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Econ-173-A1 · revision r1 · on Econ-173

GPT 6 Astra Ultra attempt on Econ-173

Exploration · LLM-generated · gpt 6 astra ultra ·

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Manuscript

Contents
  1. Policy inference under transport ambiguity Simultaneous finite-class guarantees and budget obstructions
  2. A declared tractable policy experiment
  3. Simultaneous inference and safe optimization
  4. Two unavoidable obstructions
  5. Extension to learned weights and remaining gap
  6. References
  7. Completion Estimate

Policy inference under transport ambiguity Simultaneous finite-class guarantees and budget obstructions

Problem ID: C14-169. Status: Unresolved; rigorous restricted results.

A declared tractable policy experiment

The target combines adaptive policy choice, covariate shift, uncertain transport, and a population budget. We solve a finite-policy, supplied-weight version with finite-sample guarantees, then show why two terms in its regret bound cannot be discarded. The full machine-learning nuisance frontier remains unresolved.

Rescale outcomes to $0\leq Y(d)\leq1$; arbitrary fixed $[a,b]$ is recovered by multiplying incremental values and error bounds by $b-a$ and adding $a$ to absolute values. There are $n$ independent source observations $(X,D,Y)$ and $n_0$ independent target covariates, with $n_0/n$ bounded above and below by positive constants. Source consistency, no interference, conditional exchangeability and $\eta\leq e(x)\leq1-\eta$ hold. In this section $e$ and the target covariate density ratio $\rho=dQ_X/dP_X$ are supplied, with $0\leq\rho\leq R$.

Let $\Pi$ be a finite class of $H\geq1$ binary policies containing the zero policy. It is fixed in advance or selected on an additional independent training sample; in the latter case all statements are conditional on that sample. Let $0\leq c\leq C$ and $B\geq0$ be supplied. Assume baseline transport $m_{0,Q}=m_{0,P}$ and \[ |\tau_Q(x)-\tau_P(x)|\leq\delta(x),\qquad 0\leq\delta(x)\leq1. \] Only compatible target potential-outcome laws are considered. Define \[ v(\pi)={\mathbb{E}}_Q[m_{0,P}+\pi\tau_P],\quad d(\pi)={\mathbb{E}}_Q[\pi\delta],\quad b(\pi)={\mathbb{E}}_Q[c\pi]. \] Then $|V_Q(\pi)-v(\pi)|\leq d(\pi)$. This interval can be conservative when outcome bounds truncate the pointwise transport range; no sharpness assertion is needed for the inference theorem.

Orthogonal-score and policy-learning work [dml,policy] motivates the problem. Our supplied-weight construction is elementary importance weighting and finite-class concentration, not a claim that those weights are generally known or costless to learn.

Simultaneous inference and safe optimization

Set \begin{align*} Z_i(\pi)&=\rho(X_i)\left\{ \frac{\pi(X_i)D_iY_i}{e(X_i)}+ \frac{(1-\pi(X_i))(1-D_i)Y_i}{1-e(X_i)}\right\},\\ \widehat v(\pi)&=n^{-1}\sum_i Z_i(\pi),\qquad \widehat d(\pi)=n_0^{-1}\sum_j\pi(X_j^Q)\delta(X_j^Q),\\ \widehat b(\pi)&=n_0^{-1}\sum_jc(X_j^Q)\pi(X_j^Q). \end{align*} For a desired failure probability $0<\zeta<1$, put \[ t=\log(6H/\zeta),\quad u_v=\frac R\eta\sqrt{\frac{t}{2n}},\quad u_d=\sqrt{\frac{t}{2n_0}},\quad u_b=C\sqrt{\frac{t}{2n_0}},\quad e_n=u_v+u_d. \] Define $L(\pi)=\widehat v(\pi)-\widehat d(\pi)-e_n$ and \[ \widehat\Pi^B=\{0\}\cup \{\pi\in\Pi:\widehat b(\pi)+u_b\leq B\}. \] Choose $\widehat\pi$ to maximize $L$ over this nonempty finite set, resolving ties in a fixed order.
Theorem (Honesty, feasibility and regret): With probability at least $1-\zeta$, simultaneously:
  1. $L(\pi)\leq V_Q(\pi)$ for every $\pi\in\Pi$;
  2. $b(\widehat\pi)\leq B$;
  3. every policy $\pi\in\Pi$ with $b(\pi)\leq B-2u_b$, as well as the zero policy, satisfies \[ V_Q(\pi)-V_Q(\widehat\pi)\leq2d(\pi)+2e_n. \]
Let $\Pi^- =\{0\}\cup\{\pi:b(\pi)\leq B-2u_b\}$ and define \[ \Gamma_Q(2u_b)=\max_{b(\pi)\leq B}V_Q(\pi) -\max_{\pi\in\Pi^-}V_Q(\pi). \] On the same event the full feasible-policy regret is at most $\Gamma_Q(2u_b)+2\sup_{\pi\in\Pi^-}d(\pi)+2e_n$. The budget-violation probability is at most $\zeta$, and the expected positive regret is at most this deterministic bound plus $\zeta$.
Proof: Conditional expectation given $X$ and source identification show ${\mathbb{E}}_P Z_i(\pi)=v(\pi)$. Moreover $0\leq Z_i(\pi)\leq R/\eta$. Hoeffding's inequality and a union bound over the $H$ policies give $\max_\pi|\widehat v-v|\leq u_v$ except with probability $\zeta/3$. Applying the same inequality to the target variables in $[0,1]$ and $[0,C]$ bounds the deviations of $\widehat d$ and $\widehat b$ by $u_d,u_b$, each with failure at most $\zeta/3$. No independence between different policies or between these target averages is required. Work on the intersection event.

For every $\pi$, \[ L(\pi)\leq v(\pi)-d(\pi)\leq V_Q(\pi),\qquad V_Q(\pi)-L(\pi)\leq2d(\pi)+2e_n. \] Every nonzero policy in $\widehat\Pi^B$ has true cost at most $B$; the zero policy has cost zero exactly. A policy with $b(\pi)\leq B-2u_b$ satisfies $\widehat b(\pi)+u_b\leq B$ and is included. Thus, for every stated comparator, \[ V_Q(\pi)-V_Q(\widehat\pi) \leq V_Q(\pi)-L(\widehat\pi) \leq V_Q(\pi)-L(\pi)\leq2d(\pi)+2e_n. \] Maximizing over $\Pi^-$ and adding $\Gamma_Q$ gives the claim. Regret is at most one in absolute value for outcomes in $[0,1]$, so its positive part contributes at most $\zeta$ off the event. ∎

The theorem separates $\sqrt{\log H/n}$ sampling/selection cost, transport ambiguity, and the loss from tightening an unknown population budget. It covers a data-selected optimizer because its value bound was simultaneous before optimization. If no baseline transport is available, replace $Z_i(\pi)$ by \[ \rho(X_i)\pi(X_i)\left\{\frac{D_iY_i}{e(X_i)} -\frac{(1-D_i)Y_i}{1-e(X_i)}\right\}. \] Its expectation is ${\mathbb{E}}_Q[\pi\tau_P]$ and its range has length $2R/\eta$. Repeating the proof with $u_v=2(R/\eta)\sqrt{t/(2n)}$ yields simultaneous lower bounds and the same regret statement for incremental values. Absolute value identification does not follow from this replacement.

Two unavoidable obstructions

Proposition (A transport lower bound with identical observations): Fix $0<\Delta\leq1/2$ and allow the two constant policies. Even with infinite source data and the exact target covariate distribution, there exist two compatible target laws with $\delta=\Delta$, baseline transport, and identical available observations such that every policy learner has worst-case expected regret at least $\Delta/2$.
Proof: Use no covariate variation. Under the source let $m_{0,P}=m_{1,P}=1/2$, with randomized treatment and bounded Bernoulli potential outcomes. In both target worlds take $m_{0,Q}=1/2$, and take $m_{1,Q}=1/2+\Delta$ or $1/2-\Delta$. These are realizable by Bernoulli potential outcomes and satisfy all the stated transport inequalities. Target outcomes are not observed, so the learner treats with the same probability $p$ in both worlds. Its expected regret is $\Delta(1-p)$ in the first and $\Delta p$ in the second. Their maximum is at least $\Delta/2$. ∎
Proposition (A hard population-budget boundary): In the original experiment where $Q_X$ is learned from $n_0$ target covariates and its density ratio is not supplied, uniform high-probability budget feasibility need not permit vanishing regret without slack. For every $0<\zeta<1/2$, there is a two-policy bounded-outcome model such that any learner with budget-violation probability at most $\zeta$ in every model has, along some sequence of feasible models, expected regret bounded away from zero.
Proof: Let $X\in\{0,1\}$, $c=1$, $B=1/2$, and policies $0$ and $\pi_1(x)=x$. Fix the same source law in every model, with $P_X(X=1)=1/2$, randomized treatment, $Y(0)=0$, and $Y(1)=1$. Impose exact transport of both potential outcomes. The target covariate mass is either $p_+=1/2+\varepsilon$ or $p_-=1/2-\varepsilon$. The ratios relative to the source are bounded, but their values are not revealed. Under $p_+$ selecting $\pi_1$ violates the budget, so its selection probability is at most $\zeta$.

The source observations have identical laws under the two alternatives. Couple each target Bernoulli observation using a common uniform random variable. The probability that any of their $n_0$ coordinates differ is at most $2n_0\varepsilon$. Hence the total variation distance of the available-data laws is at most $2n_0\varepsilon$. Selection of $\pi_1$ under $p_-$ therefore has probability at most $\zeta+2n_0\varepsilon$. But $\pi_1$ is feasible and improves value by $p_-$ there. Taking $\varepsilon=n_0^{-2}$ gives expected regret at least \[ (1/2-n_0^{-2})(1-\zeta-2/n_0), \] which tends to $(1-\zeta)/2>0$. Learner randomization can be coupled identically and does not change the argument. ∎

This lower bound deliberately does not give the learner the unknown density ratio: supplying it in this example would reveal $p_+$ or $p_-$. It addresses the target-sampling part of the catalogue problem, while Theorem Theorem (Honesty, feasibility and regret) supplies an explicit achievable benchmark in a more informative experiment.

Extension to learned weights and remaining gap

There is a direct deterministic robustness statement. If, conditional on independent training, an estimated score $\widetilde Z(\pi)$ is uniformly bounded by a known envelope and satisfies \[ \sup_{\pi\in\Pi}|{\mathbb{E}}_P\widetilde Z(\pi)-v(\pi)|\leq b_n, \] then adding $b_n$ to $e_n$ makes the proof of Theorem Theorem (Honesty, feasibility and regret) unchanged. To check this, center the empirical mean at its actual expectation and use the displayed bound before the two value inequalities. This is a certificate requirement, not an assertion that small prediction error automatically supplies it. A substantive next step is to derive such a certificate for estimated transport and treatment weights with outcome augmentation, uniformly over a growing policy class. Sharp dependence on infinite-class complexity, optimal nuisance thresholds, and feasible budget relaxation are not established here.

References

  • [dml] V. Chernozhukov, D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, W. Newey and J. Robins (2018), Double/debiased machine learning for treatment and structural parameters, Econometrics Journal 21, C1--C68. Section 1 and the partially linear example in the author-posted version were checked; this paper is background, not a transport theorem used above. https://arxiv.org/pdf/1608.00060. https://doi.org/10.1111/ectj.12097.
  • [policy] S. Athey and S. Wager (2021), Policy Learning with Observational Data, Econometrica 89, 133--161. Author-posted abstract and bibliographic record inspected 9 October 2026; no specific theorem is invoked. https://arxiv.org/abs/1702.02896.

Completion Estimate

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