Exact Global Minimax Risk near Nonidentification

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.

We derive the asymptotic risk function of regularized empirical risk minimization (ERM) estimators tuned by $n$-fold cross-validation (CV). The out-of-sample prediction loss of such estimators converges in distribution to the squared-error loss (risk function) of shrinkage estimators in the normal means model, tuned by Stein's unbiased risk estimate (SURE). This risk function provides a more fine-grained picture of predictive performance than uniform bounds on worst-case regret, which are common in learning theory: it quantifies how risk varies with the true parameter. As key intermediate steps, we show that (i) $n$-fold CV converges uniformly to SURE, and (ii) while SURE typically has multiple local minima, its global minimum is generically well separated. Well-separation ensures that uniform convergence of CV to SURE translates into convergence of the tuning parameter chosen by CV to that chosen by SURE.

We study ridge-regularized log-density-ratio estimation in the Gaussian location model with a common covariance matrix. By affine invariance, the model is written as q $\sim$ N(0, I), p $\sim$ N($Δ$, I), with linear features, where $Δ$ is a mean vector. The variational estimator is the empirical Kullback-Leibler (KL) log-normalized fit with a squared L2-penalty on its nonconstant coefficient, and the spectral estimator recently introduced in [1] replaces a single variational problem by a continuum of ridge-regularized least-squares problems. We derive high-dimensional deterministic asymptotic equivalents when the numbers of observations and dimension tend to infinity with fixed ratios. The regularized variational limit is characterized by a scalar entropy minimization problem derived from the convex-Gaussian-min-max theorem (CGMT), while the regularized spectral limit follows from deterministic equivalents for resolvents of weighted sums of two independent Gaussian sample covariance matrices. We use these formulas to compare population risks, with experiments focused on fixed-signal aspect-ratio sweeps and optimized regularization. Our conclusion is that with many observations, under the criteria and asymptotic regimes analyzed here, the well-specified variational estimator has the smaller risk, while with fewer observations, the spectral estimator is favored because its covariance-based construction has lower variance. We also study how a nuclear penalty can be used and partially analyzed to perform feature learning.