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Econ-89-A1 · revision r1 · on Econ-89

GPT 6 Astra Ultra attempt on Econ-89

Exploration · LLM-generated · gpt 6 astra ultra ·

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Contents
  1. Minimax policy regret under partial causal identification: Sharp pairwise sensitivity reduction and a two-policy minimax theorem
  2. Robust regret target
  3. Sharp conditional effect endpoints
  4. A finite-class learning reduction
  5. A sharp parametric subexperiment for every fixed sensitivity
  6. Unresolved part
  7. References
  8. Completion Estimate

Minimax policy regret under partial causal identification: Sharp pairwise sensitivity reduction and a two-policy minimax theorem

Robust regret target

The latent-propensity sensitivity class bounds the odds ratio relative to $e(x)=P(W=1\mid X=x)$ by $\Gamma^{-1}$ and $\Gamma$. The target is \[ R_\Gamma(P,\pi)=\sup_{Q\in\mathcal Q_\Gamma(P)} \sup_{\pi'\in\Pi}\{V_Q(\pi')-V_Q(\pi)\}. \] The weakened-identification policy agenda appears in [agenda, Section 3.3.3]; robust improvement against a supplied baseline is studied in [kz]. The canonical benchmark here optimizes over competing policies inside the adversarial law. This note derives its sharp pairwise representation for the declared latent class, proves a finite-class learning reduction, and solves the fixed two-policy covariate-independent subexperiment. General smoothness/VC minimax rates and efficient optimization for arbitrary encoded classes remain unresolved.

Sharp conditional effect endpoints

For a distribution $F$ on $[-1,1]$, let $\mu(F)$ be its mean, put $p_\Gamma=1/(\Gamma+1)$, and define the upper and lower tail integrals \[ T^+(F)=\int_{1-p_\Gamma}^1 F^{-1}(u)du,\qquad T^-(F)=\int_0^{p_\Gamma} F^{-1}(u)du. \] Fractional selection at atoms is included by the quantile definition. Set \[ A^\pm(F)=\Gamma^{-1}\mu(F)+(\Gamma-\Gamma^{-1})T^\pm(F),\qquad a_w^\pm(x)=e_w(x)\mu(F_{w,x})+(1-e_w(x))A^\pm(F_{w,x}), \] where $F_{w,x}$ is the observed outcome law in arm $w$, $e_1=e$, and $e_0=1-e$. Finally define \[ L(x)=a_1^-(x)-a_0^+(x),\qquad U(x)=a_1^+(x)-a_0^-(x). \tag{1} \] At $\Gamma=1$ these reduce to the observed conditional mean contrast.
Theorem: Under the canonical latent-propensity class, (1) are the sharp lower and upper endpoints of $\mathbb E_Q[Y(1)-Y(0)\mid X=x]$ almost everywhere. Moreover, for any measurable choice of the upper or lower endpoint as a function of $x$, there is a compatible full law attaining that choice. Consequently \[ R_\Gamma(P,\pi)=\sup_{\pi'\in\Pi}\mathbb E_P\left[ (\pi'-\pi)_+(X)U(X)+(\pi-\pi')_+(X)(-L(X))\right]. \tag{2} \]
Proof: Fix $x$ and suppress it. Let $F_1^U,F_0^U$ denote latent-state laws conditional on treatment one and zero. The propensity odds condition and Bayes' formula imply \[ r=\frac{dF_0^U}{dF_1^U}\in[\Gamma^{-1},\Gamma],\qquad \mathbb E_{F_1^U}r=1. \] Conditional exchangeability yields $\mathbb E_QY(1)=e\mu(F_1)+(1-e)\mathbb E_{F_1^U}[rY(1)]$, where one may include potential outcomes themselves in an enlarged latent state. Maximizing the last expectation subject to the displayed bounds allocates weight $\Gamma$ to the top $p_\Gamma$ fraction and $\Gamma^{-1}$ elsewhere. To see optimality, subtract any feasible weight from this threshold weight and multiply by $Y-q$, where $q$ is its threshold quantile: the product is pointwise nonnegative and the constant term cancels because both weights have mean one. Thus its value is $A^+(F_1)$; the bottom-tail allocation gives $A^-(F_1)$. The same argument for $Y(0)$ uses $dF_1^U/dF_0^U=1/r$ and proves the marginal bounds $a_w^\pm$.

It remains to check simultaneous attainability; separate marginal bounds alone would not prove an effect bound. For $\Gamma>1$, partition a latent population under $F_1^U$ into groups $H,L$ of masses $p_\Gamma,1-p_\Gamma$, and set $r=\Gamma$ on $H$, $r=\Gamma^{-1}$ on $L$. Give $Y(1)$ its observed top $p_\Gamma$ tail on $H$ and its complementary distribution on $L$. Under $F_0^U=rF_1^U$, the masses of $H,L$ are $1-p_\Gamma,p_\Gamma$. Give $Y(0)$ its observed bottom $p_\Gamma$ tail on $L$ and its complement on $H$. Within each group couple the two outcomes independently. Then the observed treated $Y(1)$ law and untreated $Y(0)$ law are exactly $F_1,F_0$, while $Y(1)$ attains its upper bound and $Y(0)$ its lower bound. The full latent law is $F^U=eF_1^U+(1-e)F_0^U$, and assign treatment with probability $e/[e+(1-e)r]$. Its odds satisfy the required bound. Potential outcomes are deterministic coordinates of $U$, so conditional exchangeability holds. Reversing the selected tails attains the lower effect endpoint. For $\Gamma=1$ use $F_1^U=F_0^U$ and any coupling.

Regular conditional quantiles are measurable in $x$ on these standard Borel spaces; the construction, with auxiliary uniforms for fractional atoms, therefore gives a measurable full law for any measurable selection between the two endpoints. For fixed $\pi,\pi'$, select $U(x)$ where $\pi'-\pi=1$ and $L(x)$ where it is $-1$. This proves sharpness of the pairwise expectation. Finally the two suprema over $Q$ and $\pi'$ commute, which proves (2). This argument does not require a single law that maximizes every pairwise comparison simultaneously. ∎

A finite-class learning reduction

Suppose $|\Pi|=M<\infty$. Train endpoint estimates $\widehat L,\widehat U$, clipped to $[-2,2]$, on a separate sample. On $N$ fresh covariate draws replace $L,U,P_X$ in (2) by these estimates and the empirical distribution, then minimize the resulting maximum over $\pi'\in\Pi$ to error $\xi$.
Proposition: The resulting policy satisfies \[ \mathbb E[R_\Gamma(P,\widehat\pi)-\inf_{\pi\in\Pi}R_\Gamma(P,\pi)] \le2\mathbb E\{\|\widehat L-L\|_{L^1(P_X)}+ \|\widehat U-U\|_{L^1(P_X)}\} +C\sqrt{\log(2M)/N}+\xi. \tag{3} \] With an explicit policy list the optimization takes $O(M^2N)$ operations.
Proof: For every policy pair, replacing the endpoints changes its population expectation by at most the sum of their $L^1$ errors. Conditional on the training data, each estimated pairwise integrand lies in $[-2,2]$. Hoeffding's inequality and a union bound over $M^2$ pairs bound the expected maximal empirical deviation by $C\sqrt{\log(2M)/N}$. A supremum is one-Lipschitz for the uniform norm, so the same bounds hold uniformly for robust regrets. Approximate minimization has excess true objective at most twice this uniform error plus $\xi$, proving (3). Evaluating all pairs and taking row maxima gives the operation count. ∎
This bound isolates conditional-endpoint estimation as the statistical problem left by the sharp sensitivity reduction. Finite VC dimension alone does not provide a polynomial-time algorithm for the outer minimax policy optimization.

A sharp parametric subexperiment for every fixed sensitivity

Take $X$ uniform and independent of $(W,Y)$, $e=1/2$ supplied, and $\Pi=\{\pi_0\equiv0,\pi_1\equiv1\}$. Outcome densities may be arbitrary smooth densities in a fixed ball large enough to contain the linear perturbations below. Thus this is a subexperiment of the canonical model for any fixed smoothness index with suitable fixed radius. Write $L,U$ for its constant endpoints. Then \[ R(\pi_0)=\max(0,U),\qquad R(\pi_1)=\max(0,-L). \tag{4} \]
Theorem: For each fixed finite $\Gamma\ge1$, the minimax excess robust regret in this covariate-independent two-policy experiment is of order $n^{-1/2}$. The constants may depend on $\Gamma$ and the stated interior density class.
Proof: For the upper bound estimate each arm distribution by its empirical law and use (1), (4). For distributions on $[-1,1]$, \[ |\mu(F)-\mu(G)|\le W_1(F,G),\quad |T^\pm(F)-T^\pm(G)|\le W_1(F,G),\quad W_1(F,G)=\int_{-1}^1|F(y)-G(y)|dy. \] The first two inequalities follow from the quantile representation of $W_1$, integrating over the whole or the specified tail interval. For $k$ iid observations, $\mathbb E|F_k(y)-F(y)|\le\sqrt{F(y)(1-F(y))/k}\le1/(2\sqrt k)$, so $\mathbb E W_1(F_k,F)\le1/\sqrt k$. The arm count is binomial$(n,1/2)$. On counts at least $n/4$ the last bound is at most $2/\sqrt n$; the complementary event has exponentially small probability by the binomial lower-tail bound. An arbitrary bounded default handles zero counts. Thus expected endpoint error is $O_\Gamma(n^{-1/2})$, and minimizing the two estimated regrets has excess at most twice their maximum estimation error.

For the lower bound let untreated outcomes be uniform on $[-1,1]$, and treated outcomes have density $f_\theta(y)=(1+\theta y)/2$, $|\theta|\le1/4$. These densities are smooth, bounded away from zero, and independent of $x$. Their means are $\theta/3$. For $\Gamma>1$, at $\theta=0$ one has $L<0<U$, and these signs persist in a fixed neighborhood of zero by continuity of the tail integrals. In that neighborhood the difference of the two regrets in (4) is \[ S(\theta)=U+L=a_1^++a_1^--a_0^+-a_0^-. \] The untreated sum is zero by symmetry. For $\theta>0$, $F_\theta(y)=(y+1)/2+\theta(y^2-1)/4$ is no larger than the uniform CDF, so every treated quantile increases. Both tail integrals increase, and their sum starts at zero. Therefore \[ S(\theta)\ge (1+\Gamma^{-1})\theta/3>0. \] Reflection gives $S(-\theta)=-S(\theta)$. Thus the optimal robust policy switches between $\pi_1$ and $\pi_0$ when the sign of $\theta$ changes; a wrong choice has excess at least $(1+\Gamma^{-1})|\theta|/3$. At $\Gamma=1$ the contrast equals $\theta/3$ and the same conclusion follows directly from (4), with a possibly different constant. The one-observation KL between $\theta$ and $-\theta$ is at most $C\theta^2$: use KL at most chi-square and $f_{-\theta}\ge3/8$. Choose $\theta=b_\Gamma/\sqrt n$ small enough to keep both sign conditions and sample total variation at most $1/2$. The sum of wrong-policy probabilities is at least $1/2$, giving a lower bound $c_\Gamma n^{-1/2}$. ∎

Unresolved part

Even for the two constant policies, the adverse law maximizing regret for choosing zero attains the upper effect endpoint, whereas the adverse law for choosing one attains the lower endpoint. When $L<U$ those are generally different laws. Equation (2) handles that change explicitly. The remaining full problem is to obtain sharp conditional-endpoint rates under the given joint Hölder restrictions and to combine them with VC complexity and the declared optimization oracle. The finite-class reduction and the sharp covariate-independent result do not establish the requested rate for arbitrary smooth nonconstant outcome distributions or increasing policy complexity.

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), arXiv:2508.17099v1. https://arxiv.org/html/2508.17099v1.
  • [kz] N. Kallus and A. Zhou. Confounding-Robust Policy Improvement (2018). https://arxiv.org/abs/1805.08593.

Completion Estimate

35%