Tail-Aware Conformal Prediction (TACP)

• Tail-Aware Conformal Prediction (TACP) is a method that modifies standard conformal calibration to address tail asymmetries and undercoverage of rare classes.
• It employs selective score transformations, including a soft variant (sTACP), to balance prediction set inclusion between frequent head and minority tail classes.
• Empirical results on benchmarks like CIFAR100-LT and ImageNet-LT show that TACP significantly reduces head-tail coverage gaps while preserving marginal validity.

Searching arXiv for the most relevant papers on Tail-Aware Conformal Prediction and closely related methods. Tail-Aware Conformal Prediction (TACP) denotes a family of conformal prediction methods that modify standard conformal calibration so that uncertainty sets are more responsive to tail asymmetries, rare classes, or other high-risk regions that are obscured by purely marginal coverage guarantees. In contemporary usage, the term has both a narrow and a broad sense. In the narrow sense, it refers directly to the long-tailed classification method introduced as “Tail-Aware Conformal Prediction” and its extension soft TACP, which target the under coverage of minority classes under long-tail label distributions (Liu et al., 15 Aug 2025). In the broader sense, it encompasses adjacent conformal methodologies that allocate miscoverage asymmetrically across tails, localize calibration to underrepresented regions, or reshape training objectives so that conformal predictors behave more evenly across rare classes or difficult strata, while typically retaining standard marginal split-conformal validity (Cuonzo et al., 16 Jun 2026, Marani et al., 14 Jan 2026).

1. Conceptual scope and problem formulation

Classical conformal prediction constructs prediction sets or intervals with marginal coverage at level $1-\alpha$, for example

$$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$

This guarantee is distribution-free under exchangeability, but it averages over the underlying population and therefore does not ensure balanced behavior across classes, subgroups, or directional tails (Liu et al., 15 Aug 2025, Cuonzo et al., 16 Jun 2026).

In long-tailed classification, this limitation becomes acute because head classes dominate calibration statistics. The paper “Conformal Prediction Meets Long-tail Classification” introduces TACP precisely in the setting where training, calibration, and test data all follow a long-tail distribution, and observes that standard conformal methods tend to over cover head classes and under cover tail classes, even while satisfying marginal coverage on average (Liu et al., 15 Aug 2025). The paper formalizes this disparity via head and tail groups $\mathcal{G}_h$ and $\mathcal{G}_t$, and evaluates

$$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$

together with the head-tail coverage gap

$$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$

In this sense, TACP is a direct response to the fact that marginal coverage can conceal systematic unreliability on minority classes (Liu et al., 15 Aug 2025).

A second line of work uses “tail-aware” in a directional sense for regression. “Conformal Prediction Intervals with Tail-Specific Guarantees” constructs lower and upper one-sided conformal intervals with separate error budgets $\alpha^-$ and $\alpha^+$, then intersects them into a two-sided interval $[L_{n+1},U_{n+1}]$ satisfying

$$\mathbb P(Y_{n+1}<L_{n+1})\le \alpha^-, \qquad \mathbb P(Y_{n+1}>U_{n+1})\le \alpha^+,$$

with $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$0 (Cuonzo et al., 16 Jun 2026). This is tail-aware in a different sense: it calibrates lower and upper tails separately rather than redistributing coverage across head and tail classes.

These two directions share a common principle: standard conformal prediction treats errors too coarsely whenever the relevant pathology is localized in rare classes, skewed tails, or difficult regions. Tail-aware approaches alter either the score, the calibration rule, or the training objective to expose and mitigate those asymmetries.

2. TACP for long-tailed classification

The most direct use of the term appears in “Conformal Prediction Meets Long-tail Classification” (Liu et al., 15 Aug 2025). The paper considers a pretrained probabilistic classifier

$$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$1

with standard split conformal prediction sets of the form

$$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$2

where $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$3 is calibrated from held-out non-conformity scores.

The head group is defined as the smallest subset of labels whose cumulative prior mass exceeds a threshold $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$4: $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$5 and the tail group is $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$6 (Liu et al., 15 Aug 2025).

The defining TACP score augments a base non-conformity score $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$7 with a selective rank penalty applied only to head labels: $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$8 where $$\mathbb{P}(Y_{n+1}\in \mathcal{C}(X_{n+1}))\ge 1-\alpha.$$9 is the rank of label $\mathcal{G}_h$0 under the model’s predictive probabilities, $\mathcal{G}_h$1 is a rank threshold, and $\mathcal{G}_h$2 is a penalty strength (Liu et al., 15 Aug 2025). This modification leaves tail labels untouched and penalizes only head labels ranked below the top $\mathcal{G}_h$3.

The practical effect is twofold. First, head labels become slightly harder to include unless they are highly ranked. Second, because the calibration threshold is recomputed on the modified scores, the global threshold can increase, which in turn makes it easier for tail labels to enter the prediction set. The paper proves that TACP preserves marginal coverage in the standard split-conformal sense,

$\mathcal{G}_h$4

and further proves that there exists $\mathcal{G}_h$5 such that the head-tail conditional coverage gap is no larger than under the corresponding standard method built from the same base score (Liu et al., 15 Aug 2025).

The framework is intentionally score-agnostic. The paper evaluates TACP with APS, LAC, TOPK, and RAPS. The underlying scores are

$\mathcal{G}_h$6

$\mathcal{G}_h$7

$\mathcal{G}_h$8

and

$\mathcal{G}_h$9

with $\mathcal{G}_t$0 used for tie-breaking (Liu et al., 15 Aug 2025).

3. soft TACP and class-wise balancing

The same paper introduces soft TACP, or sTACP, as an extension of the binary head-tail adjustment into a continuous class-prior-weighted adjustment (Liu et al., 15 Aug 2025). The motivation is that a hard partition at threshold $\mathcal{G}_t$1 is appropriate for reducing the head-tail gap but is too coarse if the goal is balanced coverage across all classes.

sTACP replaces the indicator $\mathcal{G}_t$2 with the estimated class prior $\mathcal{G}_t$3: $\mathcal{G}_t$4 Frequent classes therefore receive larger regularization than rare classes, but the adjustment now varies smoothly over the long-tail frequency spectrum rather than being switched on or off by group membership (Liu et al., 15 Aug 2025).

The paper evaluates sTACP using the class-conditional coverage gap

$\mathcal{G}_t$5

together with overall coverage and average set size (Liu et al., 15 Aug 2025). The paper does not state a separate theorem for reduced class-conditional gap under sTACP, but presents it as a more class-balanced variant of the same score-reshaping principle.

This distinction between TACP and sTACP is conceptually important. TACP addresses a groupwise asymmetry between head and tail classes. sTACP moves toward class-conditional balancing without resorting to fully classwise thresholds, which can become unstable in extreme long-tail regimes because rare classes may have very few calibration points.

4. Empirical behavior on long-tailed benchmarks

The empirical evidence for TACP and sTACP is concentrated on CIFAR100-LT and ImageNet-LT, with imbalance settings such as $\mathcal{G}_t$6 for CIFAR100-LT and Pareto power $\mathcal{G}_t$7 for ImageNet-LT (Liu et al., 15 Aug 2025). The pretrained models are taken from MetaSAug, and the methods are compared to STANDARD, Partition-Wise, CLASSWISE, CLUSTER, and RC3P depending on the experiment.

The central empirical claim is that TACP sharply reduces $\mathcal{G}_t$8. Representative APS results include ImageNet-LT at $\mathcal{G}_t$9, where STANDARD gives $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$0 and average set size $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$1, while TACP gives $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$2 and average set size $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$3 (Liu et al., 15 Aug 2025). On CIFAR100-LT at $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$4, APS gives STANDARD $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$5 and TACP $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$6, with average set sizes $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$7 and $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$8, respectively (Liu et al., 15 Aug 2025). For TOPK on ImageNet-LT at $$\text{Cov-head}=100\times \frac{\sum_{i\in \mathcal{I}_h} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_h|}, \qquad \text{Cov-tail}=100\times \frac{\sum_{i\in \mathcal{I}_t} \mathbb{I}( y'_i\in \mathcal{C}(\bm{x}'_i))}{|\mathcal{I}_t|},$$9, STANDARD yields $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$0 while TACP yields $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$1 (Liu et al., 15 Aug 2025).

sTACP is evaluated primarily on class-conditional coverage balance. On ImageNet-LT at $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$2 and $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$3, APS under STANDARD gives $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$4, coverage $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$5, and average set size $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$6, while sTACP gives $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$7, coverage $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$8, and average set size $$\text{CovGap-HT} = |\text{Cov-head}- \text{Cov-tail}|.$$9 (Liu et al., 15 Aug 2025). The pattern suggests that smoother frequency-aware regularization improves classwise balance, though sometimes at the cost of larger sets.

The same paper also underscores the practical weakness of fully classwise calibration in fully long-tailed settings. On ImageNet-LT at $\alpha^-$0, CLASSWISE with APS produced average set size $\alpha^-$1 and coverage $\alpha^-$2, illustrating the severe conservatism induced by scarce per-class calibration data (Liu et al., 15 Aug 2025). This is a recurrent theme in long-tail conformal prediction: exact classwise validity is often too expensive in efficiency when the calibration sample for rare classes is tiny.

5. Related long-tail and class-adaptive conformal methods

TACP sits within a broader literature on long-tail-aware conformal prediction, but its mechanism is distinctive. “Conformal Prediction for Long-Tailed Classification” develops two complementary strategies: a prevalence-adjusted softmax score and a label-weighted conformal calibration rule (Ding et al., 9 Jul 2025).

The prevalence-adjusted softmax score is

$\alpha^-$3

motivated by an oracle result showing that thresholding $\alpha^-$4 optimizes the trade-off between expected set size and macro-coverage (Ding et al., 9 Jul 2025). This is tail-aware in the sense that dividing by prevalence boosts rare classes directly in the score. The same paper also defines label-weighted class-specific thresholds

$\alpha^-$5

with $\alpha^-$6, thereby interpolating between standard and classwise conformal prediction (Ding et al., 9 Jul 2025). In contrast to TACP, which keeps one global threshold but modifies scores for head classes, PAS and label-weighted CP alter either the score or the effective threshold in a prevalence-aware manner.

“Class Adaptive Conformal Training” addresses the same pathology from the training side rather than the post-hoc calibration side (Marani et al., 14 Jan 2026). CaCT replaces the single global set-size penalty used by prior conformal training methods with class-specific multipliers learned via an augmented Lagrangian method. Its key constraint shift is from a marginal size target to class-wise targets,

$\alpha^-$7

and the learned class-wise multipliers are updated using

$\alpha^-$8

with class-dependent penalty coefficients $\alpha^-$9 (Marani et al., 14 Jan 2026). This does not modify the conformal calibration rule into an explicitly tail-aware procedure, but it reshapes the learned representation so that head-vs-tail disparities in set size and coverage are reduced downstream. The paper explicitly notes that CaCT “shapes prediction sets class-conditionally without making any distributional assumptions” (Marani et al., 14 Jan 2026). A plausible implication is that CaCT and TACP-like post-hoc methods are complementary rather than mutually exclusive.

These related methods help delimit the meaning of TACP. In the narrow sense, TACP refers to the selective head-penalty score transformation of (Liu et al., 15 Aug 2025). In the broader sense, it includes long-tail-aware score design, label-weighted calibration, and class-adaptive conformal training, all aimed at mitigating the same failure of marginal conformal prediction under severe imbalance.

6. Tail-specific guarantees in regression

A different but closely related use of tail-awareness appears in regression, where the issue is not head versus tail classes but lower versus upper predictive tails. “Conformal Prediction Intervals with Tail-Specific Guarantees” constructs lower and upper one-sided conformal intervals

$\alpha^+$0

and then intersects them to obtain

$\alpha^+$1

(Cuonzo et al., 16 Jun 2026).

The one-sided lower and upper guarantees are

$\alpha^+$2

which are equivalent to

$\alpha^+$3

By a union bound, the induced two-sided interval satisfies

$\alpha^+$4

(Cuonzo et al., 16 Jun 2026).

The paper gives multiple one-sided score choices. For residual-based lower and upper scores,

$\alpha^+$5

For standardized residuals,

$\alpha^+$6

The paper also emphasizes signed one-sided quantile scores,

$\alpha^+$7

which avoid the tie-induced conservatism of truncated one-sided quantile scores in skewed settings (Cuonzo et al., 16 Jun 2026).

This framework is directly tail-aware in the directional sense, and can be read as a regression analogue of TACP. The trade-off is explicit: intersecting two independently calibrated one-sided intervals yields strong directional guarantees but can be conservative relative to a single globally calibrated two-sided interval. The paper notes this efficiency cost and proves that in the equal-allocation case $\alpha^+$8, the final coverage can lie between $\alpha^+$9 and approximately $[L_{n+1},U_{n+1}]$0 (Cuonzo et al., 16 Jun 2026). This illustrates a general tension in tail-aware conformal design: stronger granularity of calibration often sacrifices global efficiency.

7. Broader interpretations, misconceptions, and limitations

A common misconception is that marginal conformal validity alone protects minority classes or both directional tails. The long-tail classification literature explicitly rejects this. Both (Liu et al., 15 Aug 2025) and (Ding et al., 9 Jul 2025) show that standard marginal coverage can coexist with systematic undercoverage on rare classes. Likewise, (Cuonzo et al., 16 Jun 2026) shows that a standard two-sided interval can satisfy global coverage while misallocating its misses across lower and upper tails.

A second misconception is that tail-aware conformal prediction necessarily requires exact class-conditional or conditional-on-$[L_{n+1},U_{n+1}]$1 guarantees. The surveyed papers do not make that claim. TACP preserves the standard marginal split-conformal guarantee while reducing the head-tail coverage gap, but does not provide finite-sample exact tail-class validity (Liu et al., 15 Aug 2025). The tail-specific regression method gives finite-sample lower-tail and upper-tail marginal guarantees, but not full $[L_{n+1},U_{n+1}]$2-conditional directional validity (Cuonzo et al., 16 Jun 2026). CaCT improves class-conditional efficiency behavior through training, but its formal guarantee remains ordinary marginal split-conformal validity (Marani et al., 14 Jan 2026).

A third misconception is that tail-awareness is only a calibration issue. CaCT argues instead that the representation and score geometry learned by the base model are part of the bottleneck in long-tailed conformal prediction (Marani et al., 14 Jan 2026). This suggests a division of labor: some methods change the score or calibration rule post hoc, whereas others alter training so that later conformalization behaves more evenly across classes.

The main limitations of current TACP-style methods follow directly from the data. Fully classwise conformal calibration is unstable when rare classes have almost no calibration examples; this is why TACP, sTACP, PAS, and label-weighted methods trade exact classwise validity for improved balance with usable set sizes (Liu et al., 15 Aug 2025, Ding et al., 9 Jul 2025). In regression, separate lower- and upper-tail calibration yields interpretable tail-specific guarantees but can widen intervals due to the conservatism of intersection (Cuonzo et al., 16 Jun 2026). More broadly, most available guarantees remain marginal or long-run-average rather than conditional.

Taken together, these papers indicate that TACP is not a single algorithmic template but a design agenda: expose the asymmetries that marginal conformal prediction averages away, then modify scores, thresholds, or training objectives so that those asymmetries are reduced without abandoning the core distribution-free calibration logic of split conformal prediction. In its narrowest and most literal form, that agenda is instantiated by the head-selective score transformation of (Liu et al., 15 Aug 2025). In its broader research context, it includes class-adaptive training (Marani et al., 14 Jan 2026), prevalence-aware long-tail calibration (Ding et al., 9 Jul 2025), and directional tail-specific interval construction (Cuonzo et al., 16 Jun 2026).