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SpeedCP: Fast Kernel-based Conditional Conformal Prediction

SpeedCP is a fast, stable algorithm that bridges the gap between the theoretical promise and prohibitive computational cost of RKHS-based conditional conformal prediction by efficiently computing the full solution path and integrating low-rank embeddings, achieving a 40-fold speedup and 30% shorter intervals compared to prior methods.

Original authors: Yating Liu, Yeo Jin Jung, Zixuan Wu, So Won Jeong, Claire Donnat

Published 2026-05-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Yating Liu, Yeo Jin Jung, Zixuan Wu, So Won Jeong, Claire Donnat

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have a very smart but mysterious "black box" computer that predicts things—like whether a patient has a tumor, how much a house will sell for, or what a molecule does. You trust this computer, but you also know it makes mistakes.

Conformal Prediction is like a safety net. Instead of just giving you a single number (e.g., "The house will sell for $500,000"), it gives you a range (e.g., "Between $480,000 and $520,000"). The goal is to make sure that range is wide enough to catch the true answer 90% of the time.

However, there's a catch. A standard safety net is "one size fits all." It might be perfect for predicting houses in the city, but it could be way too wide (useless) for houses in the countryside, or too narrow (dangerous) for luxury mansions. This is called conditional coverage: we want the safety net to be the right size for every specific situation, not just on average.

The Problem: The "Super-Expensive" Safety Net

Scientists recently figured out how to build these custom, situation-specific safety nets using a fancy mathematical tool called a Reproducing Kernel Hilbert Space (RKHS). Think of this tool as a super-precise sculptor that can carve a safety net perfectly to fit the shape of the data.

But there's a huge problem: It's incredibly slow.
The previous method (by Gibbs et al., 2025) was like trying to sculpt that perfect net by hand, over and over again, testing thousands of different chisels. If you wanted to do this for a large dataset, it would take days or even weeks. It was too expensive to be practical.

The Solution: SpeedCP (The "Fast-Forward" Sculptor)

This paper introduces SpeedCP, a new method that does the exact same job but 40 times faster.

Here is how they did it, using a simple analogy:

The Old Way (Binary Search):
Imagine you are trying to find the perfect temperature for a cake. The old method was like checking the oven at 300°, then 350°, then 325°, then 337°... You have to bake a whole new cake (run a complex calculation) for every single guess to see if it's right. This is slow.

The SpeedCP Way (The Solution Path):
SpeedCP is like a smart baker who realizes that the cake batter changes in a straight line as you turn the dial. Instead of guessing random temperatures, SpeedCP calculates the entire path of how the cake changes as you turn the dial from cold to hot.

It doesn't stop to bake a cake at every step.
It only stops when something interesting happens (like the batter suddenly changing texture).
Because it knows the math behind the "batter," it can skip all the boring steps and jump straight to the perfect temperature.

Two Key Tricks SpeedCP Uses

1. The "Elbow" Shortcut
In the math, there is a group of data points that are "sitting right on the edge" of the safety net. The authors call this the Elbow Set.

The Insight: Most of the time, this group is tiny.
The Trick: Instead of doing heavy math on all the data points (which is slow), SpeedCP only does the heavy lifting on this tiny "Elbow" group. It's like solving a puzzle by only focusing on the corner pieces first, rather than trying to fit every single piece at once. This makes the calculation lightning fast.

2. The "Low-Rank" Map (For High Dimensions)
Sometimes, the data is so complex (like having 10,000 different features for a molecule) that it's impossible to find patterns. It's like trying to find a needle in a haystack that is the size of a mountain.

The Trick: SpeedCP first squashes that giant mountain of data down into a small, manageable "map" (a low-dimensional space) that keeps the most important shapes.
It then builds the safety net on this small map. This is much easier and more accurate than trying to build it on the giant, messy mountain.

What Did They Find?

The authors tested SpeedCP on real-world problems, including:

Molecular properties: Predicting how chemicals behave.
Brain tumor MRI scans: Distinguishing between healthy and tumor tissue.
Synthetic data: Made-up scenarios to test the math.

The Results:

Speed: It is roughly 40 times faster than the previous best method.
Accuracy: It produces safety nets that are 30% tighter (more precise) than the old method, meaning you get a better answer without losing safety.
Reliability: It maintains the promised 90% coverage rate across all different types of data, even when the data is tricky or uneven.

In Summary

SpeedCP is a new algorithm that takes a powerful but slow mathematical idea for creating "smart safety nets" and makes it fast enough to use in the real world. It does this by finding a clever "shortcut" through the math (the solution path) and by simplifying complex data into a manageable map before building the net. It allows computers to give you precise, reliable predictions for specific situations without waiting days for the answer.

Technical Summary: SpeedCP

Problem Statement

Conformal prediction offers a framework for constructing prediction sets with finite-sample marginal coverage guarantees under minimal distributional assumptions. However, marginal coverage does not ensure conditional coverage—the guarantee that the prediction set contains the true label with probability $1-\alpha$ for every specific input $x$ . In high-stakes applications, systematic under-coverage on critical subgroups can lead to unreliable outcomes.

While prior work by Gibbs et al. [2025] addressed conditional validity by relaxing the requirement to a user-specified function class $\mathcal{F}$ within a Reproducing Kernel Hilbert Space (RKHS), their approach suffers from prohibitive computational costs. Specifically, their method requires fitting an RKHS quantile regression for every candidate value of an imputed test score $S$ via binary search, and often relies on a pre-specified grid for hyperparameters. This makes the method intractable for large-scale or high-dimensional settings ( $p \gg n$ ), where distance-based similarities in raw covariate spaces become less discriminative.

Methodology: SpeedCP

The authors propose SpeedCP, a fast, stable algorithm that computes the full solution path of the regularized RKHS conformal optimization problem. The method operates in two primary stages:

1. Low-Rank Latent Embeddings

To handle high-dimensional covariates ( $X \in \mathbb{R}^p, p \gg n$ ), SpeedCP first maps inputs to a low-dimensional latent space $\hat{\pi}(X) \in \mathbb{R}^K$ ( $K \ll p$ ) using a symmetric low-rank embedding map (e.g., PCA, latent Dirichlet allocation, or neural network layer embeddings). The RKHS kernel $\psi^*$ is defined on this latent space, improving the signal-to-noise ratio and enabling meaningful conditional coverage on the latent representation rather than the raw sparse covariates.

2. The $(\lambda, S)$ -Path Algorithms

The core innovation is the derivation of piecewise-linear solution paths for the RKHS quantile regression parameters, avoiding the need for repeated refitting or binary search.

$\lambda$ -Path (Smoothness Control): The algorithm traces the solution of the regularized quantile regression as the regularization parameter $\lambda$ decreases. Inspired by the LARS algorithm and Li et al. [2007], the solution coefficients evolve as piecewise-linear functions of $\lambda$ . The algorithm only updates the solution when the "elbow set" (the set of observations with exactly zero residuals) changes. This allows for efficient hyperparameter tuning (selecting optimal $\lambda$ and kernel bandwidth $\gamma$ ) via cross-validation without a full grid search.
$S$ -Path (Prediction Set Construction): Once optimal hyperparameters are fixed, the algorithm traces the solution as the imputed test score $S$ increases. The authors prove that the mapping $S \mapsto \hat{\upsilon}_S$ (where $\hat{\upsilon}$ are the RKHS coefficients) is non-decreasing and piecewise-affine. This structure allows the algorithm to directly compute the maximum score cutoff $S^*$ such that the prediction set condition holds, eliminating the binary search used in Gibbs et al. [2025].
Randomization: To ensure exact conditional guarantees and mitigate coverage inflation, the method employs a randomized cutoff $S_{rand}$ derived from a uniform variable $U \sim \text{Unif}(-\alpha, 1-\alpha)$ .

Theoretical Guarantees

The paper establishes finite-sample guarantees for approximate conditional coverage with respect to the estimated latent embedding $\hat{\pi}(X)$ .

Coverage Validity: The method satisfies a marginal guarantee over any measurable function $f$ in the RKHS class $\mathcal{F}^*$ , which implies conditional coverage on latent neighborhoods.
Robustness to Estimation Error: The coverage guarantee is robust to errors in the low-rank projection $\hat{\pi}(\cdot)$ . While estimation error may affect the tightness (size) of the prediction sets, it does not compromise the validity of the coverage guarantee.
Group-Conditional Coverage: The framework naturally extends to group-conditional coverage when the linear component $\Phi^*(X)$ encodes cluster assignments derived from the latent embedding.

Key Contributions

Algorithmic Efficiency: The authors introduce a stable, fast solution-path algorithm for RKHS-based conformal prediction. By leveraging the affine structure of the solution path over both the regularization parameter $\lambda$ and the imputed score $S$ , they achieve a 40-fold speedup compared to Gibbs et al. [2025] while maintaining the same theoretical framework.
High-Dimensional Adaptation: The method extends conditional conformal prediction to high-dimensional settings by conditioning on learned low-rank embeddings. This approach yields better-calibrated prediction sets, particularly in low-density data regions where raw-space conditioning fails.
Hyperparameter Tuning: The algorithm provides a closed-form mechanism for simultaneous hyperparameter selection (smoothness $\lambda$ and kernel bandwidth $\gamma$ ) via cross-validation, avoiding the computational bottleneck of joint grid searches.
Theoretical Characterization: The paper characterizes how embedding estimation errors affect coverage validity and provides bounds on the coverage gap, showing it decreases with the regularization parameter $\lambda$ .

Experimental Results

The authors evaluate SpeedCP on synthetic admixture data, molecular property prediction (QM9, QM7b, ESOL), and brain tumor MRI analysis.

Coverage: SpeedCP achieves uniform coverage close to the nominal level (0.9) across various latent space partitions and subgroups, outperforming baselines like SplitCP (which undercovers locally) and PCP/RLCP (which often overcover).
Efficiency: In synthetic experiments, SpeedCP runs nearly 50 times faster than CondCP (Gibbs et al. [2025]). In molecular datasets, it achieves speedups of 40–60x.
Prediction Set Size: SpeedCP produces significantly tighter prediction sets than CondCP (improving interval length by ~30%) while maintaining valid coverage. It adapts to local heterogeneity in the score distribution, whereas SplitCP relies on a single global cutoff.
Robustness: The method remains effective even with low-quality base predictors (e.g., linear regression) and in high-dimensional feature spaces where other methods fail to exploit locality.

Significance and Claims

The paper claims that SpeedCP makes flexible conditional conformal prediction computationally feasible for high-dimensional data. By replacing repeated RKHS refitting and binary search with active set path tracking, the method enables the use of complex, data-driven latent spaces for calibration without sacrificing computational tractability.

The authors emphasize that their approach provides reliable conditional coverage across modern black-box predictors while significantly reducing the computational burden. They note that while the method assumes exchangeability and symmetric embedding estimation, it is robust to model misspecification and embedding estimation errors, making it suitable for practical applications where distributional assumptions are minimal. The work bridges the gap between the theoretical promise of RKHS-based conditional validity and the practical constraints of large-scale machine learning.