Split-Localized Conformal Prediction (SLCP)

• SLCP is an advanced prediction method that constructs locally adaptive intervals using weighted calibration based on feature proximity.
• It adjusts interval widths to reflect local heteroskedasticity and covariate shifts, offering practical improvements over standard methods.
• SLCP maintains finite-sample, distribution-free marginal coverage while converging to conditional validity as sample sizes increase.

Split-Localized Conformal Prediction (SLCP) is a generalization of split conformal prediction that enables the construction of prediction sets (or intervals) with finite-sample, distribution-free marginal coverage guarantees, while adaptively calibrating interval widths to the local feature distribution of new test points. By utilizing weighted quantile computations based on proximity and local density in feature space, SLCP achieves enhanced local adaptivity compared to standard conformal methods. This approach is motivated by the need to approximate conditional coverage in heterogeneous or covariate-shifted settings, where standard conformal procedures may be overly conservative or fail to account for feature-dependent uncertainty (Guan, 2019, Han et al., 2022, Guan, 2021).

1. Problem Setting and Motivation

Let $(X_i, Y_i)$, $i=1,\ldots,n$, be independent draws from an unknown distribution $\mathbb{P}$ on $\mathcal{X} \times \mathcal{Y}$. The task is, for a user-specified miscoverage level $\alpha \in (0,1)$, to construct a prediction set (or interval) $C_{n,\alpha}(X_{n+1}) \subseteq \mathcal{Y}$ for a new test point $X_{n+1}$ such that: $$\Pr\left\{ Y_{n+1} \in C_{n,\alpha}(X_{n+1}) \right\} \ge 1-\alpha$$ marginally over all distributions $\mathbb{P}$ and in finite samples.

Traditional split conformal methods provide this guarantee in marginal coverage, averaging over the entire joint distribution of $(X, Y)$. However, in practice, practitioners may seek conditional coverage: $$\Pr(Y_{n+1} \in C(x) \mid X_{n+1} = x) \ge 1-\alpha\quad\forall x,$$ so that intervals adapt to heteroskedasticity and other feature-dependent effects. SLCP was developed to address the limitations of unconditional coverage by leveraging local information near the test point to better capture the conditional distribution of residuals (Guan, 2019, Han et al., 2022).

2. SLCP Procedure

The Split-Localized Conformal Prediction algorithm enhances standard split conformal prediction through a locality-aware weighting scheme in the calibration step. The typical SLCP instantiation in regression comprises the following steps (Guan, 2019, Han et al., 2022, Guan, 2021):

1. Data splitting: Partition indices into a proper training set $\mathcal{I}_{\mathrm{train}}$ and a calibration set $\mathcal{I}_{\mathrm{cal}}$.
2. Model fitting: Fit a prediction model (e.g., regression function $\mu(\cdot)$) using the training set.
3. Conformity score computation: For each $i \in \mathcal{I}_{\mathrm{cal}}$, calculate the conformity score $R_i = s(X_i, Y_i)$, commonly as $|Y_i - \mu(X_i)|$.
4. Localization kernel and weights: Define a similarity kernel $H:\mathcal{X} \times \mathcal{X} \to [0, \infty)$, e.g., $H(x, x') = \exp(- \|x - x'\|^2 / h^2 )$ with bandwidth $h>0$. Compute weights:

$$w_i = \frac{H(X_i, X_{n+1})}{\sum_{j \in \mathcal{I}_{\mathrm{cal}}} H(X_j, X_{n+1})}$$

so that points close to $X_{n+1}$ in feature space have higher influence in calibration.

1. Weighted quantile calculation: Form the weighted empirical distribution over $\{R_i : i \in \mathcal{I}_{\mathrm{cal}}\}$ using $\{w_i\}$. The $(1-\alpha)$-level weighted quantile is:

$$\hat q_{1-\alpha} = \inf\left\{ q : \sum_{i\in \mathcal{I}_{\mathrm{cal}}} w_i\,\mathbf{1}\{R_i \le q\} \ge 1-\alpha \right\}$$

2. Prediction region: For a new $y\in\mathcal{Y}$, define the conformity score $R_{n+1}(y) = s(X_{n+1}, y)$. The SLCP prediction region is:

$$C_{n,\alpha}(X_{n+1}) = \{ y : R_{n+1}(y) \le \hat q_{1-\alpha} \}$$

As a result, SLCP outputs intervals or prediction sets tailored to the local calibration scores around the test sample.

3. Theoretical Guarantees

SLCP retains the finite-sample, fully distribution-free marginal coverage guarantee that characterizes standard split conformal prediction. Under the assumption that the calibration set and $(X_{n+1}, Y_{n+1})$ are exchangeable, and with $H(x,x) = 1$ for all $x$ (ensuring test points have positive self-weight), one has: $$\Pr\left\{ Y_{n+1} \in C_{n,\alpha}(X_{n+1}) \right\} \ge 1-\alpha$$ for all distributions $\mathbb{P}$ regardless of the score function or kernel used (Guan, 2019, Guan, 2021). The proof uses a weighted version of the exchangeability argument; if each calibration and test point is assigned a weight proportional to $H(X_i, X_{n+1})$, then the probability that the test conformity score $R_{n+1}$ falls below the weighted $(1-\alpha)$ empirical quantile is at least $1-\alpha$.

Furthermore, under additional regularity conditions and as the sample size grows ($n\to\infty$ and $h_n \to 0$ at appropriate rates), the coverage of SLCP intervals converges to the conditional coverage level at any fixed test point $x_0$: $$\lim_{n\to\infty} \Pr\{ Y_{n+1} \in C(x_0) | X_{n+1} = x_0 \} = 1-\alpha$$ (Guan, 2021, Han et al., 2022). These results rely on smoothness of the conditional distribution and the feature density, and demonstrate SLCP's asymptotic local validity.

4. Comparison with Standard Split-Conformal Methods

A central feature of SLCP is its reduction to ordinary split conformal prediction when the localization kernel is uniform, $H(x,x') \equiv 1$, making all calibration weights equal ($w_i = 1/m$). In this regime, SLCP and standard split conformal are identical and do not adapt to heterogeneity in the residuals (Guan, 2019, Guan, 2021).

In the presence of feature-dependent heteroskedasticity or covariate shift, SLCP adaptively down-weights calibration points distant from the test feature in the feature space. This adaptivity yields prediction intervals that are narrower in low-variance or well-calibrated regions, and wider in high-variance or data-sparse regions. Standard split conformal methods pool all calibration scores equally, which can result in unnecessarily conservative intervals in easy regions and potentially under-covering intervals in difficult or underrepresented regions (Guan, 2019, Guan, 2021, Han et al., 2022).

5. Extensions and Practical Considerations

SLCP is compatible with many choices of conformity scores and localization kernels, and is applicable to both regression and classification settings. The kernel $H$ can be chosen as a Gaussian RBF, a $k$-nearest neighbor indicator, or another appropriate similarity measure. Bandwidth $h$ or $k$ may be selected via cross-validation or using the median of pairwise feature distances (Guan, 2019, Han et al., 2022).

Computationally, SLCP adds minimal overhead compared to standard split conformal (typically $O(m \log m)$ for sorting calibration scores; $O(m)$ for computing weights). In high-dimensional settings, feature embeddings $\vv f(x)$ from the penultimate layer of a neural network may be used in place of raw features. Mini-batching, using a subset of the training data for local density estimation, allows scaling to large $n$ and high dimensionality with little performance degradation (Han et al., 2022).

The SLCP framework generalizes to score functions constructed via kernel conditional distribution estimation (Nadaraya–Watson), enabling decorrelation of nonconformity scores from features and promoting conditional validity (Han et al., 2022). This approach unifies several localized conformal methods under a quantile-of-quantile correction architecture.

6. Empirical Behavior and Applications

Empirical studies confirm that SLCP adapts the size of prediction intervals to local uncertainty structure:

• On synthetic heteroskedastic data, SLCP yields intervals that are shorter in regions of low noise and wider where noise increases, consistently meeting the target coverage rate.
• On tabular datasets with complex feature-dependent error distributions, SLCP achieves the shortest average interval lengths and higher correlation between interval width and actual coverage indicators compared to contemporary split-conformal methods such as CQR, Dist-Split, and MAD-Split.
• In high-dimensional (e.g., image-based) age regression using ResNet features, SLCP attains target coverage across age groups with narrower intervals, outperforming LCP, CD-Split, CQR, and MAD-Split (Han et al., 2022).

A typical simulation example involves a two-regime heteroskedastic Gaussian regression, where SLCP automatically modulates interval width depending on local noise scale, whereas global split conformal yields a constant-width interval, which is suboptimal at both ends of the feature range (Guan, 2019, Guan, 2021).

Further, SLCP exhibits robustness to model misspecification, moderate covariate shift (via down-weighting of rare calibration points distant from the test feature), and maintains computational efficiency under mini-batch approximation (Han et al., 2022).

7. Summary Table: Key Elements of SLCP

Component	Description	Typical Choices
Conformity Score	$s(x,y)$, model residual or predictive error	$|y - \hat{\mu}(x)|$
Localization Kernel $H$	Measures similarity/proximity in feature space	Gaussian RBF, $k$-NN
Calibration Weights	$w_i \propto H(X_i, X_{n+1})$	Function of kernel, sum to 1
Prediction Region	$$C_{n,\alpha}(X_{n+1}) = \{ y : s(X_{n+1}, y) \le \hat{q}_{1-\alpha} \}$$	Depends on scores/weights

This procedure leverages the locality of calibration scores, achieving adaptive conformal inference with finite-sample marginal coverage and asymptotic conditional validity across a broad range of structured prediction tasks (Guan, 2019, Han et al., 2022, Guan, 2021).