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Econ-183 · economics · Economics

Minimax heterogeneous-effect estimation outside Hölder classes

LLM status Open unverified

Statement

CI-48 |  Econometric theory and causal inference

Research agenda formalization

Definitions and assumptions

Use the binary causal baseline with independent observations, bounded outcomes, consistency, conditional exchangeability, and fixed overlap. Let $X$ be uniform on $[0,1]^d$, and suppose the conditional average treatment effect has the sparse additive form \[ \tau(x)=b+\sum_{j\in S}h_j(x_j),\qquad |S|\leq k,\quad \int_0^1h_j(u)\,du=0, \] where $|b|\leq L$, $\|h_j\|_\infty\leq L$, and each univariate function has distributional total variation at most $L$. Impose $\|\tau\|_\infty\leq M$, $m_0\in\mathcal H_d^{s_m}(L)$, and $e\in\mathcal H_d^{s_e}(L)$. Then $m_1=m_0+\tau$ need not belong to the same Hölder ball. Conditional outcome variances are bounded above by a fixed constant; the model includes a submodel with variance bounded away from zero.

Formal question

For the resulting class $\mathcal P_{d,k}$, characterize \[ R_{n,d,k}^\star =\inf_{\widehat\tau}\sup_{P\in\mathcal P_{d,k}} \E_P\!\left[\int_{[0,1]^d} (\widehat\tau(x)-\tau(x))^2\,dx\right]. \] Determine how unknown coordinate selection, bounded variation of the active components, and propensity/outcome nuisance estimation combine. Give matching rates first for fixed $d,k$, then for specified sequences $d=d_n$, $k=k_n$, recording the conditions necessary for risk to converge to zero. Compare the unknown-propensity experiment with randomized assignment having a supplied constant propensity.

Scope and status

This is a concrete alternative to smooth CATE models. It separates structure imposed on $\tau$ from structure imposed on its nuisances; applying a regression rate to all three functions without this distinction would change the question. No optimal rate or present-day unresolved status is attributed to this particular subclass.

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, New parameters and New models. 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 1

Econ-183-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-183

Sparse additive BV treatment effects An explicit selection bound and two causal lower bounds

Problem ID: C21-179. Status: Unresolved; rigorous restricted results.

Aim and declared experiment

The target combines unknown active coordinates, nonsmooth univariate components and causal nuisance estimation. We give a finite, computationally expensive estimator with a dimension-explicit upper bound under supplied propensity, a transfer bound for learned propensity, and lower bounds that distinguish component estimation from coordinate selection. These establish useful restricted benchmarks, but do not close every logarithmic or sparsity gap and do not determine the optimal nuisance frontier.

Assume the binary causal baseline, $X\sim\operatorname{Unif}([0,1]^d)$, $|Y|\leq B$, fixed overlap, and \[ \begin{gathered} \tau(x)=b+\sum_{j\in S}h_j(x_j),\qquad |S|\leq k\leq d,\qquad \int_0^1h_j=0,\\ |b|\leq L,\qquad \|h_j\|_\infty,\ |Dh_j|((0,1))\leq L, \qquad\|\tau\|_\infty\leq M. \end{gathered} \] Take $M\leq2B$ without loss by replacing it with the smaller effective bound. Smoothness restrictions on $m_0$ and $e$ are retained for lower-bound compatibility; the supplied-propensity upper bound does not need to estimate $m_0$. The motivation is the non-Hölder minimax direction of [challenge, Section 3.5.3].

A finite sparse dictionary

Lemma (Centered component approximation): For fixed $L$ and $0<\epsilon<1/2$, there is a finite collection $\mathcal G_\epsilon$ of mean-zero step functions such that each admissible $h_j$ has an approximation $g_j$ with \[ \|h_j-g_j\|_2^2\leq C\epsilon^2, \quad \|g_j\|_\infty\leq2L, \quad \log|\mathcal G_\epsilon| \leq C\epsilon^{-1}\log(C/\epsilon). \]
Proof: Write a BV representative as $h=c+u-v$, where $u,v$ are cumulative positive and negative derivative measures and their total increases sum to at most $L$. Round $c,u,v$ to value mesh $\epsilon$. Each rounded monotone part has at most $\lceil L/\epsilon\rceil$ jumps of height $\epsilon$. Round jump locations to a grid of mesh $1/Q$, with $Q$ of order $\epsilon^{-3}$. Outside a set of measure at most $2\lceil L/\epsilon\rceil/Q$, the error is at most $3\epsilon$; on that set all functions are uniformly bounded by constants depending on $L$. Squared integrated error is therefore at most $C\epsilon^2$. Clip to $[-L,L]$, which reduces the error. The possible rounded constants and jump lists number at most $(C/\epsilon)(Q+1)^{2\lceil L/\epsilon\rceil}$, giving the entropy bound.

Subtract the integral from every resulting step function. The supremum is then at most $2L$. Since $h$ has mean zero, Cauchy--Schwarz bounds the removed integral by the previous $L^2$ error; subtracting it in fact projects the error onto the mean-zero subspace and cannot increase its $L^2$ norm. This proves the centered dictionary claim. ∎

Form a dictionary by selecting at most $k$ coordinates, assigning a member of $\mathcal G_\epsilon$ to each, and adding a constant from an $\epsilon$ grid of $[-L,L]$. Clip each total function to $[-M,M]$; call the dictionary $\mathcal F_{d,k,\epsilon}$.

Lemma (Selection and approximation costs): With constants depending only on $L,M$, \begin{align*} \inf_{f\in\mathcal F_{d,k,\epsilon}}\|f-\tau\|_2^2 &\leq C(k+1)\epsilon^2,\\ \log|\mathcal F_{d,k,\epsilon}| &\leq k\log(ed/k)+C(k+1)\epsilon^{-1}\log(C/\epsilon). \end{align*}
Proof: Before clipping, the approximation error is a constant plus a sum of centered functions on distinct independent uniform coordinates. The cross inner products are zero. Its squared norm is therefore the squared constant error plus the sum of component squared errors, at most $C(k+1)\epsilon^2$. Clipping cannot increase distance to the bounded target.

The number of supports is at most $(ed/k)^k$: with $t=k/d\leq1$, $t^k\sum_{j=0}^k\binom dj\leq(1+t)^d\leq e^k$. Each support contributes at most $|\mathcal G_\epsilon|^k$ functions, and there are at most $C/\epsilon$ constants. Lemma Lemma (Centered component approximation) and taking logarithms prove the result. Clipping may identify functions, which only reduces their number. ∎

Risk bounds with and without a supplied propensity

For supplied $e$, define $Z=WY/e(X)-(1-W)Y/(1-e(X))$, whose regression is $\tau$ and whose magnitude is at most $B/\varepsilon$. Let $\widehat\tau$ minimize empirical squared error of $Z$ over $\mathcal F_{d,k,\epsilon}$, with a fixed tie rule.
Theorem (Dimension-explicit constructive upper bound): For all $1\leq k\leq d$ and $n\geq3$, a choice $\epsilon$ of order $(\log n/n)^{1/3}$ gives \[ \sup_P{\mathbb{E}}\|\widehat\tau-\tau\|_2^2 \leq C\left\{(k+1)(\log n/n)^{2/3} +\frac{k\log(ed/k)}n\right\}\wedge4M^2.\tag{1} \] Here $C$ depends only on fixed $L,M,B,\varepsilon$, not on $d,k,n$. If an independent sample instead supplies $\widehat e\in[\varepsilon/2,1-\varepsilon/2]$, replace $e$ in $Z$ by $\widehat e$. The same bound holds with an additional term $C{\mathbb{E}}\|\widehat e-e\|_2^2$ inside the braces.
Proof: We record the finite-regression argument to make the bound self-contained. For responses and dictionary functions bounded by a fixed $K$, let $h={\mathbb{E}}[Z\mid X]$, $r_f=\|f-h\|_2^2$, and take a best dictionary member $g$. The loss difference $D_f=(Z-f)^2-(Z-g)^2$ has mean $r_f-r_g$, magnitude at most $8K^2$, and variance at most $32K^2(r_f+r_g)$. Bernstein and a union bound give, with probability $1-e^{-u}$, \[ (P-P_n)D_f\leq C K\sqrt{(r_f+r_g)(u+\log|\mathcal F|)/n} +C K^2(u+\log|\mathcal F|)/n \] for every member. For the empirical minimizer $P_nD_f\leq0$. Absorbing the square-root term into $(r_f+r_g)/2$ gives $r_{\widehat f}\leq3r_g+C K^2(u+\log|\mathcal F|)/n$. Integrate over $u$ to obtain expected risk at most $3r_g+C K^2\log(e|\mathcal F|)/n$. Lemma Lemma (Selection and approximation costs) and the chosen $\epsilon$ prove (1). The cap follows because target and estimator are both in $[-M,M]$.

For fitted propensity, conditional on training the regression is $h=\tau+b_e$, where \[ b_e=(e-\widehat e)\left\{\frac{m_1}{\widehat e} +\frac{m_0}{1-\widehat e}\right\}, \qquad\|b_e\|_2^2\leq C\|\widehat e-e\|_2^2. \] Approximate $\tau$ by the dictionary, then use $\|g-h\|_2^2\leq2\|g-\tau\|_2^2+2\|b_e\|_2^2$ in the finite-regression bound and the analogous inequality for $\widehat f-\tau$. Average over training. This proves the additional error term without imposing smoothness on the jumping treated mean. ∎

Thus randomized assignment permits consistency along every sequence with \[(k_n+1)(\log n/n)^{2/3}+k_n\log(ed_n/k_n)/n\to0.\] This is a sufficient, not necessary, sequence condition. Under unknown propensity one must also establish a vanishing prediction term at the appropriate scale. A fixed-dimensional Hölder regression estimate can supply such a term, but in growing dimension its constants and radius must be tracked; a dimension-free rate is not assumed here.

Two lower bounds within compatible causal models

In both lower bounds use $e=1/2$, $m_0=0$, and $Y\in\{-B,B\}$ with treated mean $\tau$. Assume the fixed variance upper bound permits $B^2$ and any prescribed variance lower bound is strictly smaller than $B^2$. Choose the effect amplitudes small enough to meet that lower bound as well as $B/2$. If necessary, replace $B$ in this lower-bound construction by a smaller positive symmetric outcome scale compatible with the variance bounds; all lower-bound constants then depend on that scale and its positive variance slack. The catalogue only requires that a nondegenerate subclass be present, not a lower variance bound for every law. These laws are generated by conditional independent potential outcomes and randomized treatment. Constant nuisance functions satisfy the smoothness restrictions whenever the stated radii include them.
Theorem (Cost of many BV components): For some fixed constants $c_0,c_1>0$ depending only on the fixed class constants, whenever $d\geq k\geq1$ and $k\leq c_0n^{1/3}$, \[ R_{n,d,k}^\star\geq c_1 k n^{-2/3}. \]
Proof: Take the first $k$ coordinates as active. Partition each univariate domain into $J$ intervals and let $\psi_\ell$ equal $+1$ on the first half of interval $\ell$, $-1$ on its second half, and zero elsewhere. These functions have mean zero, squared norm $1/J$, disjoint supports up to endpoints, and variation at most four. For independent signs $\omega_{j\ell}$ set \[ h_j(u)=a\sum_{\ell=1}^J\omega_{j\ell}\psi_\ell(u), \qquad a=\kappa/J. \] Each component has variation at most $4aJ=4\kappa$, magnitude $a$, and mean zero. Choose fixed $\kappa$ small relative to $L$. The total effect has magnitude at most $ka$, so the condition $k\leq c_0n^{1/3}$ with sufficiently small $c_0$ ensures the bounds $M$, $B/2$, and the specified variance slack when $J$ is of order $n^{1/3}$.

Changing one sign changes the treated mean by $2a\psi_\ell(x_j)$. The Bernoulli divergence inequality gives $n$-sample neighbor KL at most $Cna^2/J$. Choose $J$ of order $n^{1/3}$ and $\kappa$ small so every neighbor total variation is at most $1/2$. The functions $\psi_\ell(x_j)$ are mutually orthogonal across all pairs $(j,\ell)$: within a coordinate this follows from disjoint supports, across coordinates from centering and the product design.

Project any estimator onto these $kJ$ orthogonal functions and decode each coefficient by its sign. Bessel's inequality makes its total squared error at least $a^2/J$ times the number of sign errors. For each bit the average error over a uniform sign prior is at least $(1-{\operatorname{TV}})/2\geq1/4$, by pairing alternatives differing only in that bit. Average risk is at least $kJa^2/(4J)=ka^2/4$, proving the claim. ∎

Theorem (Coordinate selection is a separate cost): For $d\geq16$ and any $k\geq1$, \[ R_{n,d,k}^\star\geq c\min\{1,\log(d)/n\}, \] with a positive constant depending only on the fixed class constants.
Proof: Let $\psi$ be $+1$ on $[0,1/2)$ and $-1$ on $[1/2,1]$. Use alternatives $\tau_j(x)=a\psi(x_j)$, $1\leq j\leq d$, and the zero reference law. Choose $a^2=c_2\min\{1,\log(d)/n\}$ with sufficiently small $c_2$ to satisfy all amplitude and variation bounds. Distinct alternatives have squared distance $2a^2$ by centering and independence. Their KL divergence to the zero law is at most $Cna^2\leq(\log d)/8$ after reducing $c_2$.

For a uniformly distributed alternative index $J_0$, mutual information between $J_0$ and the data is at most the average KL to any reference law, hence at most $(\log d)/8$. For any decoded index with error probability $p$, the entropy chain rule gives $H(J_0\mid\text{data})\leq\log2+p\log d$: reveal whether an error occurred, then at most $d$ possibilities remain on error. Since $H(J_0)=\log d$, it follows that $p\geq1-1/8-\log2/\log d\geq1/2$. Decode an arbitrary function estimator by the nearest $\tau_j$ in $L^2$. On a decoding error its squared distance from the true alternative is at least $(\sqrt{2}a/2)^2=a^2/2$, by the triangle inequality. Its average risk is at least $a^2/4$, proving the bound. ∎

The gap is explicit

For fixed $d,k\geq1$, the nonparametric exponent $2/3$ is matched up to logarithms. For growing $k$, the upper bound records both component and support costs; the lower bounds prove a $kn^{-2/3}$ term in an amplitude-compatible regime and a separate $\log d/n$ obstruction. They do not establish the full $k\log(ed/k)/n$ lower term or remove logarithms. The fixed uniform bound on the sum of components matters: it restricts how many independently large components a lower-bound construction may contain. Finally the additive propensity prediction term is a property of inverse weighting and not a minimax necessity. Establishing matching unknown-propensity rates that exploit smooth $m_0$ while permitting jumps in $m_1$ remains the main causal gap.

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, "New parameters" and "New models," inspected 9 October 2026. https://arxiv.org/html/2508.17099v1#S3.SS5.SSS3.

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