Exact Global Minimax Risk near Nonidentification

Kaiyue Yang; Yunan Wu

stat.MLOct 23, 2026

Exact Global Minimax Risk near Nonidentification

Authors: Kaiyue Yang, Yunan Wu

Organizations: Qiuzhen College, Tsinghua University, Haidian District, Beijing 100084, P. R. China · Yau Mathematical Sciences Center, Tsinghua University, Haidian District, Beijing 100084, P. R. China

Abstract

We study squared-error estimation when a nuisance parameter is unidentified on a submodel but affects a target that vanishes there. For models with higher-order profile expansions, we give sufficient conditions under which the largest critical-path minimax risk is the exact asymptotic minimax risk over the full parameter space. An efficient observable feature, negligible calibration error and stable recovery yield one estimator attaining the leading constant uniformly, with the profile coordinates unknown. The estimator uses calibration estimates, a local feature mean and an empirical recovery feature. The constant is expressed through a bounded-normal decision problem. We verify the conditions in two models. In centered Gaussian logistic errors-in-variables regression with one surrogate and two unknown variances, we determine the fifth-order global coefficient risk and its exact constant. A sparse extension separates variance-decomposition uncertainty from direction estimation and specifies when the scalar constant is preserved. In a degree-balanced replicated-network model, we determine the sixth-order global edge-contrast risk when block proportions are unknown. Triangles identify a parameter combination; a bowtie supplies the additional information needed for the contrast.

Figures & tables

Figure 1: Finite-sample endpoint risks at fixed profile coordinates. Each point uses 200 replications at each endpoint. Bars are pointwise 95% bootstrap intervals for the maximum of the two empirical risks. Dotted lines give the corresponding endpoint risk of the zero estimate. These curves assess finite-sample recovery, rather than convergence along least-favourable critical paths.

Component	Setting
Tuning thresholds	Logistic $(r,M,v_{0},v_{\rm clip})=(0.5,5,0.5,0.45)$ , with target projection $[-4,4]$ ; network $(r,M,v_{0},v_{\max})=(0.08,0.40,0.155,0.16)$ , with $\widehat{v}$ restricted by projection to absolute values at most $v_{0}$ .
Bounded-normal action	Observed standardized interval width, without binning; standardized numerical gap tolerance $10^{-3}$ ; minimum parameter and validation grids $101$ and $2001$ ; Gauss–Hermite order $120$ ; optimization tolerance $10^{-12}$ ; solver parameter max_iterations=500; minimum- $\ell_{2}$ tie break.
Profile and inverse	Logistic bracketed solves use Gauss–Hermite order $256$ ; network uses exact finite graph sums and the clipped bowtie-moment ratio.
Outer candidate	Logistic Gaussian-kernel bandwidth $1$ ; $16$ Fourier frequencies ( $67$ features); $33$ values of $\theta$ from $-4$ to $4$ in steps of $0.25$ ; $c\in\{2,2.5,3,3.5,4\}$ ; $9$ values of $\lambda$ per $c$ ; Gauss–Hermite order $160$ ; $928$ admissible candidates. Network uses the closed-form clipped bowtie-moment ratio with the unclipped calibration coordinate $\widetilde{v}$ .
Monte Carlo uncertainty	Logistic uses $5000$ within-cell bootstrap repetitions for endpoint maxima and the tail slope. Network uses $4000$ for the tail slope; its pointwise MSE bars use $t_{0.975,199}=1.9719565$ .

Table 1.: Reference settings for the unchanged fixed-coordinate simulations and archived candidate-weight checks. The asymptotic sequences are specified separately in the main paper.

$n$	$v_{n}$	$n^{1/k}\widehat{R}_{n,G}/C_{k}$	$95\%$ MC interval
Logistic, $k=5$
$10^{9}$	0.243399	1.2262	$[1.2103,1.2562]$
$10^{10}$	0.192775	1.1343	$[1.1206,1.1632]$
$10^{11}$	0.152781	1.0838	$[1.0647,1.1107]$
$10^{12}$	0.121145	1.0638	$[1.0433,1.0892]$
$10^{13}$	0.096097	1.0412	$[1.0275,1.0687]$

Table 2.: Critical-path results after moment calibration. Logistic uses seven

\lambda

values and networks thirteen. The interval constructions are specified in the text; neither is a confidence statement about the full parameter-space minimax risk. Exact inputs and all cell risks are included in Supplement B.

Figure 1: Critical-path checks with joint moment calibration. Left panels show grid MSE against

C_{k}n^{-1/k}

; right panels show normalized risk. The original model, path coefficient and parameter ranges are retained. Both axes and normalization use total

n

, although generation preserves the original reference-path index

N_{\rm ref}

. Intervals reflect Monte Carlo uncertainty, and finite-profile bias remains visible for logistic regression. These curves are separate from the fixed-coordinate study in the main paper.

$n$	Hermite feature	Finite likelihood
$10^{12}$	$1.051564$	$1.048522$
$3\times 10^{12}$	$1.041216$	$1.038763$
$10^{13}$	$1.032225$	$1.030332$

Table 3.: Normalized average endpoint risks

n^{1/5}\{\mathbb{E}_{0}(\delta-\theta_{0})^{2}+\mathbb{E}_{1}(\delta-\theta_{1})^{2}\}/(2C_{5})

, estimated using the mixture identity. These averages are distinct from the grid maxima in Table 2 .

Figure 2: Sixth-order information. Exact information and $I_{M}(p)v^{12}=15\begin{pmatrix}M\cr 5\end{pmatrix}\{p(1-p)\}^{-6}v^{12}$ .

Figure 3: Fifth-order geometry. Information and contact-path KL, with their leading normalizations.

Figure 4: Fixed-fraction direction experiment. Pilot MSE, direction loss, normalized losses, and run sizes.

$n$	$s$	$\lambda_{\rm emp}$	$d_{n,s}$	$N^{1/5}\widehat{R}_{\rm AD}$	$N^{1/5}(\widehat{R}_{\rm AD}-\widehat{R}_{\rm OD})$
$10^{5}$	1	1.5	0.00187	6.976 $[3.085,10.866]$	0.002 $[-0.004,0.009]$
$10^{5}$	16	3.0	0.01840	5.543 $[2.997,8.089]$	0.098 $[0.014,0.182]$
$3\times 10^{5}$	1	1.5	0.00089	8.709 $[-0.362,17.779]$	0.000 $[0.000,0.000]$
$3\times 10^{5}$	16	1.5	0.00881	7.304 $[-1.109,15.718]$	0.165 $[-0.492,0.821]$

Table 4.: Selected settings for sparse estimation. Simultaneous

95\%

Bonferroni intervals; the larger adaptive endpoint is shown.

Figure 5: Sparse estimation. Adaptive and oracle risks, paired risk differences, and the sample-splitting benchmark.

Figure 6: Combination of four estimators. Weights, estimator selection, and normalized losses.

Quantity	Comparison	Maximum change or gap
Profiled information	GH 96 versus 128	$6.6\times 10^{-15}$ relative
Contact-path KL	GH 96 versus 128	$2.7\times 10^{-7}$ relative
Profile inversion	GH 96 with bracketed roots	$6.2\times 10^{-14}$ maximum root residual
Likelihood surface	cubic versus direct evaluation; GH 128/192	$3.41\times 10^{-12}$ for $g$ ; $6.27\times 10^{-13}$ for the score derivative
$R_{G}(z_{\star})$	GH 80, 120, 160, and 240	$2.9\times 10^{-15}$ absolute
$K_{5}$ computation	Exchange; 2001-point risk grid	endpoint support at $z_{\star}$ ; numerical gap $6.47\times 10^{-4}$

Table 5.: Quadrature and grid comparisons; gaps are numerical.

Figure 7: Bounded-normal computation. Computed Bayes and mesh-corrected rule risks, before and after scaling by $z^{-8/5}$ . The dot is the value $(z_{\star},K_{5})$ computed using the prior supported at the endpoints; gaps are numerical.

Figure 8: Gaussian approximation. Centering, variance, shape, bias, and the $21$ -grid LAN remainder.

Figure 9: Local risk. Normal approximation, bounded-normal rules, and $1005$ exact-model statistics; bars are pointwise $95\%$ intervals.

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We characterize the sharp structure-agnostic minimax risk for coefficient estimation in the partial linear model when the outcome and treatment nuisances are learned by two distinct black-box learners, which resolves the open problem in double machine learning posed by Gu (2025). For each nuisance

q\in\{μ,π\}

, we characterize the available learner by an approximation-error budget

a_q

and a stochastic-error budget

s_q

, with the latter controlled through localized Rademacher complexity. Writing

\mathcal E_n

for the minimax mean-squared error, we show that

\mathcal E_n\asymp1\wedge\left\{\frac1n+\left(a_μa_π+\min\left\{a_πs_μ+s_π^2,\,a_μs_π+s_μ^2\right\}\right)^2\right\}.

The main new ingredient is a novel lower bound for the general two-learner problem. Our proof constructs four finite-mixture testing experiments using orthogonal code functions. Across these experiments, the hidden perturbations are placed outside both learner classes, outside only the treatment learner class, outside only the outcome learner class, or inside both learner classes. These four configurations capture, respectively, the interaction between the two approximation errors, the two asymmetric interactions between one learner's approximation error and the other learner's learning error, and the joint estimation difficulty of learning both nuisances. Combining the four resulting lower bounds yields the displayed rate, which matches the latest upper bound in Gu (2026). Our result shows that standard double machine learning can overstate the intrinsic difficulty of target estimation and provides a target-specific principle for learner selection: approximation error and stochastic complexity must be jointly balanced across the two nuisance learners rather than optimized separately.

Abstract

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