CROiMS: Conditional Robust Optimization

• CROiMS is a framework for covariate-aware robust optimization that integrates localized conformal prediction with individualized model selection.
• It minimizes conditional decision risk using kernel-weighted empirical risk minimization to tailor predictions and actions based on covariate values.
• The method achieves asymptotic conditional robustness and optimality under smoothness conditions, outperforming traditional global selection strategies.

Searching arXiv for the main CROiMS paper and a closely related conformal prediction result on conditional coverage. CROiMS, short for Conformalized Robust Optimization with Individualized Model Selection, is a framework for decision-making under uncertainty in which the predictive model itself is selected as a function of the test covariate $x$. It is introduced in the context of Contextual Robust Optimization (CRO), where a decision $z\in\mathcal{Z}$ is chosen by minimizing worst-case loss over a prediction set for the uncertain outcome $Y$. CROiMS extends conformalized robust optimization by combining localized conformal prediction, kernel-weighted empirical risk minimization, and covariate-aware model choice so as to target conditional decision risk rather than only average decision risk. The framework is designed to achieve asymptotic conditional robustness and asymptotic conditional optimality under mild smoothness assumptions, while improving decision efficiency in settings with strong covariate heterogeneity (Bao et al., 7 Jul 2025).

1. Conceptual setting and motivation

The underlying decision problem observes covariates $X\in\mathcal{X}\subseteq\mathbb{R}^d$, then an uncertain label or outcome $Y\in\mathcal{Y}$ is realized, and a decision $z\in\mathcal{Z}$ incurs loss $\phi(Y,z)$. In Contextual Robust Optimization, uncertainty is represented not by a full conditional distribution for $Y\mid X=x$, but by a prediction set $\mathcal{U}(x)\subseteq\mathcal{Y}$. The CRO decision solves

$$z(x)=\argmin_{z\in\mathcal{Z}}\max_{c\in\mathcal{U}(x)}\phi(c,z),$$

so that the chosen action hedges against all labels contained in the uncertainty set.

Within this setting, conformal prediction supplies prediction sets using a score or nonconformity function $z\in\mathcal{Z}$0. For i.i.d. labeled data $z\in\mathcal{Z}$1, split conformal prediction forms

$z\in\mathcal{Z}$2

with threshold

$z\in\mathcal{Z}$3

Under exchangeability, this gives marginal coverage

$z\in\mathcal{Z}$4

which in turn yields marginal robustness at the decision level.

The motivation for CROiMS arises from the observation that practical deployments rarely involve a single predictive model. Instead, one often has a finite family of candidate models $z\in\mathcal{Z}$5, each inducing its own conformal prediction set

$z\in\mathcal{Z}$6

Although these sets can all satisfy valid marginal coverage, their downstream effect on robust decisions can differ substantially. Some models induce conservative sets and overly pessimistic decisions; others produce tighter, decision-aligned sets that lead to lower decision loss. Standard conformal model selection typically emphasizes set size or width rather than downstream loss $z\in\mathcal{Z}$7, and it usually optimizes average rather than covariate-specific efficiency.

The paper identifies several limitations of prior approaches. Standard CRO with conformal prediction offers no guidance for choosing among candidate models and does not adapt model choice to the specific test covariate. Decision-focused or end-to-end training may optimize empirical decision risk, but it does not provide finite-sample coverage or robustness guarantees and may not adapt well to localized conditional structure. CROMS, E-CROMS, and F-CROMS address average decision risk while retaining marginal robustness, yet the chosen model remains global. CROiMS is therefore motivated by three aims: to perform covariate-aware individualized model selection, to minimize conditional decision risk at each $z\in\mathcal{Z}$8, and to retain conformal-style robustness guarantees (Bao et al., 7 Jul 2025).

2. Formal objective and robustness criteria

For a fixed model $z\in\mathcal{Z}$9, the paper defines the oracle marginal quantile

$Y$0

the oracle prediction set

$Y$1

and the corresponding oracle CRO decision

$Y$2

Its average decision risk is

$Y$3

The best average model is

$Y$4

CROiMS replaces this global criterion by a conditional one. It introduces the conditional oracle quantile

$Y$5

the conditional oracle set

$Y$6

and the conditional oracle decision

$Y$7

The corresponding conditional decision risk is

$Y$8

and the individualized oracle model is

$Y$9

The robustness notions are central. Marginal robustness requires

$X\in\mathcal{X}\subseteq\mathbb{R}^d$0

CROiMS instead targets the conditional analogue: $X\in\mathcal{X}\subseteq\mathbb{R}^d$1 It also seeks asymptotic conditional optimality: $X\in\mathcal{X}\subseteq\mathbb{R}^d$2

These definitions formalize the distinction between optimizing a single model globally and choosing models pointwise in covariate space. A plausible implication is that CROiMS is specifically suited to settings where the best predictive model varies substantially across subpopulations, covariate regimes, or local shifts in the data-generating process.

3. Algorithmic construction

CROiMS uses localized conformal prediction together with kernel-weighted empirical risk minimization. Let $X\in\mathcal{X}\subseteq\mathbb{R}^d$3 be a kernel, for example Gaussian, and define

$X\in\mathcal{X}\subseteq\mathbb{R}^d$4

For each candidate model $X\in\mathcal{X}\subseteq\mathbb{R}^d$5, the localized conformal quantile at $X\in\mathcal{X}\subseteq\mathbb{R}^d$6 is the weighted $X\in\mathcal{X}\subseteq\mathbb{R}^d$7-quantile

$X\in\mathcal{X}\subseteq\mathbb{R}^d$8

which determines the localized prediction set

$X\in\mathcal{X}\subseteq\mathbb{R}^d$9

For each training covariate $Y\in\mathcal{Y}$0, the same construction gives a localized set

$Y\in\mathcal{Y}$1

and an associated auxiliary robust decision

$Y\in\mathcal{Y}$2

These auxiliary decisions are used to estimate local decision risk at a test covariate $Y\in\mathcal{Y}$3: $Y\in\mathcal{Y}$4

The individualized model selector is then

$Y\in\mathcal{Y}$5

The final prediction set and robust decision are

$Y\in\mathcal{Y}$6

$Y\in\mathcal{Y}$7

The defining feature is that the model index is itself a function of covariates,

$Y\in\mathcal{Y}$8

Different regions of $Y\in\mathcal{Y}$9 may therefore use different models.

The implementation details stated in the paper include the following. The significance level $z\in\mathcal{Z}$0 plays the same role as in conformal prediction and determines the robustness level. The kernel $z\in\mathcal{Z}$1 is typically Gaussian,

$z\in\mathcal{Z}$2

or a box kernel. Distances may be Euclidean or task-specific, such as embedding distance for images. The bandwidth $z\in\mathcal{Z}$3 scales like $z\in\mathcal{Z}$4 in theory; in practice, it is chosen so that an effective sample size within the kernel neighborhood is at least a preset threshold, for example 50. The theoretical error depends on $z\in\mathcal{Z}$5, so moderate finite candidate sets are assumed. No sample splitting is needed; all labeled data contribute to both quantile estimation and risk estimation (Bao et al., 7 Jul 2025).

4. Theoretical properties

The theoretical analysis assumes i.i.d. data $z\in\mathcal{Z}$6, bounded loss $z\in\mathcal{Z}$7, and regularity conditions on the score distribution and decision map. In particular, the density $z\in\mathcal{Z}$8 of $z\in\mathcal{Z}$9 is bounded below by $\phi(Y,z)$0; the conditional score CDFs $\phi(Y,z)$1 are Lipschitz in $\phi(Y,z)$2; the conditional score densities are bounded below by $\phi(Y,z)$3 near the $\phi(Y,z)$4-quantile; the decision map is Lipschitz in the score threshold; and the conditional expectation of the induced loss is Lipschitz in $\phi(Y,z)$5.

Under these conditions, Theorem 4.1 establishes asymptotic conditional robustness. If $\phi(Y,z)$6 and $\phi(Y,z)$7, then almost surely

$\phi(Y,z)$8

The conditional miscoverage deficit therefore vanishes as $\phi(Y,z)$9, yielding asymptotic conditional robustness in the decision sense.

Theorem 4.2 gives a corresponding result for conditional decision efficiency. Under Assumptions 3.2, 4.1, and 4.2 and the same bandwidth regime,

$Y\mid X=x$0

is bounded by a rate term that, when $Y\mid X=x$1 and $Y\mid X=x$2, simplifies to nearly $Y\mid X=x$3 up to log factors. This is presented as the usual nonparametric rate for kernel regression, up to logarithmic factors.

The theoretical distinction between marginal and conditional guarantees is important. E-CROMS and F-CROMS guarantee robustness only in a marginal sense over the joint distribution of $Y\mid X=x$4, so rare covariate regions can exhibit undercoverage or inefficient decisions even when average performance is satisfactory. CROiMS targets conditional performance at each $Y\mid X=x$5. The paper also notes that perfect distribution-free conditional coverage is impossible in finite samples, so asymptotic conditional guarantees under smoothness assumptions are positioned as the relevant attainable objective (Bao et al., 7 Jul 2025).

5. Relation to CROMS, E-CROMS, and F-CROMS

The paper presents CROiMS as an extension of model-selection procedures for conformalized robust optimization that operate at the average rather than conditional level.

E-CROMS builds a standard global conformal set for each model,

$Y\mid X=x$6

computes the corresponding CRO decision $Y\mid X=x$7 for each training point, and selects

$Y\mid X=x$8

It then uses $Y\mid X=x$9 for the final decision. The stated guarantees are asymptotic optimality with respect to the average oracle risk $\mathcal{U}(x)\subseteq\mathcal{Y}$0 and marginal robustness up to an $\mathcal{U}(x)\subseteq\mathcal{Y}$1 deficit.

F-CROMS instead uses full conformal-style symmetry on an augmented dataset with an imputed label $\mathcal{U}(x)\subseteq\mathcal{Y}$2 for the test point. It defines a symmetric model-selection rule $\mathcal{U}(x)\subseteq\mathcal{Y}$3 and prediction set

$\mathcal{U}(x)\subseteq\mathcal{Y}$4

Its guarantees are exact finite-sample marginal robustness and asymptotic average optimality under a gap condition separating the best and sub-optimal models.

The comparison can be summarized as follows.

Method	Selection target	Robustness guarantee
E-CROMS	Average decision risk $\mathcal{U}(x)\subseteq\mathcal{Y}$5	Asymptotic marginal robustness
F-CROMS	Average decision risk $\mathcal{U}(x)\subseteq\mathcal{Y}$6	Exact finite-sample marginal robustness
CROiMS	Conditional decision risk $\mathcal{U}(x)\subseteq\mathcal{Y}$7	Asymptotic conditional robustness

CROiMS differs from these variants in three ways. First, its objective is local rather than global. Second, it uses kernel-weighted empirical risk depending on the test covariate $\mathcal{U}(x)\subseteq\mathcal{Y}$8, whereas CROMS variants use unweighted empirical averages. Third, while all three procedures avoid sample splitting, only CROiMS is designed to achieve conditional rather than merely marginal guarantees. The paper states that CROiMS is preferable when conditional behavior matters, when candidate models excel in different regions of $\mathcal{U}(x)\subseteq\mathcal{Y}$9, or when asymptotic conditional guarantees are desired (Bao et al., 7 Jul 2025).

6. Empirical behavior, applications, and limitations

The empirical evaluation includes synthetic classification and regression experiments in which different models are trained on different covariate subsets or under covariate shift. The comparisons involve CROiMS, E-CROMS, F-CROMS, a naive model choice with standard conformal prediction, and Naive-LCP, defined as random model selection combined with localized conformal prediction. Reported metrics include average loss, marginal miscoverage, marginal misrobustness, worst-case conditional miscoverage and misrobustness over local neighborhoods $$z(x)=\argmin_{z\in\mathcal{Z}}\max_{c\in\mathcal{U}(x)}\phi(c,z),$$0, and group conditional loss for covariate groups $$z(x)=\argmin_{z\in\mathcal{Z}}\max_{c\in\mathcal{U}(x)}\phi(c,z),$$1. The findings reported are that CROiMS consistently achieves the lowest average loss, often substantially below global methods and naive baselines, and that its conditional miscoverage approaches $$z(x)=\argmin_{z\in\mathcal{Z}}\max_{c\in\mathcal{U}(x)}\phi(c,z),$$2 as $$z(x)=\argmin_{z\in\mathcal{Z}}\max_{c\in\mathcal{U}(x)}\phi(c,z),$$3 grows. Naive-LCP attains good conditional coverage but higher decision loss, which the paper interprets as evidence that decision-aware individualized model selection matters.

A medical imaging case study uses the COVID-19 Radiography Database with chest X-rays labeled as Normal, Pneumonia, COVID-19, and Lung Opacity. Decisions correspond to No Action, Antibiotics, Quarantine, and Additional Testing, and a loss matrix encodes clinical utility. Four candidate CNN models are trained on datasets with different label distributions, simulating hospitals with different casemix, while labeled and test data are drawn from a balanced distribution. The similarity kernel is based on a feature extractor over the four models’ softmax outputs, so localization occurs in prediction space. The reported outcomes are that CROiMS achieves the lowest average loss across all methods, has worst-case conditional misrobustness close to target $$z(x)=\argmin_{z\in\mathcal{Z}}\max_{c\in\mathcal{U}(x)}\phi(c,z),$$4, and attains the lowest group conditional risk in all analyzed groups.

A second real-data study uses HAM10000 dermoscopic skin lesion images, with candidate models trained on different age ranges. Decisions are No Action, Additional Test, and Disease, with a loss matrix reflecting the costs of false reassurance and unnecessary intervention. The paper reports that CROiMS significantly reduces average and group-conditional loss, especially in certain age groups, with reductions exceeding 40% in some groups, while maintaining conditional robustness near the nominal level.

The broader context given in the paper places CROiMS at the intersection of conformal prediction with conditional guarantees, contextual or distributionally robust optimization, and decision-focused or end-to-end optimization. Potential application domains listed are precision medicine and treatment recommendation, portfolio management and asset allocation, inventory control or newsvendor problems under regional or seasonal covariate shifts, and personalized risk prediction in insurance or credit scoring.

The stated limitations are equally specific. Scalability can be problematic because the method solves many CRO problems per model and per training point. High-dimensional covariates raise the usual kernel-method curse of dimensionality, making alternative localization schemes such as learned metrics or representation learning a natural direction. Extending the framework to structured outputs, sequences, or image-valued labels would require more complex robust optimization and scoring rules. The current theory assumes i.i.d. or exchangeable data, so online, time-series, bandit, or adversarial settings remain open problems. These limitations suggest that the main contribution of CROiMS is not a universal replacement for global conformalized robust optimization, but a covariate-adaptive framework for settings in which local conditional structure is central to decision quality and robustness (Bao et al., 7 Jul 2025).