Subdata Selection for Measurement-Constrained Linear Regression under Model Misspecification
This paper tackles measurement-constrained linear regression, where responses are expensive to collect and must be selected from a large candidate pool before being observed. The authors propose Expected Loss Optimal Selection (ELOS), which derives an A-optimality-type criterion that depends only on covariates and remains valid under potential model misspecification. The core insight is that averaging the expected prediction loss over uniformly random misspecification directions recovers the same trace-inverse criterion as under a correctly specified model, yielding a principled response-free selection rule.
The paper presents a clean theoretical connection between prediction-optimal design and misspecification-robust subset selection: under uniform averaging over misspecification directions, minimizing the expected predictive loss reduces to minimizing $\Psi_A(\bm{\Delta})=\operatorname{tr}[(\bm{U}^\top \bm{\Delta} \bm{U})^{-1}]$. The authors position this as a novel result with "no counterpart in the classical theory" (Section 1). The derivation is elegant and the empirical comparisons across six baselines are extensive. However, the guarantee is inherently average-case rather than worst-case, the experiments rely on a specific convection of structured and bootstrapped misspecification, and the primary solver (IBOSS++) is drawn from a not-yet-published companion paper, which limits immediate reproducibility.
The theoretical backbone is sound and well-motivated. Theorems 1 and 2 cleanly decompose the expected prediction loss under correct and misspecified models, and Theorem 3 shows that after averaging $E_{\bm{a}}\mathbb{E}[\mathcal{L}(\bm{\Delta})]$ over the unit sphere, the minimizer is invariant to $(\sigma^2,\tau^2)$ and reduces to the classical A-optimal trace-inverse criterion $\Psi_A(\bm{\Delta})$. This gives a principled justification for using A-optimality even when the linear model is misspecified, provided one buys the uniform-direction prior. The empirical evaluation is also admirably broad, comparing SRS, leverage sampling, IBOSS, MCG, LowCon, and coreset methods under multiple covariate distributions, sample sizes, and misspecification strengths.
The central theoretical guarantee is explicitly average-case. The authors acknowledge that "This is an average-case guarantee rather than optimality for every fixed or structured departure" (Section 5), which means ELOS does not protect against worst-case or highly structured misspecification such as omitted variables with strong covariate correlation patterns. Empirically, the simulated misspecification $f(\bm{x}_i)=\tau\sin(\sum_j x_{ij}^2)$ and the residual bootstrap evaluation in real data (Appendix D.1) do not obviously satisfy the uniform spherical symmetry assumed in Theorem 3, so the experiments partially test a different scenario than the one in which the criterion is formally optimal. Moreover, several competitors are omitted in key settings—MCG and LowCon are dropped in multiple real-data runs due to cost—so the empirical ranking is incomplete. Under multivariate normal covariates (Table 2), the response-dependent Coreset often achieves lower relative error than ELOS at moderate and large $r$, tempering claims of uniform predictive dominance.
The simulation evidence supports that ELOS matches or improves upon response-free competitors under heavy-tailed covariates, but under normal covariates Coreset frequently wins despite being response-dependent. The real-data experiments use a residual bootstrap that "does not preserve residual–covariate dependence" (Appendix D.1), which the authors candidly note may understate structured misspecification. This is a honest limitation, yet it means the real-data evaluation is not a stringent test of robustness to the kinds of structured departures practitioners fear. Comparisons to related work are generally fair, though the distinction between ELOS and LowCon could be sharpened: both target misspecification, but LowCon uses a design-based decorrelation strategy rather than an averaging argument, and their relative merits beyond the simulation settings remain unclear.
While the authors state that "Complete implementation and reproducibility details are provided in Appendix F" (Section 4), the main text does not provide a public code or data repository link, and the IBOSS++ optimizer is taken from Yang et al. (2026), cited only as an accepted NeurIPS paper that may not yet be publicly accessible. This makes exact reproduction of the selection pipeline difficult until that companion work appears. Hardware (Apple MacBook Pro M4 Pro), software (Julia 1.12.0), and BLAS settings are reported, but random seeds for the 100 replications are not stated. Hyperparameter choices for competing methods (e.g., iteration caps for MCG) are partially discussed in Appendix C but full configuration files are absent.
In many modern applications, obtaining response variables is costly, time-consuming, or technically challenging, making it impractical to measure outcomes for all available units. In such settings, covariates may be available for a large pool of candidates, but responses can only be collected for a small, carefully selected subset, making the choice of which units to measure a consequential decision. Most existing subsampling methods are developed for massive datasets, emphasizing the trade-off between statistical accuracy and computational cost, and typically relying on the assumption of a correctly specified model. These features limit their suitability in measurement-constrained settings, where statistical efficiency is paramount and model misspecification is often unavoidable. To address these challenges, we develop a new subset selection framework for linear regression that accommodates potential model misspecification. The proposed approach targets predictive expected loss and yields an A-optimality-type criterion that depends only on covariate information, requiring no response observations at the selection stage. We refer to this method as Expected Loss Optimal Selection (ELOS). By focusing on predictive risk rather than model-based optimality, ELOS remains well-defined and robust under misspecification. Simulation studies and real data applications demonstrate that ELOS achieves strong predictive performance and compares favorably with existing methods across a range of scenarios.
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