  $n\to \infty$ . The function classes  $\{\mathcal{F}_j\}_{j}$  comprise of  $L_{j}$ -Lipschitz contractions of  $\pi$ .
We will assume that we are using doubly-robust/orthogonalized estimation as in proposition 4, hence state results depending on estimation error of nuisance vector  $\eta$ .
Theorem 1 (Variance-Based Oracle Policy Regret). Suppose that the mean-squared-error of the nuisance estimates is upper bounded w.p.  $1 - \delta /2$  by  $\chi_{n,\delta}^2$ , over the randomness of the nuisance sample:  $\max_l\{\mathbb{E}[(\hat{\eta}_l - \eta_l)^2 ]\}_{l\in [L]}\coloneqq \chi_n^2$ . Let  $r = \sup_{Q\in \mathcal{Q}}\sqrt{\mathbb{E}[v_{DR}(z;\pi)^2]}$  and  $\epsilon_{n} = \Theta (\kappa (r,\mathcal{F}_{\Pi}) + r\sqrt{\frac{\log(1 / \delta)}{n}})$ . Moreover, let
$$
\mathcal {Q} _ {*} (\epsilon) = \left\{Q \in \Delta [ \Pi ]: V \left(Q _ {*} ^ {0}\right) - V (Q) \leq \epsilon , h \left(Q _ {*} ^ {0}\right) - h (Q) \leq d + \epsilon \right\}
$$
denote an  $\epsilon$ -regret slice of the policy space. Let  $\tilde{\epsilon}_n = O(\epsilon_n + \chi_{n,\delta}^2)$  and
$$
V _ {2} ^ {0} = \sup  \left\{\operatorname {V a r} \left(v _ {D R} ^ {0} (O; Q) - v _ {D R} ^ {0} \left(O; Q ^ {\prime}\right)\right): Q, Q ^ {\prime} \in \mathcal {Q} _ {*} (\tilde {\epsilon} _ {n}) \right\}
$$
denote the variance of the difference between any two policies in an  $\epsilon_{n}$ -regret slice, evaluated at the true nuisance quantities. (Define  $V_{2}^{j}$  analogously for the variance of constraint moments). Then, letting  $\gamma(Q) \coloneqq M h(Q)$  denote the constraint values, the policy distribution  $Q_{2}$  returned by the out-of-sample regularized ERM, satisfies w.p.  $1 - \delta$  over the randomness of  $S$ :
$$
V (\hat {Q}) - V (Q ^ {*}) = O (\kappa (\sqrt {V _ {2} ^ {o b j}}, \operatorname {c o n v} (\mathcal {F} _ {\Pi})) + n ^ {- \frac {1}{2}} \sqrt {V _ {2} ^ {o b j} \log (3 / \delta)} + \chi_ {n, \delta} ^ {2})
$$
$$
(\gamma_ {j} (\hat {Q}) - c _ {j}) - (\gamma_ {j} (Q ^ {*}) - c _ {j}) = O (\kappa (\sqrt {V _ {2} ^ {j}}, \mathrm {c o n v} (\mathcal {F} _ {j})) + n ^ {- \frac {1}{2}} \sqrt {V _ {2} ^ {j} \log (3 / \delta)} + \chi_ {n, \delta} ^ {2})
$$
The benefits are that 1) the constants are improved from an absolute, structure-agnostic bound to depending on the variance of low-regret policies, which also reflects improved variance from using doubly-robust estimation as in proposition 4, and 2) less-conservative out-of-sample fairness constraint satisfaction.
Due to space constraints, in the main text we only present a case study based on the PSA-DMF for supervised release [38]. In the appendix we include additional experiments and robustness checks, including a case study of fully-randomized recommendations and non-adherence.
PSA-DMF case study. Our case study is on a dataset of judicial decisions on supervised release based on risk-score-informed recommendations [38]. The PSA-DMF (Public Safety Assessment Decision Making Framework) uses a prediction of failure to appear for a future court data to inform pretrial decisions, including our focus on supervised release (i.e. electronic monitoring in addition to release) where judges make the final decision. Despite a large literature on algorithmic fairness of pretrial risk assessment, to the best of our knowledge, recommendations about supervised release are not informed by empirical evidence. There are current policy concerns about disparities in increasing use of supervised release given mixed evidence on outcomes [38]; e.g. Safety and Justice Challenge [41] concludes "targeted efforts to reduce racial disparities are necessary". First, we acknowledge data issues. We work with publicly available data that was discretized for privacy [38]. The final supervision decision does not include intensity, but different intensities are recommended in the data, which we collapse into a single level. The PSA-DMF is an algorithmic recommendation so here we are appealing to overlap in treatment recommendations, but using parametric extrapolation in responsivity. Finally, unconfoundedness is likely untrue, but sensitivity analysis could address this. So, this analysis should be considered exploratory, to illustrate the relevance of the methods. Future work will seek proprietary data for a higher-fidelity empirical study.
Next in Figure 1 we provide descriptive information illustrating heterogeneity (including by protected attribute) in adherence and effectiveness. We observe wide variation in judges assigning supervised release beyond the recommendation. We use logistic regression to estimate outcome models and treatment response models. The first figure shows estimates of the causal effect for different groups, by gender (similar heterogeneity for race). The outcome is failure to appear, so negative scores are beneficial. The second figure illustrates the difference in responsiveness: how much more likely decision-makers are to assign treatment when there is vs. isn't an algorithmic recommendation to do so. The last figure plots a logistic regression of the lift in responsiveness on the causal effect  $\mu_1(x,a) - \mu_0(x,a)$ . We observe disparities in how responsive decision-makers are conditional on the same treatment effect efficacy. This is importantly not a claim of animus because decision-makers didn't have access to causal effect estimates. Nonetheless, disparities persist.
In Figure 2 we highlight results from constrained policy optimization. The first two plots in each set illustrate the objective function value and  $A = a$  average treatment cost, respectively; for  $A$  being race (nonwhite/white) or gender (female/male), respectively. We use costs of 100 for  $Y = 1$  (failure to appear, 0 for  $Y = 0$ , and 20 when  $T = 1$  (set arbitrarily). On the x-axis we plot the penalty  $\lambda$  that we use to assess the solutions of Proposition 9. The vertical dashed line indicates the solution achieving  $\epsilon = 0$ , i.e. parity in treatment take-up. Near-optimal policies that reduce treatment disparity can be of interest due to advocacy concerns about how the expansion of supervised release could increase the surveillance of already surveillance-burdened marginalized populations. We see that indeed, for race, surveillance-parity constrained policies can substantially reduce disparities for nonwhites while not increasing surveillance on whites that much: the red line decreases significantly at low increase to the blue line (and low increases to the objective value). On the other hand, for gender, the opportunity for improvement in surveillance disparity is much smaller. See the appendix for further experiments and computational details.
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Figure 1: Distribution of treatment effect by gender, lift in treatment probabilities  $p_{11a} - p_{01a} = P(T = 1 \mid R = 1, A = a, X) - P(T = 1 \mid R = 0, A = a, X)$ , and plot of  $p_{11a} - p_{01a}$  vs.  $\tau$ .
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A Additional discussion
A.1 Additional related work
Principal stratification and mediation analysis in causal inference [32] studies an optimal test-and-treat regime under a no-direct-effect assumption, that assigning a diagnostic test has no effect on outcomes except via propensity to treat, and studies semiparametric efficiency using Structural Nested-Mean Models. Though our exclusion restriction is also a no-direct-effect assumption, our optimal treatment regime is in the space of recommendations only as we do not have control over the final decision-maker, and we consider generally nonparametric models.