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Econ-243 · economics · Economics

Policy learning with explicit resource and fairness constraints

LLM status Open unverified

Statement

CI-31 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Observe independent $(X,S,W,Y)$ with $X\in[0,1]^d$, a binary group label $S$, binary treatment $W$, and bounded outcome. Invoke the binary causal assumptions conditional on $(X,S)$, with group probabilities and treatment overlap bounded below. A randomized policy $\pi(X,S)\in[0,1]$ specifies a Bernoulli treatment independently of potential outcomes conditional on $(X,S)$. Let $c(X,S)\in[0,1]$ be a known cost and let $\Pi$ be a pointwise separable class of finite Vapnik--Chervonenkis subgraph dimension. Define \[ V_P(\pi)=\E_P[m_0(X,S)+\pi(X,S)\tau(X,S)]. \] The budget constraint is $\E_P[c\pi]\leq b$; demographic-parity tolerance is \[ |\E_P[\pi\mid S=1]-\E_P[\pi\mid S=0]|\leq\zeta, \] where $b,\zeta>0$. Assume a policy in $\Pi$ satisfies both constraints with a fixed positive slack, and require, separately for each group, $e\in\mathcal H_d^\alpha(L)$ and $m_w\in\mathcal H_d^\beta(L)$ for fixed $\alpha,\beta>0$.

Formal question

Construct a learned policy satisfying both population constraints with probability at least $1-\eta$, uniformly over the model. Determine sharp worst-case expected regret relative to the supremum of $V_P$ over population-feasible policies in $\Pi$, accounting for nuisance estimation, policy complexity, unknown group proportions, and feasibility slack. Identify any necessary cost of conservative empirical constraints, and give matching lower bounds when the optimum lies on a constraint boundary.

Scope and status

The stated fairness criterion concerns treatment allocation, not equal outcomes or a causal definition of fairness. Different fairness definitions produce different mathematical problems. Exact feasibility cannot be inferred from empirical feasibility without uncertainty control. This specific constrained minimax problem is an editorial refinement of the published agenda; it does not imply that all budget-constrained or fair policy learning lacks established solutions.

References for the source of the problem

Carlos Cinelli, Avi Feller, Guido Imbens, Edward Kennedy, Sara Magliacane, and Jose Zubizarreta, Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1 [stat.ME], 23 August 2025, §3.3.3, "New optimization goals." https://arxiv.org/html/2508.17099v1#S3.SS3.SSS3.

Common conventions: Source mathematical conventions

Well-defined objects. All probability spaces and measurable variables are specified or inherited from a local statistical model. Expectations used as finite targets require the stated integrability. A decision rule or estimator is measurable with respect to its available information; randomized rules may use an auxiliary independent random variable. Norms without a subscript are Euclidean in finite-dimensional spaces. An optimizer is requested only when attainment is assured; otherwise the question concerns the optimal value and $\varepsilon$-optimal admissible rules for every $\varepsilon>0$.

Identification. A structural model $m\in\mathcal M$ implies an observable law $P_m^{\rm obs}$ and target $\theta(m)$. The target is point identified on $\mathcal M$ if \[ P_{m_1}^{\rm obs}=P_{m_2}^{\rm obs} \quad\Longrightarrow\quad\theta(m_1)=\theta(m_2) \qquad(m_1,m_2\in\mathcal M). \] When this fails, the sharp identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M, P_m^{\rm obs}=P\}. \] Specifying an intervention or estimand does not establish identification. Positivity, exclusion, causal sufficiency, measurement restrictions, and transport assumptions are substantive local inputs, never automatic consequences of notation.

Minimax risk and nontrivial inference. For a statistical experiment with data law $P^{(n)}$, target $\theta(P)$, and nonnegative loss $\ell$, define \[ \mathcal R_n(\mathcal P,\ell) =\inf_{\widehat\theta_n}\sup_{P\in\mathcal P} \E_{P^{(n)}}\ell(\widehat\theta_n,\theta(P)). \] The infimum is over measurable estimators. A sharp rate is a sequence $r_n$ for which $0<c\leq C<\infty$, independent of $n$, satisfies $c r_n\leq\mathcal R_n\leq C r_n$ for all sufficiently large $n$. Squared-error parametric risk is of order $n^{-1}$; the corresponding root-mean-squared error is of order $n^{-1/2}$. These different conventions are distinguished locally. A uniform asymptotic confidence region $C_n$ must satisfy \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P} \Pr_{P^{(n)}}\{\theta(P)\in C_n\}\geq 1-\alpha, \qquad 0<\alpha<1, \] together with an informative diameter or expected-width requirement. Coverage alone permits the vacuous whole parameter space. Testing questions similarly impose both size control and power against a declared separated alternative class.

Binary causal baseline. When an entry explicitly invokes this baseline, one observes independent copies of $O=(X,W,Y)$, with $W\in\{0,1\}$ and potential outcomes $Y(0),Y(1)$. Assume consistency $Y=Y(W)$, no interference, conditional exchangeability $(Y(0),Y(1))\perp W\mid X$, and overlap $\epsilon\leq e(X)\leq1-\epsilon$ almost surely for a fixed $0<\epsilon<1/2$, where \[ e(x)=\Pr(W=1\mid X=x),\qquad m_w(x)=\E[Y\mid W=w,X=x],\qquad \tau(x)=m_1(x)-m_0(x). \] These assumptions identify conditional mean effects almost everywhere under the covariate law. Pointwise or uniform effects require appropriate regularity and a specified support. Continuous treatment, longitudinal data, adaptive experiments, and network interference require their own assumptions in place of this baseline. Bounded outcomes, densities, and smoothness are additional assumptions only where stated.

Function classes. On $[0,1]^d$, $\mathcal H_d^s(L)$ is the isotropic Hölder ball of radius $L>0$: put $k=\lceil s\rceil-1$ and $\gamma=s-k\in(0,1]$, bound derivatives through order $k$, and require the order-$k$ derivatives to be $\gamma$-Hölder with constant at most $L$. Thus integer orders use a Lipschitz final derivative convention. The class $\operatorname{BV}_d(L)$ consists of functions $f\in L^1((0,1)^d)$ whose distributional derivative is a finite vector measure and for which $\|f\|_{L^1}+|Df|((0,1)^d)\leq L$. An $L^\infty$ bound or a particular pointwise version is imposed separately when needed. Sparse and neural-network classes require a declared dictionary or architecture and coefficient bounds.

An anisotropic Hölder class with declared block smoothness indices means coordinatewise Hölder regularity in each block, uniformly in the other blocks, with all corresponding derivative and Hölder bounds at most the stated radius. Derivative orders and the integer-order convention in each block are those just given. Mixed-derivative bounds are additional restrictions only when explicitly stated.

Decision and computational questions. Policy regret compares admissible feasible policies under the same law and constraints. In computational entries, finite data and thresholds have rational encodings; running time is measured in their total bit length. Real-valued equilibrium search uses the explicitly declared algebraic or FIXP encoding. An approximation ratio for maximization is written $\operatorname{OPT}/\operatorname{ALG}$ and for minimization as $\operatorname{ALG}/\operatorname{OPT}$, with zero optima and infeasible instances handled locally. Historical approximation bounds are not automatically current bounds.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-243-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-243

Policy learning with explicit resource and fairness constraints: a feasibility discontinuity and a convex repair

Scope

For randomized policies $\pi(X)\in[0,1]$, define value, budget, and group treatment disparity by \[ V_P(\pi)=E_P[\pi(X)\tau(X)],\quad B_P(\pi)=E_P[c(X)\pi(X)], \quad D_P(\pi)=E[\pi\mid S=1]-E[\pi\mid S=0]. \] Feasibility means $B_P(\pi)\le b$ and $|D_P(\pi)|\le\zeta$. Section 3.3.3 of [agenda] raises policy learning with new optimization goals. A strictly feasible policy is useful, but without a closure property of the policy class it does not suffice for uniform learnability under exact population feasibility.
Proposition (Strict slack alone permits constant minimax regret): Fix $0<\eta<1/2$. There is a finite policy class of VC dimension one, a policy with fixed strict budget and fairness slack, and a smooth bounded-overlap statistical submodel for which every learner taking values in that class and satisfying \[ P^n\{\widehat\pi\text{ is population feasible}\}\ge1-\eta \] for every law has worst-case expected regret bounded below by a positive constant, uniformly in $n$.
Proof: Take $X\in[0,1]$, $A=[0,1/2]$, $\Pi=\{0,1_A\}$, cost $c=1$, and budget $b=1/2$. Let $S$ be an independent fair group label and choose any fixed fairness tolerance $\zeta>0$. The zero policy has slack at least $\min(1/2,\zeta)$. Set propensity $e=1/2$ and constant treatment effect $\tau=a>0$, implemented by two bounded Bernoulli outcome means in the interior of $[0,1]$.

Let the covariate density be $p_\theta(x)=1+\theta\sin(2\pi x)$, $|\theta|\le1/2$. These densities are smooth and bounded away from zero; the other nuisances are constant. Since $\int_A\sin(2\pi x)dx=1/\pi$, $P_\theta(A)=1/2+\theta/\pi$. Thus $1_A$ is feasible at $\theta_-=-h/\sqrt n$ and infeasible at $\theta_+=h/\sqrt n$. Both policies have zero fairness disparity.

Only the covariate distribution differs between the two experiments. The inequality $\operatorname{KL}(p,q)\le\int(p-q)^2/q$ bounds their $n$-sample relative entropy by $C h^2$, and their total variation by $C'h$. At $\theta_+$, feasibility requires selection of zero with probability at least $1-\eta$. Choose $h>0$ so small that total variation is at most $(1-\eta)/2$. At $\theta_-$ the learner therefore chooses zero with probability at least $(1-\eta)/2$. The optimal feasible policy there is $1_A$, with value $a(1/2-h/(\pi\sqrt n))$, bounded below by a positive constant after reducing $h$. The expected regret is consequently bounded below uniformly in $n$. ∎

This obstruction concerns the exact feasibility requirement and a policy class that does not contain attenuated versions of $1_A$. It persists despite a known safe policy. It does not assert that convex randomized policy classes have the same obstruction.

Proposition (A finite convex policy class admits a repair): Suppose $\Pi$ is the convex hull of $M$ supplied policies $\pi_1,\ldots,\pi_M$, $|\tau|\le T$, $0\le c\le1$, and $P(S=s)\ge s_0>0$. Suppose a supplied $\pi^\circ\in\Pi$ satisfies $B_P(\pi^\circ)\le b-\sigma$ and $|D_P(\pi^\circ)|\le\zeta-\sigma$. Assume on an event $E$ that empirical linear estimates obey, uniformly over $\Pi$, \[ |\widehat V-V|\le t_V,\quad |\widehat B-B|\le t, \quad |\widehat D-D|\le t,\qquad 2t\le\sigma. \] Let $\widehat\pi$ maximize $\widehat V$, up to error $\xi$, subject to $\widehat B\le b-t$ and $|\widehat D|\le\zeta-t$. On $E$ the empirical feasible set is nonempty, $\widehat\pi$ is population feasible, and \[ \sup_{\pi\in\Pi:\,\pi\text{ feasible}}V(\pi)-V(\widehat\pi) \le2t_V+\frac{4Tt}{\sigma}+\xi. \]
Proof: For any population feasible $\pi$, mix it with the safe policy using $\pi_\lambda=(1-\lambda)\pi+\lambda\pi^\circ$, $\lambda=2t/\sigma$. Convexity keeps it in $\Pi$. Linearity of budget gives a population margin $2t$. Convexity of absolute disparity gives $|D(\pi_\lambda)|\le(1-\lambda)\zeta+\lambda(\zeta-\sigma)=\zeta-2t$. The empirical errors leave margin $t$ in both constraints, so $\pi_\lambda$ is empirically admissible. Conversely the conservative empirical constraints imply true feasibility on $E$. Empirical optimality and the two value errors show $V(\widehat\pi)\ge V(\pi_\lambda)-2t_V-\xi$. Finally $|V(\pi)-V(\pi_\lambda)|\le2T\lambda$, a conservative bound since each value lies in $[-T,T]$. Take the supremum over population feasible $\pi$; attainment is unnecessary. ∎
Lemma (The required event is attainable): With bounded outcomes, supplied propensity in $[\epsilon,1-\epsilon]$, iid observations, and the preceding group-probability bound, empirical inverse-probability values, empirical budgets, and empirical within-group treatment rates satisfy the required event with probability at least $1-\alpha$ for \[ t_V\le C\epsilon^{-1}\sqrt{\frac{\log(8M/\alpha)}n},\qquad t\le C\sqrt{\frac{\log(8M/\alpha)}{ns_0}}, \] provided $ns_0\ge C\log(8/\alpha)$; constants depend on the outcome bound.
Proof: Hoeffding's inequality and a union bound control the bounded value and budget scores for all $M$ base policies. Chernoff's inequality makes each group count at least $ns_0/2$ except with probability at most $\alpha/4$, after adjusting $C$. Conditional on group counts, the covariates within each group are iid from their group distribution. Hoeffding and a union bound over policies and the two groups therefore control both conditional treatment rates, and their difference. Each estimate and each population functional is linear in the convex mixture weights, so a uniform error bound over base policies extends to their whole convex hull. ∎
Choose a safe fallback when the conservative program is empty. For large enough $n$, $2t\le\sigma$; the preceding event proves feasibility with probability $1-\alpha$ and the regret bound on that event. Taking $\alpha\le\min(\eta,n^{-1})$ yields a vanishing expected-regret upper bound since off-event regret is bounded by $2T$.

Remaining gap

The negative result answers a genuine obstruction in the stated broad class, while the positive result adds convex closure and a supplied safe policy. Sharp dependence on general VC complexity, unknown nuisances, vanishing slack, and fairness constraints with rare groups is not characterized. No full constrained-policy minimax theorem is claimed.

Completion Estimate

50%

This is a post hoc, heuristic estimate for the full catalogue target. The attempt proves that strict feasibility slack alone permits constant regret and gives conservative feasible learning for a finite convex policy hull. Sharp rates for the full VC class with unknown nuisances, constraint-boundary optima and rare groups remain uncharacterized beyond that obstruction and repaired subclass.

References

  • [agenda] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane, and J. Zubizarreta. Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference (2025). https://arxiv.org/html/2508.17099v1.

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