Introduction

This paper identifies the exact regular local asymptotic minimax squared-error constant for estimating a treatment effect when both treatment assignment and outcome are locally private. The experiment consists of independently sampled subjects in a randomized binary trial, with a known assignment probability strictly between zero and one, interior outcome probabilities in both arms, and a fixed positive privacy budget. Each subject contributes one private assignment–outcome record. Release rules may depend on preceding private outputs and may take values in arbitrary measurable spaces. Within this model, the stationary optimal contrast variance is the reciprocal of a finite nuisance-profiled staircase optimization. Fixing any measurable strong-saddle selector satisfying the stated conditions on the interior parameter domain, a private-pilot estimator attains the constant and supports pointwise Wald inference.

Randomization identifies the average treatment effect as the difference between the treatment-arm and control-arm outcome probabilities. Joint local privatization determines how much information about that difference reaches the analyst. A release must satisfy the privacy inequality across all four assignment–outcome pairs, including pairs that differ in both coordinates. The resulting design problem concerns the scalar treatment-effect contrast within a two-parameter model: both arm means vary, and their common shift is a nuisance direction. Efficient release selection therefore depends on how the channel preserves information about the contrast after accounting for that nuisance variation.

Our first result connects this statistical objective to finite channel geometry. Under the stated interiority and fixed-privacy conditions, Theorem 1 characterizes the smallest stationary contrast variance over all measurable output spaces. Its reciprocal is obtained by maximizing nuisance-profiled information over weights on fourteen nonconstant staircase patterns. For each fixed profiling direction, information is linear in these weights; minimization over the common-mean direction produces the saddle objective. Equivalently, the optimized information is the minimum of a finite upper envelope of convex quadratics. This representation gives an exact finite optimization at each specified pair of arm means, assignment probability, and privacy budget.

The stationary optimum also governs the sequential experiment. Conditional on a fixed measurable selector satisfying the strong-saddle conditions in Algorithm 1, Theorem 2 establishes a local asymptotic squared-error lower bound for every sequence of sequential private protocols and transcript-measurable estimators. The criterion takes the worst risk over bounded admissible local perturbations, then the lower asymptotic limit, and finally the supremum over perturbation radii. The same constant is attained within the regular class by a private-pilot construction with pilot size equal to the integer part of the square root of the total sample size. Thus history adaptation and arbitrary measurable releases are incorporated into the exact efficiency comparison.

The construction first estimates the arm means using four-category randomized response and then selects a staircase channel for the remaining subjects. Its treatment-effect estimate combines the pilot contrast with an affine correction computed from the main-sample releases. Under a fixed measurable strong-saddle selector, Theorem 3 establishes exact conditional centering given the private pilot history, regular asymptotic normality, consistent empirical variance estimation, and pointwise asymptotic Wald coverage. These conclusions hold for every pilot schedule satisfying Assumptions 10 and 11. They apply throughout the interior parameter domain, including points where the optimal channel support changes.

The optimal output alphabet has a corresponding statistical structure. Theorem 4 guarantees an attaining stationary staircase release with at most five active outputs at every interior design point. At assignment probability $3/10$ and privacy budget $\log 3$, Theorem 7 identifies open neighborhoods of arm means $(1/2,1/2)$ and $(1/10,1/10)$ in which the minimum attaining cardinalities among all stationary channels are five and three, respectively. Under balanced assignment and complementary arm means, Theorem 8 instead yields an optimal two-output randomized-response release. These results establish specific optimal release cardinalities under contrast-specific efficiency at selected outcome prevalences and designs.

The analysis connects two established strands of private inference. Steinberger (2024) develops sequential asymptotic efficiency theory under his stated measurable-space and information-limit conditions, and identifies scalar functionals of inverse information as objectives in multiparameter models. Here the four-record trial model permits an exact nuisance-profiled contrast calculation over arbitrary measurable sequential output spaces. The staircase geometry follows the extremal-mechanism framework of Kairouz et al. (2016); nuisance profiling gives the present optimization its concave weight objective and associated support structure. The exact fixed-privacy constant complements the statistical-rate analyses of Duchi et al. (2018); Rohde et al. (2020); Duchi et al. (2024).

For a direct causal benchmark, we evaluate the known-assignment custom-Laplace release and sample-mean estimator of Ohnishi et al. (2025). At assignment probability $1/2$, both arm means $1/2$, and privacy budget one, Theorem 9 establishes a strict variance improvement for the efficient procedure. The implied asymptotic half-length ratio is approximately $2.69$, with the custom-Laplace sample mean in the numerator; the corresponding sample-size ratio at equal precision is approximately $7.26$. We illustrate these quantities through a prospective binary endpoint defined by meeting the $80\%$ attendance threshold of the Colombian cash-transfer program studied by Barrera-Osorio et al. (2011).

The paper next reviews related work, defines the private trial model, derives the efficiency bound and finite optimization, develops private-pilot inference, studies optimal release cardinality, and gives the interval comparison and prospective design calculation. Discussion is followed by the proof exposition in Section 9 and the appended proofs.

Related work

The closest asymptotic efficiency framework is that of Steinberger (2024). For parametric models differentiable in quadratic mean, his sequential theory establishes local asymptotic mixed normality along subsequences with convergent conditional information, under countably generated input and output sigma-fields. The convolution theorem additionally imposes positive-definiteness conditions on the conditional information matrices and their limiting accumulation points. His scalar optimization maximizes output Fisher information, and the private two-step procedure uses a preliminary estimate to select an asymptotically optimal mechanism under the stated regularity conditions. Remark 3.6 explicitly identifies scalar functionals of inverse information as objectives for multiparameter models. Our analysis treats a treatment-effect contrast in a two-parameter randomized binary trial with binary outcomes and known interior assignment probability, as specified in Definition 1. At interior arm means and a fixed positive privacy budget, Theorems 1 and 2 connect finite channel optimization with the exact efficiency bound for the stated class of regular procedures using arbitrary measurable sequential output spaces. The scalar target therefore retains a nuisance parameter throughout the optimization.

The finite reduction is closely related to the staircase mechanisms of Kairouz et al. (2016). Their utility class consists of sums over output columns of a sublinear function, meaning a positively homogeneous, subadditive function of the column. For this class, they establish an optimal staircase mechanism with output alphabet no larger than the input alphabet and express the optimum as a finite linear program. The present fourteen-pattern staircase program, characterized in Definition 4, Definition 5, and Theorem 1, combines this finite channel geometry with nuisance profiling. It also parallels contrast-specific optimal-design geometry: Elfving (1952) represents optimal designs through a convex set generated by signed regression vectors, and Pukelsheim (2006) develops the corresponding information-matrix framework. For a fixed parameter direction, directional Fisher information here is additive over staircase outputs and linear in their weights. Contrast information minimizes that directional information over directions normalized to have unit treatment-effect derivative. This minimization makes the objective concave in the staircase weights and gives the optimization its saddle structure. The relevant support geometry is consequently governed jointly by channel normalization and profiling. Under the conditions stated in the support results, Theorem 4 supplies a five-output upper bound, while Theorem 5 identifies five-output optimality through its certificate and score-separation conditions. Comparison by randomization provides the broader experiment-ordering context (Blackwell, 1953); the results characterize contrast-specific efficiency within this framework.

The nuisance projection and local risk formulation follow the local-asymptotic tradition of Le Cam et al. (2000) and van der Vaart (1998). The sequential lower bound uses the Bayesian information-inequality lineage of Van Trees (2001) and Gill et al. (1995). These references provide intellectual antecedents; the formal verification contract records no external theorem as a Lean proof dependency.

The direct causal benchmark is Ohnishi et al. (2025). Their randomized-experiment model permits potential outcomes in the unit interval. Lemma 4 gives a worst-case squared-error lower bound for privacy budgets between zero and one. In the known-assignment custom scenario, each subject forms the inverse-probability-weighted observed outcome and adds independent Laplace noise calibrated to the sum of the reciprocal treatment and control assignment probabilities. The sample mean of these releases is unbiased and consistent; Theorem 7 supplies asymptotic normality, a variance estimate, confidence intervals, and the privacy-dependent convergence rate. These statements provide both a rate benchmark and a concrete inferential procedure. For binary outcomes, our comparison uses precisely this release and sample-mean estimator, specified in Definition 14. The comparison in Theorem 9 evaluates its asymptotic variance against the optimized contrast bound at the stated trial parameters and privacy budget.

These exact variance comparisons complement the literature on statistical rates under local privacy. In the bounded scalar-mean problem, Duchi et al. (2018) obtain squared-error order $\min \{1,(n{\epsilon }^2{)}^{-1}\}$, up to numerical constants, in the high-privacy regime $0<\epsilon \le 1$. Ohnishi et al. (2025) obtain the same benchmark order for their bounded-outcome randomized-experiment model. This rate describes the joint sample-size/privacy scaling; the fixed-privacy efficiency constant derived here resolves the model- and design-specific numerical factor that rate notation suppresses. Rohde et al. (2020) characterize the difficulty of functional estimation through total-variation moduli, including rate-optimal binary releases for linear functionals over convex models. Instance-specific local minimax complexity is developed further by Duchi et al. (2024). Information-based approaches also include the Fisher-information contraction inequalities of Barnes et al. (2020) and the communication-based estimation lower bounds of Duchi et al. (2019). Here the finite binary-trial experiment permits an exact optimization of the asymptotic contrast variance at a fixed privacy budget.

Exact scalar mechanisms provide a further point of comparison. For a Gaussian mean with known unit variance, Kalinin et al. (2025) identify randomized response applied to the centered sign as an information-maximizing mechanism in their high-privacy regime, and attain efficiency through a private preliminary estimate. The private-pilot construction in Algorithm 1 and Theorem 3 uses the same broad principle of learning the parameter before selecting the main-sample release. Its channel selection solves the nuisance-profiled contrast problem, with pilot size diverging and occupying a vanishing fraction of the sample under Assumptions 10 and 11.

Recent work further develops extremal-channel optimization. Amorino et al. (2025) study factorization through extremal mechanisms and Fisher-information maximization in discrete and continuous settings, while Amorino et al. (2026) analyze symmetry and asymmetry in staircase information across privacy regimes. For binary hypothesis testing between two known finite-alphabet distributions, Ghazi et al. (2026) optimize output $f$-divergences through likelihood-ratio ordering, contiguous partitions, and randomized response. These objectives illuminate the role of staircase structure. Our contrast objective additionally accounts for variation in both arm means, and its support certificates characterize the resulting release cardinality.

Finally, causal efficiency connects the analysis to Hahn (1998); Hirano et al. (2003), whose work studies efficient average treatment-effect estimation under unconfoundedness. In the private setting, Kormos et al. (2026) construct suitable identifying mechanisms that transfer rate-double-robust and semiparametric inferential properties to noisy observations, including for causal targets. The present binary randomized-trial model makes the release mechanism itself an object of efficiency optimization: the channel, nuisance adjustment, and attainable treatment-effect variance are characterized together under the stated sequential privacy conditions.

Trial model and private releases

Throughout, $n$ denotes the positive sample size, subjects are indexed by $i=1,\dots ,n$, record categories by $j\in \{0,1,2,3\}$, and treatment arms by $a\in \{0,1\}$. Thus subject indices are one-based while the four-record alphabet is zero-based. Vectors are columns, ${}^{\top }$ denotes transpose, and $\Vert \cdot \Vert$ is the Euclidean norm. Generic positive constants are denoted by $C$; $O_p$ and $o_p$ denote stochastic order, and $\Rightarrow$ denotes convergence in distribution.

Data generation

For subject $i$, let $Y_i(0)$ be the control potential outcome and $Y_i(1)$ the treatment potential outcome. The binary assignment $W_i$ indicates treatment, and $Y_i$ denotes the observed outcome. The subject’s private record is the ordered assignment–outcome pair $X_i$, written $X_i=(W_i,Y_i)$.

Write $p$ for the treatment-assignment probability and $q$ for the control-assignment probability, with $q=1-p$. For arm $a$, the population mean ${\mu }_a$ is ${\mu }_a=E[Y_i(a)]$; thus ${\mu }_0$ and ${\mu }_1$ are the control-arm and treatment-arm means. Collect these means in the parameter vector $\theta$, with $\theta =({\mu }_0,{\mu }_1{)}^{\top }$. The average treatment effect $\tau$ and its contrast vector $c$ satisfy $$c=(-1,1{)}^{\top },\tau =E[Y_i(1)-Y_i(0)]={\mu }_1-{\mu }_0=c^{\top }\theta .$$ The following data-generation conditions connect this causal target to the distribution of the observed records.

Independent superpopulation sampling makes each subject a draw from the same population and gives the sample a product structure (Ohnishi et al., 2025). Within each subject, the joint distribution of the two potential outcomes may accommodate dependence. Treatment assignment is governed by the next two conditions.

Random assignment makes the potential-outcome distribution in each assigned arm representative of the corresponding population distribution (Ohnishi et al., 2025). Its probability is a known feature of the trial design.

The known Bernoulli assignment probability condition treats $p$ as a design quantity when estimating the arm means (Ohnishi et al., 2025). To specify the record that enters the privacy mechanism, we next connect assignment to the observed outcome.

Potential-outcome consistency identifies the observed response with the potential outcome under the assigned treatment (Ohnishi et al., 2025). The outcome restriction then gives the record a finite alphabet.

The binary potential outcomes condition describes trials with a dichotomous response, such as whether a specified event occurs (Ohnishi et al., 2025). Under Assumptions 4 and 5, the private record takes values in the ordered four-point alphabet $\mathcal{X}$: $$\mathcal{X}=\{x_0,x_1,x_2,x_3\},x_0=(0,0),x_1=(0,1),x_2=(1,0),x_3=(1,1).$$ Let the record law $P_{\theta }$ have probability vector ${\pi }_{\theta }$, whose coordinates are ${\pi }_{\theta ,j}=P_{\theta }(X_i=x_j)$. Random assignment and consistency give $${\pi }_{\theta }=\left(\begin{array}{c}q(1-{\mu }_0)\\ q{\mu }_0\\ p(1-{\mu }_1)\\ p{\mu }_1\end{array}\right).$$ Thus the distribution of the observed record is determined by the two arm means and the known assignment probability.

Interiority

The analysis uses an interior parameter domain. The next two conditions ensure positive assignment probabilities and positive record probabilities throughout that domain.

Strict treatment overlap assigns positive probability to each arm, so both arm means enter the observed-record distribution (Ohnishi et al., 2025). We also impose interiority on those means.

The interior Bernoulli parameter condition gives each outcome category positive probability within its arm and provides an open parameter domain for information calculations (Steinberger, 2024). Together with strict overlap, it makes all four coordinates of ${\pi }_{\theta }$ positive. We collect the resulting product experiments in the following model.

In Definition 1, the assignment probability remains fixed as the arm means vary. This separates the known trial design from the two-dimensional outcome parameter whose contrast is the treatment effect.

Private releases

A stationary locally private channel, denoted by $Q$, maps each private record to a randomized output on a measurable space $(\mathcal{Z},\mathcal{G})$. Stationarity means that the same release rule is used for each subject, with fresh randomization. For every measurable event $A$ and every pair of records $x,x^{\prime }\in \mathcal{X}$, its local privacy requirement is $$Q(A\mid x)\le e^{\epsilon }Q(A\mid x^{\prime }).$$ Here $\epsilon$ is the privacy budget. Because the inputs are assignment–outcome pairs, the comparison protects assignment and outcome jointly: $x$ and $x^{\prime }$ may differ in either or both coordinates.

The privacy budget is held fixed across the sequence of experiments.

The fixed local privacy budget condition keeps the per-subject protection level constant as sample size grows (Steinberger, 2024). It therefore supplies a common privacy constraint for comparing release rules.

Sequential releases allow the rule for a new subject to depend on earlier private outputs. Let $Z_i$ denote subject $i$’s released observation, taking values in the measurable output space $({\mathcal{Z}}_i,{\mathcal{G}}_i)$. The preceding-release history $H_{i-1}$ is $$H_{i-1}=(Z_1,\dots ,Z_{i-1}),$$ with the empty history at $i=1$. Conditional on the current record and this history, the release rule $Q_i$ generates $Z_i$ according to $Q_i(\cdot \mid X_i,H_{i-1})$. Each subject contributes one release; adaptation proceeds through the outputs of preceding subjects. The privacy requirement applies uniformly over the full history space.

Sequentially interactive local differential privacy permits history-dependent release rules while retaining the same recordwise protection at each stage (Steinberger, 2024). The quantification over every history makes the privacy guarantee a property of the release rule throughout its domain, including histories with zero probability under a particular parameter. The protocol class ${\mathfrak{Q}}_{n,\epsilon }^{\mathrm{seq}}$ combines this guarantee with measurability and nonanticipation.

Nonanticipation in Definition 2 restricts each rule to the current subject’s record and preceding releases. A stationary channel yields a member of this class by taking $Q_i(\cdot \mid x,h)=Q(\cdot \mid x)$. More generally, earlier private releases may determine the channel used for later subjects.

Output information and cardinality

Efficiency comparisons use the output Fisher-information matrix $I_{\theta }(Q)$ of a stationary channel and the active-output cardinality $\kappa (Q)$. The following definition fixes these quantities for measurable output spaces and specifies how singular information is treated when evaluating the treatment-effect contrast.

For a stationary channel $Q$ on $(\mathcal{Z},\mathcal{G})$, let $$\nu =\sum _{j=0}^{3}Q(\cdot \mid x_j),k_j=\frac{dQ(\cdot \mid x_j)}{d\nu },f_{\theta }=\sum _{j=0}^3{\pi }_{\theta ,j}{k}_j.$$ Holding $Q$ and $p$ fixed, define $$I_{\theta }(Q)={\int }_{\{f_{\theta }>0\}}\frac{({\nabla }_{\theta }{f}_{\theta })({\nabla }_{\theta }{f}_{\theta }{)}^{\top }}{f_{\theta }}d\nu .$$ The contrast variance is $c^{\top }{I}_{\theta }(Q{)}^{+}c$ when $c\in \mathrm{range}(I_{\theta }(Q))$, and $+\infty$ otherwise. Let $P_{\theta }Q$ be the output law and $(P_{\theta }Q{)}^{\ast }$ its induced outer measure. For every finite underlying output space, write $$\kappa (Q)=\#\left\{z\in \mathcal{Z}:(P_{\theta }Q{)}^{\ast }(\{z\})>0\right\};$$ for every infinite underlying output space, set $\kappa (Q)=+\infty$. On a finite power-set alphabet each singleton is measurable, so $(P_{\theta }Q{)}^{\ast }(\{z\})=(P_{\theta }Q)(\{z\})$, recovering the ordinary count of positive-probability output labels. The outer-measure form also gives the stated convention when a finite carrier has a coarser sigma-field.

This density formulation expresses information directly in terms of the released observation. The range condition accommodates channels that retain information about the treatment-effect contrast while their information matrix is singular in the full arm-mean parameter. Its extended-value convention makes estimability part of the channel comparison.

For finite power-set alphabets, $\kappa (Q)$ counts output labels with positive probability. Since every input record has positive probability under Definition 1, an output label is active exactly when at least one of the four conditional release probabilities is positive. The count is therefore independent of the interior arm means and measures the number of active labels used by a fixed finite release rule.

Exact efficiency bound and finite optimization

The contrast-variance convention from the preceding section provides a common criterion for stationary private channels, including channels with singular information matrices that identify the treatment effect. We now characterize its optimum by a finite staircase program and connect that optimum to local squared-error risk for the sequential protocols in Definition 2. Throughout this section, assignment is interior, the two arm means are interior, and privacy is fixed as specified in Assumptions 6, 7, and 8.

Staircase channels and nuisance projection

A staircase release associates each output with a nonconstant pattern on the four-record alphabet. Its conditional release probabilities take two values whose ratio is $e^{\epsilon }$. The following enumeration fixes the fourteen patterns $S_j$ used to index channel weights and, subsequently, active supports.

The channel is determined by a nonnegative weight vector $\alpha$. Row normalization restricts these weights to a polytope, while the record probabilities in Definition 1 determine the information carried by each pattern. The next definition gathers the normalization constraints and the directional information objective.

The privacy increment $d$ determines the two entries of each pattern vector $b_S$. The feasible set ${\mathcal{A}}_{\epsilon }$ enforces all four channel row sums simultaneously. For each pattern, the denominator $h_S(\theta )$ is its probability factor, and the vector $g_S$ is the derivative of that factor with respect to the arm means. These quantities give a direct construction of the staircase channel $Q_{\alpha }$ and its projected output score ${\rho }_S(\theta ,t)$.

For $\alpha \in {\mathcal{A}}_{\epsilon }$, the stationary staircase channel on $\mathcal{S}$ is $$Q_{\alpha }(S\mid x_k)={\alpha }_S{b}_S(k),\underset{\theta }{\mathrm{Pr}}(S)={\alpha }_S{h}_S(\theta ).$$ For an active pattern $S$, its output score and projected score are $$s_S^{\mathrm{out}}(\theta )=\frac{g_S}{h_S(\theta )},{\rho }_S(\theta ,t)=\frac{g_S^{\top }v(t)}{h_S(\theta )}.$$ Thus $I_{\theta }(Q_{\alpha })=I_{\theta }(\alpha )$ and $$\kappa (Q_{\alpha })=\#\{S\in \mathcal{S}:{\alpha }_S>0\}.$$ Write $\mathrm{vert}({\mathcal{A}}_{\epsilon })$ for the vertices of the feasible weight polytope.

Feasibility in Definition 4 makes these release probabilities sum to one for every record. Within an active output, their largest-to-smallest ratio is $e^{\epsilon }$. The output score $s_S^{\mathrm{out}}(\theta )$ therefore yields the staircase information matrix $I_{\theta }(\alpha )$ directly from the released pattern.

Nuisance projection enters through the direction $v(t)$. Since $c^{\top }v(t)=1$, these directions parameterize all arm-mean perturbations that change the treatment effect by one unit. Varying $t$ adds a common shift to both arm means, which leaves their difference unchanged. Minimizing the directional information $F_{\theta }(\alpha ,t)$ over $t$ thus profiles out that common-mean nuisance direction. When the contrast is estimable, the resulting minimum is the reciprocal of the contrast variance under the preceding convention.

Optimizing this profiled value over feasible releases defines the information benchmark $J^{\ast }(\theta ,p,\epsilon )$ and its reciprocal variance benchmark $V^{\ast }(\theta ,p,\epsilon )$.

The stationary comparison ranges over every measurable output space, with contrast estimability incorporated into its extended-value criterion. The following result identifies its optimum with the finite staircase benchmark and gives an equivalent optimization over the vertices of the weight polytope.

The finite reduction reflects two features of the objective: it is linear in the channel weights for a fixed profiling direction, and it is quadratic in the profiling coordinate for fixed weights. Accordingly, Theorem 1 expresses the optimized information as the minimum of a finite upper envelope of convex quadratics. This gives a one-dimensional representation of the efficiency surface at each specified pair of arm means, assignment probability, and privacy level. The contrast-variance convention also permits a singular channel to attain the optimum when its information range contains the contrast.

The staircase refinement underlying the comparison with arbitrary stationary output spaces is presented in Section 9. For the sequential lower-bound construction, we also record the smooth scalar localization density used there.

The density in Definition 15 combines compact localization with vanishing density and derivative at the endpoints. Its prior-information contribution is explicit and decreases as the localization radius grows. The directional construction and sequential lower-bound apparatus using this density are deferred to Section 9.

Local alternatives and regular procedures

We first define the local perturbations used to compare procedures near an interior baseline. A perturbation of order $n^{-1/2}$ changes the treatment-effect contrast on the same scale as the estimator’s limiting error.

The corresponding scaled error centers the estimate at the contrast under this local parameter, so fixed perturbations can share a common limiting error law.

Regularity combines a common local limiting law with squared-error tail control on bounded local neighborhoods. The next definition packages the admissible perturbations, procedure sequence, scaled error, and regular class used by the minimax theorem.

The common limit makes local error behavior comparable across fixed perturbations, while uniform integrability connects convergence in distribution to squared-error risk. We now turn to the pilot allocation used for attainment. Its first requirement supplies an increasing number of private observations for learning the arm means (Kalinin et al., 2025).

The second pilot requirement appears next. Together, the two conditions let the pilot learn the arm means while reserving an asymptotically full fraction of subjects for the selected channel.

Private pilot, channel selection, and sequential efficiency

The pilot fraction vanishes, subject to the finite-sample restrictions that give both stages observations once $n\ge 2$.

The preliminary release is four-category randomized response on the zero-based record alphabet. Its probabilities are affine in the record probabilities, so their inverse estimates the two success cells used for the arm means.

Inverting these probabilities and clipping the result supplies the interior pilot estimate $\tilde{\theta }$. Under interior assignment and positive privacy, a measurable strong-saddle selector exists; the construction fixes any such selector. It jointly returns feasible channel weights, a minimizing profiling coordinate, and the optimized information value. Conditional on the pilot, the selected channel and projected-score correction are fixed.

Selection depends only on released pilot observations, so the construction belongs to the history-dependent protocol class in Definition 2. The theorem below uses distinct quantifiers for its two conclusions: the lower bound applies to every sequential private procedure sequence with measurable estimators, while attainment and the infimum equality are stated within the regular class. Both conclusions allow arbitrary measurable release spaces and history-dependent channels, and attainment retains the selector condition in Algorithm 1.

The equality in Theorem 2 connects the stationary channel problem to the full sequential experiment: the same variance constant governs the local risk lower bound and is attained by the regular private-pilot construction. Intuitively, the diverging pilot learns the design point, while its vanishing sample fraction leaves the first-order allocation to the selected staircase release. With the fixed measurable strong-saddle selector, the schedule $m_n=\lfloor \sqrt{n}\rfloor$ implements this allocation and attains the benchmark.

The risk criterion first takes the worst squared error over admissible perturbations at a fixed radius, then its lower asymptotic limit, and finally the supremum over radii. Under these quantifiers, Theorem 2 identifies $V^{\ast }({\theta }_0,p,\epsilon )$ as the exact regular local asymptotic squared-error constant at the specified interior design point.

Private-pilot estimation and Wald inference

The efficiency benchmark in Theorem 2 is attained by a procedure that uses private pilot releases to select the main-sample channel. The construction combines the pilot treatment-effect estimate with an affine correction computed from the remaining releases. Under interior assignment and positive privacy, a measurable strong-saddle selector exists. The procedure fixes any such selector satisfying the interface in Algorithm 1: at every interior parameter, it returns feasible staircase weights, a minimizing profiling coordinate, and the optimal information value, with the selected weights also maximizing the objective at that coordinate.

Two restrictions govern the pilot allocation: $$m_n⟶\infty ,\frac{m_n}{n}⟶0,$$ as specified in Assumptions 10 and 11. The first supplies an increasing number of private observations for learning the arm means; the second allocates an asymptotically full fraction of subjects to the selected main-sample channel. The attainment result below applies to an arbitrary schedule satisfying these restrictions. The schedule $m_n=\lfloor \sqrt{n}\rfloor$ exhibited for the minimax equality in Theorem 2 is one such choice.

To describe the estimator, we first specify the selected affine correction $\tilde{\varphi }(S)$. It combines the selected profiling direction with the pattern-specific score and normalizes by the selected information value.

The pilot determines both the release probabilities and the numerical correction attached to each main-sample pattern. Conditional on the private pilot history $H_{m_n}=(R_1,\dots ,R_{m_n})$, these choices are fixed. The remaining subjects then supply releases under the selected channel. The following boxed algorithm collects the implementation in Algorithm 1, displaying the steps for sample sizes with $1\le m_n<n$, as eventually implied by the two pilot-size restrictions.

Clipping places the pilot estimate in the interior parameter space where the selector conditions apply. The main-sample correction then targets the discrepancy between the pilot contrast and the true treatment effect. The conditional unbiasedness clause in Theorem 3 gives this centering an exact interpretation: the estimation error has mean zero conditional on the private pilot history. Consequently, the theorem accommodates every pilot schedule satisfying the two stated limits.

For studentization, use the empirical variance estimator ${\hat{V}}^{\ast }$. It centers the main-sample corrections at their empirical mean, so the pilot contrast, which is fixed conditional on the pilot history, contributes no term to this empirical variance.

For a nonempty main sample, this is the empirical second central moment with divisor $n-m_n$. Its zero-based transcript indexing corresponds to the one-based main-sample indices in the algorithm. Under the attainment conditions, it consistently estimates the asymptotic variance constant $V^{\ast }(\theta ,p,\epsilon )$ defined in Definition 5. The resulting standard error is $\sqrt{{\hat{V}}^{\ast }/n}$, reflecting the theorem’s normalization by the total sample size.

Fix the Wald error probability $\gamma$ in $(0,1)$, and let $\Phi$ denote the standard normal distribution function. The normal critical value $z_{1-\gamma /2}$ uses the generalized-inverse convention $$z_{1-\gamma /2}=\inf \{z\in \mathbb{R}:1-\gamma /2\le \Phi (z)\}.$$ Together, the selected estimator and empirical variance yield the interval $$\left[{\hat{\tau }}^{\ast }-z_{1-\gamma /2}\sqrt{{\hat{V}}^{\ast }/n},{\hat{\tau }}^{\ast }+z_{1-\gamma /2}\sqrt{{\hat{V}}^{\ast }/n}\right].$$ The next result establishes privacy, conditional centering, regular attainment, variance consistency, and pointwise coverage for this construction.

The construction separates learning the release rule from centering the treatment-effect estimate. The increasing pilot informs channel selection, while the affine correction exactly centers the estimator conditional on that selection. A vanishing pilot fraction then leaves the main sample to determine the first-order variance. Under the measurable strong-saddle conditions, these features deliver the optimized variance for every admissible pilot schedule, including the exhibited schedule used in Theorem 2.

Regularity strengthens the fixed-parameter normal approximation by giving the same limiting error law under every fixed local perturbation, with centering at the corresponding treatment effect. The additional uniform-integrability clause controls squared-error tails over bounded admissible perturbation sets. Together, these conclusions connect the estimator’s distributional behavior to the regular squared-error benchmark in Theorem 2.

The variance-consistency and coverage conclusions apply pointwise at every interior arm-mean parameter satisfying the stated conditions, including parameter points at which the optimal channel support changes. At such points, the measurable strong-saddle selector remains the governing selection rule, and the empirical variance estimates the same optimized variance constant appearing in the normal limit. Thus the procedure supplies a selected private release rule, a treatment-effect estimate, and an asymptotically valid Wald interval throughout this interior domain.

Optimal release cardinality

Output cardinality is a property of the stationary release design that attains the optimized treatment-effect information. Although the optimization built from the fourteen patterns in Definitions 3 and 4 uses all candidate rays, an efficient design can always be chosen with at most five active outputs. The following result establishes this bound throughout the interior trial model.

The fourteen patterns provide a common representation for the optimization, while the active weights determine the alphabet used by a particular release. Theorem 4 therefore gives a uniform bound on the number of output symbols needed to attain the optimized profiled information. Its practical content is an existence guarantee: at each admissible parameter point, the efficient stationary experiment admits a release with at most five active symbols.

To identify where particular supports attain the optimum, we use certificates that combine channel normalization, minimization over the profiling coordinate, and dual optimality for the weights. The fixed pattern enumeration in Definition 3 gives the support indices a common meaning across parameter points.