Singleton-Optimized Conformal Prediction (SOCOP)

• Singleton-Optimized Conformal Prediction is a method that produces clear, single-element prediction sets while retaining rigorous marginal coverage guarantees.
• It employs a geometric reformulation and per-instance Lagrangian minimization to select top-k classes efficiently under a constrained optimization framework.
• Empirical evaluations on datasets like ImageNet show that SOCOP can boost the singleton prediction rate by over 20% with minimal impact on average set size.

Singleton-Optimized Conformal Prediction (SOCOP) is a variant of conformal prediction that prioritizes the production of unambiguous (singleton) prediction sets, while retaining the finite-sample marginal coverage guarantees inherent to conformal methods. Traditional conformal prediction frameworks optimize efficiency in terms of average set size, but in many practical deployments—such as decision automation, diagnostics, and interactive AI—singleton outputs are substantially more actionable and less costly to process than ambiguous or multi-element prediction sets. SOCOP directly targets this operational desideratum by modifying the nonconformity score and the corresponding conformal selection rule to minimize the probability of non-singleton output, subject to the usual validity constraint.

1. Motivation and Objective

The core motivation behind SOCOP is the recognition that standard efficiency metrics such as expected set size, $E[|C(X)|]$, are only proxies for the operational goal of generating clear, singleton decisions. In applications, ambiguous sets (i.e., $|C(x)| > 1$) can trigger costly workflows or require human adjudication. SOCOP is designed to directly minimize $P(|C(x)| > 1)$—the probability of non-singleton outputs—while maintaining the marginal coverage guarantee $P(Y \in C(X)) \geq 1 - \alpha$.

Formally, SOCOP addresses the following constrained optimization:

$$\min_{C \in \mathcal{M}}\, F_\lambda(C) = P_X[|C(X)| > 1] + \lambda\,\mathbb{E}_X[|C(X)|] \quad\text{subject to}\quad P(Y \in C(X)) \geq 1 - \alpha$$

with a non-negative regularization parameter $\lambda$ that controls the trade-off between singleton frequency and average set size. The objective captures both the singleton rate and a regularized penalty for overly large sets to prevent pathological solutions.

2. Methodological Framework

The SOCOP method constructs its nonconformity score by leveraging a reduction of the constrained optimization problem to a per-instance Lagrangian minimization. For each $x$ with estimated class probabilities $\gamma = \hat{p}(\cdot|x)$ over $K$ classes (sorted so that $\gamma_{y_1} \geq \cdots \geq \gamma_{y_K} > 0$), the relevant per-instance cost is:

$$\ell_{\gamma, \lambda}(S; \eta) = \mathbb{I}(|S| > 1) + \lambda|S| - \eta \sum_{y \in S} \gamma(y)$$

where $S \subseteq \mathcal{Y}$ and $\eta \geq 0$ is the Lagrange multiplier corresponding to the coverage constraint.

Key structural results:

• The solution $$S_{\eta,\gamma} = \arg\min_S \ell_{\gamma,\lambda}(S;\eta)$$ is always of the form $\{y_1, ..., y_k\}$ for some $k$, i.e., the "top-$k$" classes by probability.
• The "nested sets" property holds: for $\eta_1 < \eta_2$, $S_{\eta_1,\gamma} \subseteq S_{\eta_2,\gamma}$.
• Define $g_k = \mathbb{I}(k > 1) + \lambda k$ and $\Gamma_k = \sum_{i=1}^k \gamma_{y_i}$. The family of candidate sets maps to points $P_k = (\Gamma_k, g_k)$ in $\mathbb{R}^2$, and the optimal $k$ index arises as the lowest value on the lower convex hull as $\eta$ varies.

Nonconformity scores are then assigned as:

$$r(x, y) = \min\{\eta \geq 0 : y \in S_{\eta, \hat{p}(\cdot|x)}\}$$

This construction is compatible with the split conformal prediction framework, enabling calibration via standard quantile procedures for the nonconformity scores across a validation (calibration) set.

3. Geometric Reformulation and Algorithmic Solution

A geometric insight is used to enable fast $O(K)$ computation of the SOCOP nonconformity score:

• For each $k=0, ..., K$, compute $P_k = (\Gamma_k, g_k)$.
• The lower convex hull of these points (constructed by monotone chain/Andrew's algorithm) yields the set of critical slopes (break points), along which the minimum over $k$ of $g_k - \eta \Gamma_k$ changes.
• The algorithm "walks" through the convex hull edges, efficiently identifying for each label $y_i$ the smallest $\eta$ for which it enters the top-$k$ set.

This reduction allows per-instance complexity to scale linearly in the number of classes. The method is thus practical for modern classification problems with large label spaces.

4. Theoretical Properties

SOCOP inherits the marginal coverage guarantee of conformal prediction by construction. That is, for error level $\alpha$, $P(Y \in C(X)) \geq 1 - \alpha$ holds—exchangeability of the calibration data being the only requirement.

The new singleton-optimized nonconformity score does not compromise coverage but does alter the geometry of the selection region: for a fixed $\alpha$, the threshold may admit slightly larger prediction sets in ambiguous cases, but the overall frequency of singleton outputs increases substantially. The user can control the $\lambda$ regularization parameter to navigate the trade-off between singleton frequency and average set size.

5. Empirical Performance

SOCOP has been evaluated on large-scale classification tasks (ImageNet-Val, ImageNet-V2) and multiple-choice question answering with LLMs (e.g., Llama-3.1-8B-Instruct on MMLU) (Wang et al., 28 Sep 2025). Key findings include:

• For ImageNet-classification at $\alpha = 0.05$, SOCOP increased the singleton prediction rate by over 20% compared to baselines (e.g., negative log-probability or cumulative probability-based nonconformity scores) with only a minor increase in average set size.
• In all settings evaluated, SOCOP maintained nominal marginal coverage.
• The singleton rate improvement is particularly pronounced in highly ambiguous or class-imbalanced tasks and for models whose raw outputs are poorly calibrated for cumulative set-size optimization.
• This indicates that optimizing for singleton rate directly, rather than average set size, is an effective strategy for practical deployment scenarios where ambiguous prediction sets are undesirable.

A representative comparison is organized below:

Method	Singleton Rate $\uparrow$	$\mathbb{E}[|C(X)|] \downarrow$	Coverage
SOCOP	Highest	Slightly higher than baseline	Nominal
Negative Log-Probability	Baseline	Lowest	Nominal
RAPS, Cumulative Mass	Intermediate	Intermediate	Nominal

The table demonstrates that SOCOP increases the frequency of singleton predictions, sometimes by over 20%, with only a marginal penalty on average set size.

6. Relation to Existing Efficiency-Optimized Methods

Traditional conformal methods, including those using negative log-probability nonconformity scores and methods based on cumulative class probability thresholds, aim to minimize $\mathbb{E}[|C(X)|]$, not $P(|C(X)| > 1)$. Modifications like RAPS introduce regularization or adjust rankings, but the underlying objective remains average size reduction. SOCOP's explicit inclusion of the singleton objective differentiates it operationally and mathematically.

Other efficiency-oriented innovations—such as nested conformal prediction with quantile regression (Gupta et al., 2019), gradient-descent surrogate optimization of set size (Bellotti, 2021), and constrained empirical risk minimization over general function classes (Bai et al., 2022)—demonstrate the breadth of methods that can be paired with the conformal prediction machinery to refine efficiency. SOCOP can be interpreted as a member of this family, but is distinct in its use of cost functions and selection rules directly targeting singleton outputs.

7. Practical and Theoretical Implications

The SOCOP framework provides a systematic method for practitioners requiring unambiguous predictions with statistical coverage guarantees. It is particularly valuable in settings where ambiguous sets impose high downstream costs or when operational constraints necessitate clear decisions. The efficient $O(K)$ algorithm ensures scalability.

SOCOP also admits natural generalization to customized objectives beyond singleton rate—for instance, optimizing for at-most-$k$ outputs (smallest ambiguous set), or refining set selection with application-specific trade-offs. A plausible implication is that future deployments in domains such as high-stakes medical triage, legal decision support, or autonomous control will preferentially adopt singleton-optimized conformal prediction as a standard operational protocol when set ambiguity is costly.

Summary

Singleton-Optimized Conformal Prediction (SOCOP) systematically reorients the conformal prediction objective from average set size to singleton frequency, leveraging a novel nonconformity score derived from a geometric reformulation of the constrained efficiency-validity trade-off. SOCOP achieves significant increases in the frequency of unambiguous predictions—sometimes by over 20%—with negligible impact on coverage and only minor increases in average set size, making it highly suitable for operational scenarios that require decision clarity and rigorous statistical guarantees (Wang et al., 28 Sep 2025).