CausalSmith · seminar slides

Interior Dose-Response Lower Bounds

Under Hölder smoothness, local positivity, bounded outcomes, and strict baseline slack, the minimax mean-squared error for an interior continuous-treatment dose-response partial mean is at least the one-dimensional treatment-regression rate.

slides for A Minimax Lower Bound for Interior Dose-Response Estimation

Overview

We study the target-dose partial mean for a continuous treatment.
The target asks: what is the average outcome if the dose is fixed at an interior value $t_0$ ?
The lower bound is over the same Hölder dose class used in the risk statement.
The obstruction comes from learning the outcome regression locally in the treatment coordinate.
The lower-bound exponent is independent of the treatment-density smoothness $\beta$ , for each fixed $\beta>0$ .

informal · Theorem T-1 Under the stated smoothness, positivity, boundedness, and strict-slack conditions, the minimax MSE is at least a constant times $n^{-2\alpha/(2\alpha+1)}$ .

Motivation

Continuous-treatment dose-response curves are common in policy, medicine, and economics.
A policymaker may ask for the mean outcome at a particular interior dose, rather than for a binary treatment contrast.
Regression adjustment estimates the conditional mean at that dose and averages over the covariate distribution.
The hard part is local: observations near $t_0$ carry the information about the regression value at $t_0$ .
The minimax question asks how small the worst-case squared error can be.

Running Example

Think of $A$ as dose intensity, $Y$ as an outcome, and $X$ as pre-treatment covariates.
We evaluate an interior dose $t_0$ , such as a moderate policy intensity or medication level.
Local positivity says each covariate group has enough probability of receiving doses near $t_0$ .
Hölder smoothness says the regression and density vary regularly near that dose.
The target averages the dose- $t_0$ regression over the population distribution of $X$ .

Target

- One observation is $O=(Y,A,X)$ . - $A$ is a continuous treatment in $[0,1]$ . - $\mu_P(a,x)$ is the conditional mean of $Y$ at dose $a$ and covariates $x$ . - $p_{X,P}$ is the covariate density. - The estimand is $\theta_P(t_0) = \int_{[0,1]^d} \mu_P(t_0,x)\,p_{X,P}(x)\,dx .$

illustrative Box-and-arrow schematic from observed data to the regression, target dose, covariate averaging, target mean, and causal dose-response interpretation under consistency, no confounding, and local positivity.

Identification

The statistical target is an observed-data partial mean.
With consistency, the observed outcome equals the potential outcome at the realized dose.
With no unmeasured confounding, treatment assignment is conditionally independent of potential outcomes given $X$ .
With local positivity, the data contain information near the target dose for every covariate value.
These conditions give the causal reading of $\theta_P(t_0)$ as the dose-response mean at $t_0$ .

Model

Outcomes are uniformly bounded by $M$ .
The dose $t_0$ has a full local window inside $(0,1)$ .
The treatment density is bounded below by $c_0$ on that window.
The treatment regression has Hölder smoothness $\alpha$ in the dose coordinate.
The treatment density has Hölder smoothness $\beta$ in the dose coordinate.
The regression, treatment density, and covariate density have Hölder smoothness $s$ in the covariate coordinate.

Baseline Slack

The lower-bound construction starts from baseline densities $p_0$ and $q_0$ .
Strict slack means these baselines sit inside the smoothness, boundedness, and positivity restrictions with positive margin.
That margin lets us add a local perturbation while staying in the same model class.
In the running example, the baseline population and dose assignment mechanism are regular enough that a small local regression change remains admissible.

Assumption A-12 (Baseline submodel slack)

There exist a covariate density $p_0$ on $[0,1]^d$ , a conditional treatment density $q_0$ on $[0,1]$ , a constant $\eta_0>0$ , and a constant $B_0$ with $0<B_0<M$ , such that $p_0(x)\ge 0$ for all $x\in[0,1]^d$ , $q_0(a)\ge 0$ for all $a\in\mathbb R$ , $\int_{[0,1]^d} p_0(x)\,dx=1 \qquad\text{and}\qquad \int_0^1 q_0(a)\,da=1,$ $\|p_0\|_{\mathcal H^s([0,1]^d)}\le M-\eta_0,$ $q_0$ has Hölder norm at most $M-\eta_0$ of order $\beta$ on $[t_0-\varepsilon_0,t_0+\varepsilon_0]$ , $q_0(a)\ge c_0+\eta_0 \qquad\text{for all }a\in[t_0-\varepsilon_0,t_0+\varepsilon_0],$ and $p_0(x)\le M-\eta_0 \qquad\text{for all }x\in[0,1]^d.$

Related Literature

Rubin (1974), Rosenbaum and Rubin (1983), and Robins (1986) provide the causal potential-outcome and adjustment foundations.
Imbens (2000), Hirano and Imbens (2004), and Imai and van Dyk (2004) develop continuous-treatment propensity and dose-response ideas.
Kennedy et al. (2017), Lee (2018), and Colangelo and Lee (2020) study nonparametric and debiased continuous-treatment estimation.
Bonvini and Kennedy (2022) give the higher-order influence-function benchmark used for comparison.
Stone (1982) and Tsybakov (2009) provide the nonparametric minimax lower-bound toolkit.

Main Result I

informal · Theorem T-1 For every fixed positive $\beta$ , the same-class minimax MSE is at least a constant times the one-dimensional treatment-regression rate under the stated slack-baseline conditions.

Theorem T-1 (Sharp Pointwise Lower Bound)

For every covariate dimension $d$ and every real constants $\alpha,\beta,s,M,c_0,\varepsilon_0,t_0$ , suppose that

(Positive smoothness.) $\alpha>0$ , $\beta>0$ , and $s>0$ .
(Regime constants.) $M>0$ , $c_0>0$ , $t_0\in(0,1)$ , $\varepsilon_0\in(0,1/2)$ , and the local window determined by $(t_0,\varepsilon_0)$ is contained in $(0,1)$ .
(Strict-slack baseline.) Assumption A-12 holds for $(d,\beta,s,M,c_0,\varepsilon_0,t_0)$ .

Then there exists a constant $c>0$ such that, for all sufficiently large sample sizes $n$ , $c\, n^{-2\alpha/(2\alpha+1)} \le R_n\!\left( \mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0), t_0 \right),$ where $\mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0)$ is the model class of Definition P-1 and $R_n$ is the minimax risk of Definition P-3.

Key Idea

Hold the covariate density and treatment density fixed.
Perturb only the outcome regression in a narrow neighborhood of $t_0$ .
The perturbation changes $\theta_P(t_0)$ because the target evaluates the regression exactly at $t_0$ .
The induced data laws remain statistically close because the perturbed region is narrow.
The best test between the two laws cannot reliably detect the target shift.

illustrative Box-and-arrow schematic showing fixed covariate and treatment densities, a local bump near

t_0

, two observed-data laws, statistical closeness, target separation, and the resulting all-

\beta

lower-bound obstruction.

Intuition

A direct plug-in view says the problem is to learn $\mu_P(t_0,x)$ from observations with doses near $t_0$ .
A very narrow neighborhood gives low bias but few effective observations.
A wider neighborhood gives more observations but cannot resolve $\alpha$ -Hölder local variation at $t_0$ .
The lower-bound construction chooses the local bump width that balances these two forces.
Because the treatment density is fixed inside the construction, $\beta$ -smoothness remains part of the class while the hard pair is governed by $\alpha$ .

Smooth Covariates

- The published benchmark rate is $\rho_n = n^{-2\alpha/(2\alpha+1)} \vee n^{-2/(1+d/(4s)+1/\alpha)} .$ - When $d\le 4s$ , the benchmark collapses to the treatment-regression term. - In that regime, the same-class lower floor has the same exponent as $\rho_n$ .

informal · Theorem T-2 In the smooth-covariate regime $d\le 4s$ , the published benchmark equals the treatment-regression rate, and the minimax risk is at least a constant times that benchmark.

Main Result II

informal · Theorem T-4 When $d\le 4s$ , the same-class lower bound is at least a constant times $n^{-2\alpha/(2\alpha+1)}$ , and this rate equals $\rho_n$ .

Theorem T-4 (Smooth Covariate Minimax Floor)

Let $d\in\mathbb N$ and let $\alpha,\beta,s,M,c_0,\varepsilon_0,t_0\in\mathbb R$ . Suppose that:

(Smoothness.) $\alpha>0$ , $\beta>0$ , and $s>0$ .
(Regime constants.) $M>0$ , $c_0>0$ , $t_0\in(0,1)$ , $\varepsilon_0\in(0,1/2)$ , and the window $[t_0-\varepsilon_0,t_0+\varepsilon_0]$ is contained in $(0,1)$ .
(Smooth-covariate regime.) $d\le 4s$ .
(Strict-slack baseline.) Assumption A-12 holds for $(d,\beta,s,M,c_0,\varepsilon_0,t_0)$ .

Then there exists a constant $c>0$ such that, for all sufficiently large $n$ , $c\,n^{-2\alpha/(2\alpha+1)} \le R_n\!\left( \mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0),t_0 \right),$ where $R_n$ is the minimax risk of Definition P-3 over the same-class model of Definition P-1. Moreover, $n^{-2\alpha/(2\alpha+1)} = \rho_n,$ with $\rho_n$ as in Definition P-4.

Low Covariate Smoothness

When $4s<d$ , the published benchmark is governed by the covariate-smoothness term.
The lower-bound construction still gives the treatment-regression floor.
The two exponents are strictly ordered in this regime.
The result identifies the algebraic gap between the same-class lower floor and the published benchmark sequence.

informal · Theorem T-5 When $4s<d$ , the minimax risk is still at least the treatment-regression lower floor, while the published benchmark has a strictly smaller exponent.

Theorem T-5 (Deficient Frontier Bracket)

Fix a covariate dimension $d\in\mathbb N$ and constants $\alpha,\beta,s,M,c_0,\varepsilon_0,t_0\in\mathbb R$ . Assume:

(Regime constants.) $\alpha>0$ , $\beta>0$ , $s>0$ , $M>0$ , $c_0>0$ , $t_0\in(0,1)$ , $\varepsilon_0\in(0,1/2)$ , and the window $[t_0-\varepsilon_0,t_0+\varepsilon_0]$ lies in $(0,1)$ .
(Deficient covariate smoothness.) $4s<d$ .
(Strict-slack baseline.) Assumption A-12 holds for $(d,\beta,s,M,c_0,\varepsilon_0,t_0)$ .

Then there exists a constant $c>0$ such that, for all sufficiently large sample sizes $n$ , $c\,n^{-2\alpha/(2\alpha+1)} \le R_n\!\left( \mathcal P_{\alpha,\beta,s}(M,c_0,\varepsilon_0,t_0),t_0 \right),$ where the minimax risk is as in Definition P-3 and the class is Definition P-1. Moreover, for the published HOIF benchmark of Definition P-4, $\rho_n = n^{-\,2/(1+d/(4s)+1/\alpha)}$ and its exponent is strictly smaller than the certified lower-bound exponent: $\frac{2}{1+d/(4s)+1/\alpha} < \frac{2\alpha}{2\alpha+1}.$

This separates the certified same-class lower floor from the published benchmark exponent in the low-covariate-smoothness regime.
The comparison is exact at the level of the displayed benchmark algebra.

Also in the Paper

informal · Gate L-17 Under the stated Bonvini-Kennedy localized-regularity conditions, the cited higher-order influence-function estimator attains conditional mean-squared error at most a constant times $\rho_n$ .

informal · Definition P-5 The beta comparison handle records the all- $\beta$ lower floor, the two benchmark regimes, and the possible same-class upper-frontier alternatives.

Proof Sketch

Build two laws with the same covariate density and the same treatment density.
Add a local Hölder-compatible regression bump near $t_0$ under one law.
Use bounded two-point outcome channels so the target shift is carried by the conditional mean.
Calibrate the bump so the $n$ -sample laws have bounded Kullback-Leibler divergence.
Apply Le Cam’s two-point argument to convert indistinguishability into a squared-error lower bound.

Takeaways

The interior dose-response partial mean has a same-class minimax lower floor at $n^{-2\alpha/(2\alpha+1)}$ .
The floor holds for every fixed $\beta>0$ under the corresponding strict-slack baseline condition.
When $d\le 4s$ , this exponent matches the published higher-order influence-function benchmark exponent.
When $4s<d$ , the benchmark exponent and the certified same-class lower-bound exponent are strictly ordered.
The comparison clarifies exactly which rate facts are established for the Hölder dose class and which upper-side questions remain for the low- $s$ regime.