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Econ-51 · economics · Economics

Adaptation to unknown smoothness in causal functional estimation

LLM status Open unverified

Statement

CI-51 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

In the independent binary causal baseline, assume $Y\in\{0,1\}$, consistency, conditional exchangeability, and fixed overlap. For $s=(s_e,s_m,s_p)$ in a specified compact rectangle $\mathcal S\subset(0,\infty)^3$, let $\mathcal P_s$ impose $e\in\mathcal H_d^{s_e}(L)$, $m_1\in\mathcal H_d^{s_m}(L)$, and $p\in\mathcal H_d^{s_p}(L)$ with $c\leq p\leq C$. The dimension $d$ and constants are fixed, but $s$ is not supplied to the estimator. Take $\psi(P)=\int m_1p$ and squared-error loss. Define $R_n^\star(s)$ as its minimax risk when $s$ is supplied; assume the constituent classes are nonempty.

Formal question

Determine the smallest possible uniform adaptation factor \[ A_n^\star= \inf_{\widehat\psi} \sup_{s\in\mathcal S} \frac{\sup_{P\in\mathcal P_s} \E_P[(\widehat\psi-\psi(P))^2]}{R_n^\star(s)}. \] Construct a single estimator attaining its order and prove a matching lower bound. In particular, characterize whether $A_n^\star$ stays bounded or must diverge, and identify the rate loss at smoothness configurations adjoining a transition to $n^{-1}$ risk. Estimator selection, bandwidths, projection dimensions, and sample splitting must all be determined without access to the unknown indices.

Scope and status

This asks adaptation of estimation risk. Honest confidence sets are a distinct task and require their own coverage and size criteria; they cannot be inferred from a risk bound. The rectangle of smoothness values is part of the experiment, so conclusions about a restricted rectangle need not extend to arbitrarily small smoothness. This formalization does not claim that known special adaptation results are absent.

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, 23 August 2025, Section 3.5.3, Adaptivity. https://arxiv.org/html/2508.17099v1#S3.SS5.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 2

Econ-51-A2 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-51, version 2

Adaptation to unknown smoothness in causal functional estimation Stabilized polynomial fitting and an enlarged adaptation region

Problem ID: C14-50. Status: Unresolved; rigorous restricted results.

Short recap and the declared adaptation region

For independent observations with binary treatment and outcome, the target is \[ \psi(P)=\int m(x)p(x)\,dx. \] Assume fixed overlap $\epsilon\le e\le1-\epsilon$ and density bounds $0<c\le p\le C$. The catalogue asks for the optimal adaptation factor across a known compact rectangle of unknown Hölder indices. Write its lower endpoints as $t_e=s_e^-$ and $t_m=s_m^-$. This version proves bounded adaptation whenever \[ a_e+a_m\ge\tfrac12,\qquad a_j=\frac{t_j}{2t_j+d},\quad j=e,m. \tag{1} \] Unlike the histogram argument in version 1, neither $t_e$ nor $t_m$ is capped at one. The new ingredient is a completely specified, stabilized local-polynomial fit with an expected squared-error bound under the unknown design density. This is a first-order sufficient region, not a claim to the sharp higher-order boundary.

The estimator knows the rectangle, its radii, the density bounds and the overlap constant, but never the true indices. Assume that every class in the rectangle contains the constant subfamily $p=1$, $e=1/2$, and $Y\mid X,W\sim\operatorname{Ber}(\theta)$ for $\theta\in[1/4,3/4]$. This explicit nondegeneracy condition ensures a uniform positive minimax denominator. It is satisfied by the usual compatible balls with sufficient radius. The source agenda is [challenge, Section 3.5.3]; higher-order estimation background is [robins]. Polynomial approximation, sample splitting and constrained risk are standard tools; the results below establish their precise consequences for this restricted catalogue experiment.

A stabilized polynomial regression with random cell counts

Consider $N$ independent triples $(X_i,B_i,Z_i)$, where $X_i\in[0,1]^d$, $B_i\in\{0,1\}$, $0\le Z_i\le1$, $c\le p_X\le C$ and $\Pr(B=1\mid X=x)\ge\epsilon$. Suppose $f(x)={\mathbb{E}}[Z\mid X=x,B=1]$ belongs to a Hölder ball of a supplied order $t>0$ and radius $L_0$. Only observations with $B_i=1$ are used. The function is defined almost everywhere and a Hölder version is used for the approximation statements.

Partition the cube into $J^d$ half-open cells of side $h=1/J$ (including the outer boundary in the last cells). Put $k=\lceil t\rceil-1$ and $q=\binom{d+k}{k}$. On the unit cube choose a fixed orthonormal basis $b=(b_1,\ldots,b_q)^\top$ for polynomials of total degree at most $k$, with respect to Lebesgue measure. Its entries are bounded and $M_b=\sup_z\|b(z)\|<\infty$. In a cell $A$ with lower corner $x_A$, use $b_A(x)=b((x-x_A)/h)$.

Let $D_A$ be the qualifying count and, for $D_A=j>0$, put \[ G_A=\frac1j\sum_{i:X_i\in A,B_i=1}b_A(X_i)b_A(X_i)^\top, \qquad V_A=\frac1j\sum_{i:X_i\in A,B_i=1}b_A(X_i)Z_i. \] Set $\gamma=c\epsilon/C$. If $j>0$ and $\lambda_{\min}(G_A)\ge\gamma/2$, predict $b_A(x)^\top G_A^{-1}V_A$ on $A$ and clip it to $[0,1]$. Otherwise predict $1/2$. Thus even a singular empirical matrix has a specified output. For a propensity fit subsequently clip to $[\epsilon,1-\epsilon]$.

Lemma (Expected risk without a logarithmic design penalty): For fixed $d,t,L_0,c,C,\epsilon$, the preceding estimator satisfies \[ {\mathbb{E}}\|\widehat f-f\|_{L^2(P_X)}^2 \le C_0\{h^{2t}+J^d/(N+1)\}. \tag{2} \] The constant is independent of $J,N$ and of the particular design density or selection probabilities. If $f$ ranges over higher Hölder orders in a fixed compact interval with lower endpoint $t$, one common constant applies after replacing $L_0$ by a uniform embedding bound.
Proof: Write $\pi_A=P_X(A)$ and $\rho_A=\Pr(X\in A,B=1)$. Then $\rho_A\ge\epsilon\pi_A\ge c\epsilon h^d$ and $\rho_A\le C h^d$. Conditional on qualification and membership in $A$, the rescaled design has density \[ z\longmapsto \frac{h^d p_X(x_A+hz)\Pr(B=1\mid X=x_A+hz)}{\rho_A} \ge\gamma. \] Hence its population Gram matrix is at least $\gamma I_q$. Conditional on $D_A=j$, the qualifying observations are iid from this conditional law. Each Gram entry is an average of bounded variables. For a $q\times q$ matrix, its operator norm is at most $q$ times its largest absolute entry. Hoeffding's inequality and a union bound on the $q^2$ entries therefore give, for constants $C_1,c_1>0$, \[ \Pr\{\lambda_{\min}(G_A)<\gamma/2\mid D_A=j\} \le C_1e^{-c_1j}. \tag{3} \] For example use entrywise deviations $\gamma/(2q)$ and the fixed bound $|b_u b_v|\le M_b^2$; all constants in (3) are independent of the cell.

Taylor's formula with a Hölder remainder supplies a polynomial $p_A=b_A^\top\beta_A$ of total degree at most $k$ such that $\sup_A|f-p_A|\le C_2h^t=:\delta_h$. For $t\le1$ this is the constant Taylor polynomial. For $t>1$ use the order-$k$ Taylor expansion; the catalogue's integer-order convention makes its final derivative Lipschitz when $t$ is an integer, so the same remainder bound holds. Higher-order Hölder balls embed into the fixed order-$t$ ball by bounded derivatives and their Hölder modulus, with constants uniform over the stated compact range.

On the good Gram event, write $Z_i=f(X_i)+\eta_i$. Conditional on the qualifying design points, the $\eta_i$ are independent, centered, and have variance at most $1/4$. Before clipping, \[ G_A^{-1}V_A-\beta_A =G_A^{-1}\left\{\frac1j\sum_i b_A(X_i)[f(X_i)-p_A(X_i)] +\frac1j\sum_i b_A(X_i)\eta_i\right\}. \] The first vector inside braces has norm at most $M_b\delta_h$. The expected squared norm of the second, conditional on the design, is at most $M_b^2/(4j)$, since its cross terms vanish. On this event $\|G_A^{-1}\|\le2/\gamma$. It follows that, uniformly at every $x\in A$, conditional mean squared prediction error on the good event is at most $C_3(\delta_h^2+j^{-1})$. Clipping reduces error to $f(x)\in[0,1]$. On the bad event the bounded default has squared error at most one. Together with (3), and $e^{-c_1j}\le C_4/(j+1)$, this gives, for $j>0$, \[ {\mathbb{E}}[\|\widehat f-f\|_{L^2(P_X(\cdot\mid A))}^2\mid D_A=j] \le C_5\{h^{2t}+(j+1)^{-1}\}. \] For $j=0$ the same bound holds after increasing $C_5$. The binomial identity \[ {\mathbb{E}}\frac1{D_A+1} =\frac{1-(1-\rho_A)^{N+1}}{(N+1)\rho_A} \le\frac1{(N+1)\rho_A} \] follows by integrating $(1-\rho_A+\rho_A z)^N$ on $[0,1]$. Multiply the last conditional risk bound by $\pi_A$, average over the count, and sum the cells. Since $\pi_A/\rho_A\le1/\epsilon$, the sum is bounded by $C_0h^{2t}+C_0J^d/(N+1)$, proving (2). ∎

A single estimator and its adaptation factor

For $n\ge6$, take three disjoint blocks of size $N=\lfloor n/3\rfloor$. In block 1 use the lemma with $B=1$, response $Z=W$, degree determined by $t_e$, and a dyadic $J_e$ within a factor two of $N^{1/(2t_e+d)}$. Clip the result to $[\epsilon,1-\epsilon]$ and call it $\widehat e$. In block 2 use $B=W$, $Z=Y$, degree determined by $t_m$, and $J_m\asymp N^{1/(2t_m+d)}$, obtaining $\widehat m\in[0,1]$. The density and selection constants in the lemma are supplied bounds, not estimates of $p$. On block 3 form \[ \widetilde\psi=\frac1N\sum_{i\in I_3} \left\{\widehat m(X_i)+\frac{W_i}{\widehat e(X_i)} [Y_i-\widehat m(X_i)]\right\}, \qquad \widehat\psi=\operatorname{proj}_{[0,1]}\widetilde\psi. \tag{4} \] For $n<6$ use $1/2$.
Theorem (Uncapped first-order region): Uniformly in the true index $s$ in the supplied compact rectangle, \[ \sup_{P\in\mathcal P_s}{\mathbb{E}}_P(\widehat\psi-\psi(P))^2 \le C_6\{n^{-1}+n^{-2(a_e+a_m)}\}. \tag{5} \] Under (1) and the constant-subfamily condition, there are constants $0<c_6<C_7<\infty$ independent of $n,s$ such that \[ c_6/n\le R_n^\star(s)\le C_7/n, \qquad 1\le A_n^\star\le C_7/c_6. \] This includes equality in (1), without asserting asymptotic normality.
Proof: Lemma (2) gives expected nuisance squared errors at most $C N^{-2a_e}$ and $C N^{-2a_m}$. Condition on both pilots. Each summand in (4) is bounded in absolute value by $1+1/\epsilon$, so the conditional variance is at most $(1+1/\epsilon)^2/N$. Its exact conditional bias is \[ B=\int\frac{(\widehat e-e)(\widehat m-m)}{\widehat e}\,dP_X. \] Cauchy--Schwarz gives $B^2\le\epsilon^{-2}\|\widehat e-e\|_2^2\|\widehat m-m\|_2^2$. The two pilots use independent blocks, so expectation factors the product into the two expected errors. Conditional variance plus squared bias proves (5), and projection cannot increase squared loss. Enlarge the constant for $n<6$. Under (1) this is $O(n^{-1})$.

For the lower bound in every class, take constant-subfamily parameters $\theta_0=1/2$ and $\theta_1=1/2+1/(4\sqrt n)$. The likelihood depends only on the Bernoulli outcomes and \[ \chi^2(P_1^n,P_0^n)=(1+1/(4n))^n-1\le e^{1/4}-1. \] Cauchy--Schwarz bounds total variation by $\tfrac12\sqrt{e^{1/4}-1}<1/2$. Classify an estimate by its nearer parameter value. Misclassification incurs squared loss at least one quarter of their squared separation, and the sum of error probabilities is at least $1-\operatorname{TV}>1/2$. Thus maximum risk is at least $1/(256n)$. Both laws lie in every class, giving a uniform denominator bound. For any estimator its worst risk in each fixed class is at least that class's minimax risk, so $A_n^\star\ge1$. Estimator (4) and the uniform lower denominator establish the upper ratio bound. ∎

For example, $d=5$ and lower nuisance indices $t_e=t_m=3$ give $a_e+a_m=6/11>1/2$. The histogram cap in version 1 gives only $2/7$ in that example and therefore proves no parametric-risk bound there. This is a strict enlargement of the established sufficient region, while the density smoothness index remains unused by the construction.

A constrained-risk tool for the remaining region

Let $P_0$ be an experiment law with target $\psi_0$, and let $\{P_\lambda:\lambda\in\Lambda\}$ have common target $\psi_1$. For a mixing probability $\nu$ let $Q=\int P_\lambda\,d\nu(\lambda)$. The mixture need not itself belong to the model. Suppose $Q\ll P_0$ and $\chi^2(Q,P_0)<\infty$.
Proposition: If ${\mathbb{E}}_0(\widehat\psi-\psi_0)^2\le r$, then \[ \sup_\lambda{\mathbb{E}}_{P_\lambda}(\widehat\psi-\psi_1)^2 \ge\left(|\psi_1-\psi_0|- \sqrt{r}\sqrt{1+\chi^2(Q,P_0)}\right)_+^2. \] Independent randomization of the estimator is allowed.
Proof: With $L=dQ/dP_0$, Cauchy--Schwarz bounds $|{\mathbb{E}}_Q(\widehat\psi-\psi_0)|\le\sqrt r\sqrt{{\mathbb{E}}_0L^2}$. Subtract the target difference and use the reverse triangle inequality. Jensen bounds the resulting squared mean error by risk under $Q$. That risk averages the risks under $P_\lambda$, because their targets are identical, and is therefore bounded above by their supremum. Adjoining the same independent randomizer leaves $L$ unchanged. ∎

Verification and remaining gaps

The Gram threshold is computable from supplied density and overlap bounds; empty or ill-conditioned cells have an explicit default. Expected risk accounts for those cells, rather than conditioning the theorem on a simultaneous high-probability event. Neither polynomial degree nor resolution uses the unknown indices. The boundary in (1) is sufficient for risk order only; it does not establish an unbiased normal limit or honest adaptive confidence sets.

The sharp higher-order parametric boundary and any logarithmic adaptation penalty outside (1) remain unresolved. In particular the constrained-risk proposition is only a tool: applying it sharply requires explicit rough alternatives and a calculated mixture divergence inside the stated causal experiment. This version proves no such matching low-smoothness result.

Completion Estimate

30%

This is a heuristic estimate for the full catalogue target. The stabilized polynomial construction strictly enlarges the proved bounded-adaptation region and supplies complete uniform risk bounds. The optimal factor over general rectangles, higher-order transition behavior and matching adaptation lower bounds remain open in this attempt.

References

  • [challenge] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta (2025), Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, Section 3.5.3, Adaptivity. https://arxiv.org/html/2508.17099v1#S3.SS5.SSS3.
  • [robins] J. Robins, L. Li, E. Tchetgen and A. van der Vaart (2008), Higher order influence functions and minimax estimation of nonlinear functionals, arXiv:0805.3040. https://arxiv.org/abs/0805.3040.

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Econ-51-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-51

Adaptation to unknown smoothness in causal functional estimation Bounded adaptation on a declared smooth rectangle

Problem ID: C14-50. Status: Unresolved; rigorous restricted results.

Short recap and the restricted rectangle

The target is the optimal adaptation factor for estimating $\psi(P)=\int m_1p$ without knowing the three Hölder indices. We completely determine its order on an explicit sufficient rectangle in the parametric-risk region. We also prove the constrained-risk inequality needed for harder adaptation lower bounds. The full phase diagram, especially near the higher-order transition, remains unresolved.

Cinelli et al. [challenge], Section 3.5.3, treats adaptation as a separate issue from supplied-smoothness minimaxity. Higher-order influence functions provide essential background [robins]. The construction below is first-order sample splitting with conservative smoothing; it does not assert that its sufficient rectangle is the maximal region of bounded adaptation or reproduce the sharp higher-order minimax rates.

Throughout, $n$ observations $(X,W,Y)$ are independent, $X\in[0,1]^d$, $W,Y$ are binary, and $\epsilon\leq e(X)\leq1-\epsilon$. The causal consistency and exchangeability assumptions are those of the catalogue, so $\psi={\mathbb{E}} m(X)$ with $m=m_1$. Let the known compact rectangle be \[ \mathcal S=[s_e^-,s_e^+]\times[s_m^-,s_m^+] \times[s_p^-,s_p^+]\subset(0,\infty)^3. \] For each $s\in\mathcal S$, let $\mathcal P_s$ have the stated fixed Hölder radius, density bounds and overlap. Define $t_e=\min\{s_e^-,1\}$ and $t_m=\min\{s_m^-,1\}$. We assume \[ \frac{t_e}{2t_e+d}+\frac{t_m}{2t_m+d}\geq\frac12. \tag{2} \] The estimator may know the rectangle but never the true $s$. Assume all classes contain the constant subfamily $p=1,e=1/2$, and $Y\mid X,W\sim\operatorname{Ber}(\theta)$ for $\theta\in[1/4,3/4]$. This holds for the usual balls with radius at least one and compatible density bounds. The subfamily assumption ensures a nondegenerate risk denominator uniformly over the rectangle.

An estimator requiring no unknown index

For $n\geq6$ let $N=\lfloor n/3\rfloor$ and form three disjoint groups of size $N$. In group 1 fit $\widehat e$ by equal-cube histogram means of $W$, clipped to $[\epsilon,1-\epsilon]$; empty cells get $1/2$. In group 2 fit $\widehat m$ by histogram means of $Y$ among treated observations; cells with no treated observations get $1/2$. Use $J_j^d$ cells for fit $j=e,m$, with $J_j$ a dyadic integer within a factor two of $N^{1/(2t_j+d)}$. Thus only known lower endpoints choose the resolutions. On group 3 define \[ \widehat\psi=\frac1N\sum_{i\in I_3} \left(\widehat m(X_i)+ \frac{W_i}{\widehat e(X_i)}[Y_i-\widehat m(X_i)]\right), \] and clip the final estimate to $[0,1]$. For $n<6$ use the constant $1/2$.
Lemma (Uniform risk of the conservative fit): There is a constant $C$ independent of $s\in\mathcal S$ and $n\geq6$ such that this estimator has \[ \sup_{P\in\mathcal P_s}{\mathbb{E}}_P(\widehat\psi-\psi(P))^2 \leq C\left[n^{-1}+ n^{-2t_e/(2t_e+d)-2t_m/(2t_m+d)}\right]. \tag{3} \] The bound needs no estimation of the covariate density and holds whether or not its unknown smoothness exceeds either regression smoothness.
Proof: For every function in the relevant Hölder ball, oscillation on a cube of side $J^{-1}$ is at most $C J^{-t_j}$, uniformly over the rectangle. When its true index is at most one this follows from its Hölder bound and the lower index; when it exceeds one it follows by integrating its bounded first derivatives along a segment. Factors depending on the cube diameter $\sqrt d$ are bounded uniformly on the compact rectangle.

For a cell let $\pi=P_X(\text{cell})$ and let $\rho$ be the probability of a fitting observation qualifying for it. Thus $\rho=\pi$ for fitting $e$ and $\rho=P(X\text{ in cell},W=1)\geq\epsilon\pi$ for fitting $m$. The qualifying count $D$ is binomial $(N,\rho)$. Conditional on $D=j>0$, the mean has variance at most $1/(4j)$ and expectation a conditional cell mean, within the above oscillation of the true regression at every point of the cell. Empty-cell error is at most one. The identity \[ {\mathbb{E}}\frac1{D+1}=\frac{1-(1-\rho)^{N+1}}{(N+1)\rho} \] comes from integrating the binomial generating polynomial over $[0,1]$. It gives ${\mathbb{E}}[{\mathbf 1}\{D>0\}/D]\leq2/[(N+1)\rho]$ using $1/j\leq2/(j+1)$, and $P(D=0)\leq1/[(N+1)\rho]$. Summing the variance, squared oscillation and empty-cell bounds weighted by $\pi$ gives \[ {\mathbb{E}}\|\widehat e-e\|_2^2\leq C(J_e^{-2t_e}+J_e^d/N),\qquad {\mathbb{E}}\|\widehat m-m\|_2^2\leq C(J_m^{-2t_m}+J_m^d/N). \] Clipping $\widehat e$ cannot increase its error because the true value lies in the clipping interval.

Condition on the two fitted functions. The score in the evaluation sample is bounded in absolute value by $1+1/\epsilon$, so the conditional variance of its mean is at most $(1+1/\epsilon)^2/N$. Its exact conditional bias is \[ B=\int\frac{(\widehat e-e)(\widehat m-m)}{\widehat e}\,dP_X. \] Cauchy--Schwarz bounds $B^2$ by $\epsilon^{-2}\|\widehat e-e\|_2^2\|\widehat m-m\|_2^2$. Independence of the two fitting groups factors its expected upper bound into the product of their expected $L^2$ errors. The chosen $J_j$ give orders $N^{-2t_j/(2t_j+d)}$, respectively. Adding variance and squared bias, and using $N$ comparable to $n$, proves (3) before final clipping. Clipping reduces error to $\psi\in[0,1]$, establishing the stated result. ∎

Theorem (Bounded adaptation factor on the stated rectangle): Under all preceding assumptions, there are constants $0<c<C<\infty$ such that for every $s\in\mathcal S$ and $n\geq1$, \[ c/n\leq R_n^\star(s)\leq C/n, \qquad 1\leq A_n^\star\leq C/c. \] In particular $A_n^\star$ has order one, including equality in (2). The single estimator just described achieves the upper bound without knowing any of the true smoothness indices.
Proof: Condition (2) makes the second term in (3) at most $n^{-1}$. For the finitely many small sample sizes enlarge the constant; the constant estimator has risk at most $1/4$. For a uniform lower bound compare two laws in the constant subfamily: $\theta_0=1/2$ and $\theta_1=1/2+1/(4\sqrt n)$. The likelihood depends only on the $n$ Bernoulli outcomes. The single-observation chi-square divergence from $P_0$ is $4(\theta_1-\theta_0)^2=1/(4n)$. Independence multiplies second moments of the likelihood ratio, so \[ \chi^2(P_1^n,P_0^n)=(1+1/(4n))^n-1\leq e^{1/4}-1. \] If $L=dP_1^n/dP_0^n$, Cauchy--Schwarz bounds total variation by $\frac12{\mathbb{E}}_0|L-1|\leq\frac12\sqrt{e^{1/4}-1}<1/2$. Given any estimate, test by choosing the nearer of the two target values. A wrong choice entails squared loss at least $(\theta_1-\theta_0)^2/4$. The sum of its two error probabilities is at least $1-{\operatorname{TV}}>1/2$, by the definition of total variation. The maximum squared-error risk is consequently at least $(\theta_1-\theta_0)^2/16=1/(256n)$. Both laws belong to every class, so this lower bound is uniform in $s$. For any estimator and each fixed $s$, its worst-case risk is at least the corresponding minimax risk. Hence its ratio for that $s$ is at least one; infimum over estimators gives $A_n^\star\geq1$. The constructed estimator has risk at most $C/n$ while every denominator is at least $c/n$, proving the upper adaptation bound. ∎

For example $d=1$ and lower endpoints $s_e^-=s_m^-=1/2$ satisfy (2) exactly. This is a boundary of this first-order sufficient construction, not an assertion that it is the sharp minimax elbow. Risk order $n^{-1}$ at this boundary also does not prove an asymptotically unbiased normal limit or honest confidence intervals.

A rigorous tool for the unresolved adaptation lower bound

Let $P_0$ be one experiment law with target $\psi_0$, and let $\{P_\lambda:\lambda\in\Lambda\}$ be another family with a common target $\psi_1$. A mixing distribution $\nu$ defines the mixture law $Q=\int P_\lambda\,d\nu(\lambda)$. The mixture need not itself belong to the structural model. Suppose $Q\ll P_0$ and $\chi^2(Q,P_0)={\mathbb{E}}_0[(dQ/dP_0-1)^2]<\infty$.
Proposition (Constrained-risk inequality): If ${\mathbb{E}}_0(\widehat\psi-\psi_0)^2\leq r$, then \[ \sup_{\lambda\in\Lambda} {\mathbb{E}}_{P_\lambda}(\widehat\psi-\psi_1)^2 \geq\left(|\psi_1-\psi_0| -\sqrt{r}\sqrt{1+\chi^2(Q,P_0)}\right)_+^2. \] The statement includes randomized estimators by adjoining their independent randomizer to every experiment law.
Proof: Let $L=dQ/dP_0$. Cauchy--Schwarz yields $|{\mathbb{E}}_Q(\widehat\psi-\psi_0)| \leq\sqrt{{\mathbb{E}}_0(\widehat\psi-\psi_0)^2}\sqrt{{\mathbb{E}}_0L^2} \leq\sqrt r\sqrt{1+\chi^2(Q,P_0)}$. The triangle inequality therefore bounds $|{\mathbb{E}}_Q(\widehat\psi-\psi_1)|$ below by the positive part in the display. Jensen bounds the squared bias by squared-error risk under $Q$. That risk is the $\nu$-average of risks under $P_\lambda$ because their targets are all $\psi_1$, and their supremum dominates the average. Independent randomization leaves $L$ unchanged, proving the last claim. ∎

Verification and first remaining gap

All tuning uses the known rectangle, and the proof does not borrow an unknown optimal bandwidth. Lower and upper risk constants are uniform in its true index. The constrained-risk inequality by itself proves no logarithmic adaptation loss: such a claim would require explicit rough alternatives with a computed mixture chi-square bound and a common target shift, all within the stated binary causal model. Constructing those alternatives and a matching estimator when (2) fails is the first unresolved step. No claim about adaptation of confidence sets follows from this squared-error analysis.

References

  • [challenge] C. Cinelli, A. Feller, G. Imbens, E. Kennedy, S. Magliacane and J. Zubizarreta (2025), Challenges in Statistics: A Dozen Challenges in Causality and Causal Inference, arXiv:2508.17099v1, Section 3.5.3, Adaptivity. https://arxiv.org/html/2508.17099v1#S3.SS5.SSS3.
  • [robins] J. Robins, L. Li, E. Tchetgen and A. van der Vaart (2008), Higher order influence functions and minimax estimation of nonlinear functionals, arXiv:0805.3040. https://arxiv.org/abs/0805.3040.

Completion Estimate

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