Measuring Full Adoption from Partial Rollout

• With interference, a unit’s outcome can depend on other units’ treatment assignments.
• We want the mean outcome change between nobody treated and everybody treated.
• The rollout reaches only a treated fraction $q<1$.
• Subtracting the baseline from the final measurement estimates the partial-rollout contrast.

Cortez et al. (2022) show how bounded-order interactions can make full-adoption effects identifiable through polynomial extrapolation.

How should we measure a partial rollout to estimate full adoption precisely?

Suppose the mean response is a polynomial of degree at most $\beta$, where $\beta$ is the maximum order.

A natural approach measures at $\beta+1$ equally spaced fractions and extrapolates the fitted polynomial:

$$p^{\mathrm{eq}}(\beta,q)_j=\frac{qj}{\beta}, \qquad j=0,\ldots,\beta .$$

Here $p^{\mathrm{eq}}_j$ is the treated fraction at measurement $j$; $q$ is the final fraction.

• Interpolation gives an unbiased estimate. That alone does not control how strongly its weights amplify noise.
• Cortez-Rodriguez et al. (2024) study clustering to reduce variance for polynomial rollout estimation.
• Choosing measurement fractions is an optimal-design problem, as in Kiefer and Wolfowitz (1959).

Which fractions control the variance cost of extrapolation?

Place more measurements near both endpoints of the observed interval.

These Chebyshev-Lobatto fractions control polynomial growth between measurements. Extrapolation beyond the interval still has an unavoidable cost.

• Choose weights that recover the target for every polynomial of degree at most $\beta$.
• Among those weights, minimize the sum of their absolute values.
• With more than the minimum number of measurements, Chebyshev-Lobatto placement attains the unavoidable exponential variance amplification up to constants.

This conclusion uses a static polynomial mean, correct round expectations, a common variance bound, and a budget bounded away from full adoption.

At $q=0.5$, measurements cover only $[0,0.5]$, while the target includes the response at one.

The Chebyshev-Lobatto measurement fractions are

$$p^{\mathrm{Ch}}_j(k,q) = \frac{q}{2}\left\{1-\cos\left(\frac{\pi j}{k}\right)\right\}.$$

Here $k$ is the number of rollout intervals, giving $k+1$ measurements; $j=0,\ldots,k$ indexes them. The superscript $\mathrm{Ch}$ names this schedule.

• At this budget, the schedule starts at zero, ends at $0.5$, and concentrates measurements near both endpoints.
• The variance-amplification base is approximately $5.83$.
• The matching bounds raise that base to $2\beta$, up to constants.

The design controls what happens inside the observed half; it cannot eliminate extrapolation to one.

For an outcome law $P$, let $m_P(u)$ be the expected population mean at treated fraction $u$.

$$m_P(u)=\sum_{\ell=0}^{\beta} a_{P,\ell}u^\ell$$

The coefficients $a_{P,\ell}$ describe the response curve. This restriction holds throughout $[0,1]$, including the unobserved endpoint.

Our target is

$$\tau_P=m_P(1)-m_P(0)=\sum_{\ell=1}^{\beta} a_{P,\ell}.$$

Thus $\tau_P$ adds the nonconstant coefficients; the intercept cancels.

At measurement fraction $p_j$, we observe the population mean $\bar Y_j$. We require $\mathbb E_\pi[\bar Y_j]=m_P(p_j)$, where $\pi$ denotes rollout randomization.

Static consistency: outcomes depend on current assignments, including others’ assignments, with no separate dependence on earlier rollout steps.

Common variance envelope: every round satisfies

$$\operatorname{Var}_\pi(\bar Y_j)\le \frac{\sigma_0^2}{n}.$$

Here $n$ is population size, and $\sigma_0^2\ge0$ sets the common variance scale. It incorporates outcome magnitudes and dependence across units.

• Cross-round covariances are otherwise unrestricted, provided they form a valid covariance matrix.
• Admissible schedules, denoted $S_{k,q}$, start at zero, increase strictly, and end at $q$.
• Budget cap: $0<q\le q_{\max}<1$, keeping full adoption outside the observed interval.
• Oversampling: $k\ge c\beta$, with fixed $c>1$, gives more than the minimum $\beta+1$ measurements.

Let $\mathcal P_\beta$ denote laws satisfying static consistency, polynomiality, correct round expectations, and the variance envelope.

For the half-rollout example, polynomiality must hold beyond $0.5$, all the way to one. Bounded-order interference motivates this restriction; we impose it.

Estimate the full-adoption contrast with

$$\hat\tau_{w,p}=\sum_{j=0}^k w_j\bar Y_j,$$

where $w_j$ is the weight on measurement $j$, and $p$ is the schedule.

The weights must recover the contrast for every polynomial term:

$$W_\beta(p) = \left\{ w\in\mathbb R^{k+1}: \sum_{j=0}^k w_j p_j^0=0 \ \text{and}\ \sum_{j=0}^k w_j p_j^\ell=1 \ \text{for every } \ell=1,\ldots,\beta \right\}.$$

Here $W_\beta(p)$ is the set of polynomial-exact weights.

• The first equation removes the intercept; $p_j^0$ means one.
• Each remaining equation recovers one nonconstant coefficient’s contribution to $\tau_P$.
• Distinct nodes and $k\ge\beta$ guarantee that these weights exist.

These equations deliver unbiasedness. What determines their variance?

Large positive and negative weights can cancel in the expected estimate while magnifying noise.

For every polynomial-exact weight vector,

$$\operatorname{Var}_\pi\!\left(\sum_{j=0}^k w_j\bar Y_j\right) \le \frac{\sigma_0^2}{n}\left(\sum_{j=0}^k |w_j|\right)^2 .$$

Estimator variance is bounded by the common round variance times the squared sum of absolute weights.

Why absolute values? The envelope permits covariance signs that align with the weight signs. A valid covariance matrix attains this bound.

Unbiasedness constrains signed sums; variance control requires small absolute sums.

First choose the least costly polynomial-exact weights at a fixed schedule:

$$A_\beta(p) = \inf_{w\in W_\beta(p)} \left(\sum_{j=0}^k |w_j|\right)^2.$$

Here $A_\beta(p)$ is the smallest worst-case variance multiplier available at schedule $p$.

Then choose measurement fractions:

$$M_{\beta,k,q} = \inf_{p\in S_{k,q}} A_\beta(p).$$

Here $M_{\beta,k,q}$ is the best multiplier across all admissible schedules with $k+1$ measurements and budget $q$.

Multiplying by $\sigma_0^2/n$ gives the optimal worst-case variance over all valid covariance matrices satisfying the round variance envelope.

$$C_-(q_{\max}) \left(\frac{(1+\sqrt{1-q})^2}{q}\right)^{2\beta} \le M_{\beta,k,q} \le C_+(c,q_{\max}) \left(\frac{(1+\sqrt{1-q})^2}{q}\right)^{2\beta}.$$

The positive constants $C_-$ and $C_+$ are independent of $q,\beta,k$; their arguments show what they depend on.

• Every admissible schedule faces the lower bound.
• Chebyshev-Lobatto placement attains the upper bound.
• Chebyshev-Lobatto placement attains the unavoidable exponential rate up to constants.

At $q=0.5$, with a fixed cap $q_{\max}\ge0.5$ below one, the common base is approximately $5.83$.

Why does measurement placement control this cost?

The weight problem has an equivalent polynomial interpretation:

$$\inf_{w\in W_\beta(p)} \sum_{j=0}^k |w_j| = \sup\left\{ |r(1)-r(0)|: r\in\mathbb R[x],\ \deg(r)\le \beta,\ \max_{0\le j\le k}|r(p_j)|\le 1 \right\}.$$

Here $r$ is a test polynomial of degree at most $\beta$; $\mathbb R[x]$ denotes real polynomials.

The right side asks: How large can the full-adoption contrast be while every measured value stays within one?

• A large hidden contrast forces large absolute weights.
• Choosing fractions therefore means controlling polynomial growth that measurements fail to constrain.

In the half-rollout example, we bound $r$ at measurement fractions in $[0,0.5]$, but its contrast involves $r(1)$.

Measurements must first control the polynomial throughout the observed interval.

Using the Chebyshev-Lobatto mesh inequality, oversampling with $k\ge c\beta$, $c>1$, gives

$$|R(x)| \le K(c) \max_{0\le j\le k}\left|R\!\left(-\cos\!\left(\frac{\pi j}{k}\right)\right)\right|.$$

Here $R$ is a polynomial of degree at most $\beta$, $x\in[-1,1]$, and $K(c)$ is a finite constant depending only on the oversampling ratio.

• Rescaling $[0,q]$ to $[-1,1]$ sends the measurement fractions to the nodes inside the maximum.
• Small values at those nodes imply small values everywhere in the observed interval, up to $K(c)$.
• Oversampling supplies control between measurements.

For the half-rollout example, this step controls the entire interval $[0,0.5]$. How much growth remains outside it?

Under the same rescaling, full adoption becomes the exterior point $x_q=2/q-1>1$.

Its growth factor is $\lambda(q)=x_q+\sqrt{x_q^2-1}$.

For polynomials bounded throughout the rescaled interval,

$$\text{for every real polynomial }R\text{ with }\deg(R)\le \beta \text{ and }\sup_{x\in[-1,1]}|R(x)|\le 1,\qquad |R(x_q)-R(-1)|\le C(q_{\max})\,\lambda(q)^\beta,$$

Here $R(-1)$ corresponds to baseline and $R(x_q)$ to full adoption. The constant $C(q_{\max})$ depends only on the budget cap.

• The mesh inequality controls the observed interval.
• This bound controls the remaining endpoint extrapolation.
• The dual identity converts endpoint growth into absolute-weight size; squaring gives variance amplification.

The growth factor equals the main result’s base, $(1+\sqrt{1-q})^2/q$.

A polynomial bounded throughout $[0,q]$ satisfies the measurement restriction for every schedule.

The Chebyshev polynomial $T_\beta$, of degree $\beta$, supplies such a polynomial after rescaling. Its endpoint contrast is at least $T_\beta(x_q)-1$.

$$T_\beta(x_q)-1\ge c(q_{\max})\,\lambda(q)^\beta .$$

This lower bound uses a positive constant $c(q_{\max})$, distinct from the oversampling ratio $c$.

• Every placement inside the rollout interval admits this test polynomial.
• The dual identity forces absolute-weight size at least proportional to $\lambda(q)^\beta$.
• Squaring gives the matching variance-amplification lower bound.

The unavoidable cost comes from extrapolation. Chebyshev-Lobatto placement prevents additional uncontrolled growth between measurements.

The envelope solution also supplies an upper bound when covariance comes from the rollout itself.

$$\inf_{w\in W_\beta(p^{\mathrm{Ch}}(k,q))} \sup_{P\in\mathcal P_\beta} \operatorname{Var}_\pi\!\left(\sum_{j=0}^k w_j\bar Y_j\right) \le \frac{\sigma_0^2}{n}\, C_+(c,q_{\max}) \left(\frac{(1+\sqrt{1-q})^2}{q}\right)^{2\beta},$$

This bounds the infimum worst-case rollout variance over polynomial-exact weights on the Chebyshev-Lobatto schedule. The ceiling rounds the interval count upward.

The reason is direct: rollout covariance satisfies the envelope, so the envelope upper bound applies.

• Covariance-specific optimality: Does Chebyshev-Lobatto placement minimize worst-case variance under the covariance restrictions of a specified nested rollout? A matching lower bound remains open.
• Clustered rollouts: How should measurement placement account for cluster size and dependence within clusters?
• Mixed designs: How should fractions be chosen jointly with saturation, encouragement, or exposure-based randomization?

• A static degree-$\beta$ mean curve over $[0,1]$, with correct round expectations, makes the full-adoption contrast estimable through polynomial-exact weights.
• Under the common variance envelope, the squared sum of absolute weights is the sharp worst-case variance multiplier.
• With $k\ge c\beta$, $c>1$, and $0<q\le q_{\max}<1$, Chebyshev-Lobatto placement controls growth between measurements and attains the unavoidable exponential amplification up to constants.

Polynomial-exact weights are unbiased, and their squared absolute-weight sum gives the sharp worst-case variance over the diagonal covariance envelope.

With oversampling and budgets bounded away from one, Chebyshev-Lobatto schedules attain matching minimax amplification bounds up to constants.

Under the rollout law restrictions, Chebyshev-Lobatto placement supplies the stated exponential upper bound for worst-case rollout variance.