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Prediction Models That Learn to Avoid Missing Values

T0 review · 2 major / 6 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Training a model with a missingness-reliance penalty makes it avoid requiring missing feature values at test time, with AUROC essentially unchanged.

desk verdict A practical extension of the MINTY idea to trees, LASSO, and ensembles, with a solid empirical study but an ambiguous baseline-selection protocol that should be clarified before the headline ρ reductions are taken at face value. read the letter →

arxiv 2505.03393 v1 pith:AA2SOWWM submitted 2025-05-06 cs.LG

Lena Stempfle , Anton Matsson , Newton Mwai , Fredrik D. Johansson This is my paper · ORCID
classification cs.LG
keywords missingvaluesmissingnessreliancetest-timeregularizeddecisiontreessparselinearmodelstreeensemblesinterpretablepredictionobserveddeterministicdatacollectionrules
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

When a deployed predictor asks for a feature that is missing, the user must impute, guess, or accept an incomplete result, each option adding bias or hiding what the model actually used. This paper proposes missingness-avoiding (MA) machine learning: add a term $\alpha \rho(h)$ to the learning objective that penalizes any prediction which, for a particular example, requires a feature whose value is missing. The authors implement this penalty for decision trees (MA-DT), sparse linear models (MA-LASSO), random forests (MA-RF), and gradient-boosted trees (MA-GBT), and show on six real datasets that these models cut the share of test predictions relying on a missing value, in the extreme case from 100% to 0%, while AUROC confidence intervals overlap with unregularized baselines in nearly every comparison. The reason to care: a model that avoids missing values preserves interpretability where it matters, because the clinician or analyst never has to supply data the model was designed not to need.

What carries the argument

The load-bearing object is the missingness-reliance penalty $\alpha \rho(h)$: a regularization term counted per training example, where $\rho(h,x)=1$ if the prediction path for $x$ touches any feature whose value is missing. It is carried through three instantiations. For decision trees, the split-selection criterion $C(\ell,D;j,\tau)$ is augmented by $\alpha$ times the empirical fraction of samples in the leaf for which feature $j$ is missing, so splits are chosen that avoid frequently missing features while still reducing impurity; for Lasso, the coefficient penalty becomes $(\lambda + \alpha m_j)|\theta_j|$, which drives frequently missing features out of the model through standard feature rescaling; and for boosted trees, per-sample weights $\sigma_{i,j}$ are updated so that later trees prefer features already used by earlier trees, preventing independently fit trees from each re-introducing reliance. The theoretical companion is the class of ODDC rules, implications of the form 'whenever variables $T$ are observed and take values in $A$, variable $j$ is observed', which characterize when zero reliance and minimal risk can coexist.

What would settle it

Run the NHANES benchmark again but alter the test-time missingness mask, for instance by dropping the survey-year feature so the deterministic collection rules no longer hold for a random subset of test rows, and measure $\rho$ and AUROC of a fixed MA-DT. If reliance rises back toward the unregularized tree's level while AUROC falls with it, the claimed trade-off holds only under exactly matched missingness, contradicting the paper's implicit deployment claim.

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Extended reading notes

Core claim

The paper's central claim is that missingness reliance can be moved from an architectural afterthought into the training objective. Reliance is defined as $\rho(h) = \mathbb{E}_p[\max_j \mathbb{1}[a_h(X,j)=1 \wedge X_j = \mathrm{na}]]$, the probability that evaluating $h$ on a random input needs at least one feature value that is absent, and the learning problem becomes minimizing $\mathbb{E}_p[L(Y,h(X))] + \alpha \rho(h)$. The paper instantiates this objective for four model classes: trees gain a split-criterion term penalizing splits on features that are missing for samples in the node, Lasso gains a per-feature penalty $(\lambda + \alpha m_j)|\theta_j|$ with $m_j$ the empirical missingness rate, random forests apply the tree penalty per estimator, and gradient boosting decays per-sample weights $\sigma_{i,j}$ so later trees reuse features earlier trees already relied on. The theoretical core is the class of observed deterministic data collection rules, implications guaranteeing that a feature is observed whenever certain observed variables take certain values, under which a minimal-risk model with zero reliance provably exists (Corollary 1) and any rule-respecting tree has $\rho(h)=0$ (Proposition 1). Empirically the trade-off is favorable: on NHANES, logistic regression and random forests rely on missing values for 100% of test predictions, while the MA versions rely on them for at most 0.4% at overlapping AUROC.

Load-bearing premise

The framework's practical benefit rests on the assumption that the missingness pattern at test time is stable and at least partly predictable from observed variables, so that a model trained to lean on features that tend to be present will still find those features present when deployed.

Editorial extensions

If this is right

  • Practitioners can deploy MA-DT in settings like clinical risk scoring and still read off the decision path: the tree reaches patients with missing MRI results without ever branching on the scan.
  • In datasets whose collection process follows ODDC rules, the framework can find models that are simultaneously Bayes-optimal and never require a missing value, so zero-reliance predictions carry no accuracy cost.
  • The speed gap matters in practice: MA-LASSO trains in under a second, while the earlier rule-based baseline MINTY takes 18 to 292 seconds on the same tasks, making low-reliance prediction feasible at scale.
  • When features are missing completely at random and independently, Proposition 2's lower bound $\rho(h) \ge \max_j \mathbb{E}[a_h(X,j)]\,p_j$ means no training trick can avoid reliance, so the benefit of MA learning is structural rather than universal.
  • Setting $\alpha$ very high forces $\rho \approx 0$ but at a real accuracy cost, for instance MA-DT on LIFE falls from AUROC 0.90 to 0.67, so $\alpha$ must be tuned as a trade-off parameter, which the paper's model-selection rule does automatically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same penalty could be carried into other model classes the paper leaves open, such as neural networks, generalized additive models, and rule lists, wherever 'requires the value of feature $j$' can be read off the architecture.
  • An alternative model-selection rule that caps the number of missing features any single prediction may rely on, rather than averaging reliance over the test set, would better match safety-critical per-patient guarantees.
  • If deployment itself decides what to measure next, the ODDC analysis suggests a natural loop: train the data-collection policy and the MA model jointly, since the model's zero-reliance structure is only as stable as the collection rules that guarantee observability.
  • A testable extension would quantify how quickly the benefit decays under shift: on a dataset with matched train/test collection rules but discontinued availability of one feature, measuring $\rho$ and AUROC of a fixed MA model would give practitioners a degradation curve.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 6 minor

Summary. The paper introduces missingness-avoiding (MA) machine learning, a framework that penalizes a model's expected reliance on missing feature values at test time. The reliance measure ρ(h) is defined as the probability that computing h(x) requires the value of at least one feature that is missing in x. The authors derive model-specific regularizers for decision trees (Eq. 4), linear models (Eq. 5), and tree ensembles (Section 4.3), and provide implementations for MA-DT, MA-LASSO, MA-RF, and MA-GBT. The theory (Section 5) gives sufficient conditions for the existence of zero-reliance optimal models under observed deterministic data collection (ODDC) rules (Prop. 1, Cor. 1) and a lower bound on reliance when features are missing completely at random (Prop. 2). Experiments on six real-world datasets report AUROC and ρ for the MA models against several baselines, finding overlapping AUROC confidence intervals in most settings and substantially lower ρ for the MA models.

Significance. The framework addresses a real practical problem: models that at deployment require features that are currently missing, forcing imputation or compromising interpretability. The paper's contribution is primarily algorithmic and empirical. The theory is acknowledged by the authors to be largely definitional, but it provides a useful vocabulary for when the approach can work. The experiments are carefully designed with bootstrap confidence intervals, multiple datasets, and both zero and MICE imputation, and the code is provided. If the empirical comparison can be made fully controlled (see major comments), the paper would offer a valuable tool for practitioners who need models that make themselves robust to missingness at test time.

major comments (2)
  1. [Section 6, Table 1] The model-selection protocol in Section 6 states that hyperparameters, including α, are chosen by selecting the candidate with the lowest ρ̂ among those with AUROC at least 95% of the maximum. The paper does not state whether the baseline models (LR, DT, RF, XGBoost, M-GAM, NeuMiss) were selected under this same rule or by standard AUROC-only selection. If baselines were selected for AUROC only, the reported reductions in ρ (e.g., 66.0 and 70.2 percentage points for MA-LASSO vs. LR and MA-RF vs. RF in Table 1) are not a controlled estimate of the MA penalty's benefit, because baselines were not given the same opportunity to trade accuracy for lower reliance. The authors should either confirm that the same rule was applied to all models and report the selected baseline hyperparameters, or add a matched comparison, such as reporting the AUROC-ρ Pareto frontier for every model class.
  2. [Section 4.3, Table 3] In Table 3, MA-GBT is restricted to 10 estimators and MA-RF to 50, while the XGBoost search space includes 100–500 estimators and the RF baseline uses the default ensemble size (100). Since the ensemble reliance in Eq. (6) is defined as the maximum over constituent trees, the number of estimators directly bounds the achievable ρ: a smaller ensemble has fewer opportunities to rely on missing features. The reported ρ comparisons between MA-GBT and XGBoost, and between MA-RF and RF, are therefore not controlled for model capacity. The authors should either match the number of estimators across methods or show how ρ and AUROC vary with ensemble size for both the MA and baseline models.
minor comments (6)
  1. [Section 4.1, Eq. (4)] The notation 'σi,j ∈ 0, 1' is missing set braces; it should be 'σ_{i,j} ∈ {0,1}'.
  2. [Section 4.2] The phrase 'encourages sparsities in linear models' should read 'encourages sparsity in linear models'.
  3. [Section 6.1] The sentence 'with an average decrease of 66.0 (MA-LASSO vs. LR), 4.4 (MA-DT vs. DT), and 70.2 (MA-RF vs. RF)' should specify that these are percentage-point decreases in ρ, not relative percentages.
  4. [Section 5.1, Corollary 1] The main-text statement of Corollary 1 omits the caveat noted in Appendix A that the Bayes-optimal model h* is assumed to belong to the hypothesis class H; for restricted classes such as linear models this is not automatic, and the statement should be qualified accordingly.
  5. [Figure 4c caption] The caption contains the typo 'MA-LASSSO'; it should be 'MA-LASSO'.
  6. [Section 4.3] The sentence 'adding models to an ensemble can only increase the average reliance on a variable' is imprecise because ρ is defined as the maximum over trees, not an average; consider rewording to 'can only increase the missingness reliance ρ(e,x)'.

Circularity Check

1 steps flagged · score 2.0 of 10

Central MA objective and experiments are self-contained; only the explicitly admitted definitional Corollary 1 is self-referential.

  1. self definitional [Section 5.1 (Corollary 1 and following paragraph); Appendix A proof of Corollary 1]
    "Corollary 1 follows by definition: if the outcome Y can be described by input features known to be observed together, a model that fits p(Y | X,M ) perfectly can be optimally predictive and have zero reliance ρ."

    The assumption of Corollary 1 is that p(Y|X) is fully determined by variables implied to be observed by ODDC rules R. The proof then defines h*=E[Y|X] and observes that ρ(h*)=0 because the Bayes-optimal function depends only on those guaranteed-observed variables. The conclusion—that a minimal-risk model with zero missingness reliance exists—is therefore a restatement of the assumption in hypothesis notation; the derivation supplies no independent mechanism. The paper's own sentence 'follows by definition' confirms this. Because the corollary is presented as motivation and explicitly flagged, it does not infect the empirical evaluation, which uses held-out ρ and AUROC with an independently defined metric.

full rationale

The central derivation chain is not circular. The missingness-avoiding objective (Eq. 2) is fitted with a penalty on ρ, but α is selected by cross-validation on training folds and ρ is measured on held-out test data (Sec. 6), so the reported reductions are empirical outcomes of an optimization, not identities. The ρ metric is formally defined in Sec. 3 and applied consistently to MA and baseline models (App. C.3). The new algorithms (MA-DT Eq. 4, MA-LASSO Eq. 5, MA-GBT Alg. 1) do not guarantee AUROC preservation by construction; indeed Table 6 shows α=∞ can collapse AUROC. The only definitional step is Corollary 1, which the paper itself labels as following by definition: the ODDC assumption already contains the existence of a zero-reliance optimal model. This is transparent and not load-bearing for the empirical claims. Self-citations to Stempfle & Johansson (2024) supply the ρ concept and the MINTY baseline, but the definition is formalized in this paper and MINTY is an independent benchmark, not an assumed theorem. No uniqueness theorem or ansatz is imported from prior work. The baseline-selection asymmetry noted by a skeptic (baselines not matched for ρ budget) is a fairness/control concern, not circularity. Overall: no significant circularity; one admitted definitional side-result.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central method depends on one hyperparameter (α) per model and one stabilization constant (β) for MA-LASSO; no new physical or mathematical entities are postulated. The key domain assumption is stationarity of the missingness mechanism, explicitly stated in Section 3.

free parameters (2)
  • α (MA regularization strength) = CV-selected per dataset from {0.001, 0.01, 0.1, 1, 10} (trees/RF/GBT) or {1, 10, 100, 1000, 10000} (LASSO)
    Controls the trade-off between loss and missingness reliance in Eqs. (2), (4), (5); selected on 3-fold CV by the 95%-AUROC rule, so the reported ρ is partly optimized.
  • β (MA-LASSO stabilization constant) = from {0.001, 0.01, 0.1, 1, 10, 100, 1000}
    Used in the implementation λ_j = ((Σ_i m_i,j + β)/n)·α (Appendix C.2); a small additive constant to avoid zero penalties.
assumptions (4)
  • domain assumption The joint distribution p(X, M, Y) is fixed and identical at train and test time.
    Stated in Section 3; if missingness patterns shift, MA models may rely on features that become missing at deployment, a limitation the authors acknowledge.
  • standard math Training data are i.i.d. from p(X, M, Y).
    Standard supervised learning assumption used throughout.
  • domain assumption For Corollary 1, the Bayes-optimal predictor h* belongs to the hypothesis class H.
    Noted in Appendix A 'A Note on the Hypothesis Class'; the corollary's conclusion depends on this membership.
  • domain assumption The greedy split criterion with the added penalty in Eq. (4) is an adequate surrogate for the global tree objective.
    The paper relies on greedy C4.5/ID3-style splitting; no guarantee that greedy search finds the regularized optimum, though this is standard practice.

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Pith. "Pith review of Prediction Models That Learn to Avoid Missing Values." pith.science (2026). https://pith.science/paper/AA2SOWWM

@misc{pith2026250503393,
  author       = {Pith},
  title        = {Pith review of: Prediction Models That Learn to Avoid Missing Values},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AA2SOWWM}},
  note         = {Machine review of arXiv:2505.03393}
}
read the original abstract

Handling missing values at test time is challenging for machine learning models, especially when aiming for both high accuracy and interpretability. Established approaches often add bias through imputation or excessive model complexity via missingness indicators. Moreover, either method can obscure interpretability, making it harder to understand how the model utilizes the observed variables in predictions. We propose missingness-avoiding (MA) machine learning, a general framework for training models to rarely require the values of missing (or imputed) features at test time. We create tailored MA learning algorithms for decision trees, tree ensembles, and sparse linear models by incorporating classifier-specific regularization terms in their learning objectives. The tree-based models leverage contextual missingness by reducing reliance on missing values based on the observed context. Experiments on real-world datasets demonstrate that MA-DT, MA-LASSO, MA-RF, and MA-GBT effectively reduce the reliance on features with missing values while maintaining predictive performance competitive with their unregularized counterparts. This shows that our framework gives practitioners a powerful tool to maintain interpretability in predictions with test-time missing values.

Figures

Figures reproduced from arXiv: 2505.03393 by the authors.

Figure 1
Figure 1. Two decision trees built to diagnose cognitive impair￾ment (CI) in adult patients. The left side illustrates a regu￾lar decision tree fit solely for accuracy, while the right shows a missingness-avoiding (MA) tree, which incorporates a regulariza￾tion parameter α to reduce reliance on missing values. The regular tree initially splits on a positive MRI scan outcome, determined by hippocampal volume (Vh). However, thi… view at source ↗
Figure 2
Figure 2. Missing values can be avoided in several ways. Sparse models (left) can be trained to not use features that are frequently missing. Disjunctive rule models (middle) can be fit to include rules that exploit redundancy in the variable set. Trees (right) can be fit so that missing values rarely occur on the decision paths. uating xj or its imputed value x I j . We use a binary indicator function ρ(h, x) ∈ {0, 1} to ind… view at source ↗
Figure 3
Figure 3. Average test performance across cross-validation folds for MA decision trees fit to LIFE using varying values of α and max_depth. A shallow tree of depth 3–4 captures the complexity of the data effectively. Missingness regularization with α ≤ 1.0 does not significantly reduce predictive performance but helps decrease missingness reliance from approximately 30 % to approximately 10 %. and α set high enough to ensure … view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: (a) and (b): Test-set AUROC and missingness reliance (ρˆ) for MA estimators in ADNI when transitioning from α = ∞ to α = 0. Removing missingness regularization improves predictive performance, but the increase in missingness reliance is much more pronounced, especially…
Figure 5
Figure 5. Figure 5: LIFE: AUROC − ρˆ vs. max. tree depth. et al., 2011). For XGBoost, we used the XGBoost Python Package (see https://xgboost.readthedocs.io/ en/stable/python/index.html). For MINTY and M-GAM, we adapted code provided by Stempfle & Johansson (2024) and McTavish et al. (202…
Figure 6
Figure 6. Figure 6: Example decision trees for α = 0, α = α ∗ , and α = ∞ fit to LIFE. The nodes are colored based on the missingness reliance ρ. The goal is to predict whether a country’s life expectancy (LE) is above or below the median life expectancy. (a): MA-DT with α = 0 behaves as …

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.