From Conformal Predictions to Confidence Regions (2405.18601v1)

Abstract: Conformal prediction methodologies have significantly advanced the quantification of uncertainties in predictive models. Yet, the construction of confidence regions for model parameters presents a notable challenge, often necessitating stringent assumptions regarding data distribution or merely providing asymptotic guarantees. We introduce a novel approach termed CCR, which employs a combination of conformal prediction intervals for the model outputs to establish confidence regions for model parameters. We present coverage guarantees under minimal assumptions on noise and that is valid in finite sample regime. Our approach is applicable to both split conformal predictions and black-box methodologies including full or cross-conformal approaches. In the specific case of linear models, the derived confidence region manifests as the feasible set of a Mixed-Integer Linear Program (MILP), facilitating the deduction of confidence intervals for individual parameters and enabling robust optimization. We empirically compare CCR to recent advancements in challenging settings such as with heteroskedastic and non-Gaussian noise.

Overview of Conformal Confidence Regions (CCR) for Finite-Sample Valid Inference

This paper addresses a significant challenge in the field of predictive modeling: constructing confidence regions for model parameters with finite sample guarantees and minimal assumptions about data distribution. While traditional conformal prediction methods have facilitated robust uncertainty quantification for model outputs, extending these techniques to the model parameter space has been problematic. In response, the authors introduce Conformal Confidence Regions (CCR), a novel method that combines conformal prediction intervals to create confidence regions for model parameters.

Key Contributions

The notable contributions of this paper are multifaceted:

1. Extension of Conformal Prediction:
   • The authors extend conformal prediction methodologies to build uncertainty sets for noise-free model outputs, providing finite-sample coverage guarantees with minimal assumptions on noise distribution.
2. Innovative Methodology:
   • A new approach, CCR, aggregates conformal prediction intervals from multiple unlabelled inputs to construct a confidence region for the ground-truth parameter $\theta_\star$. This aggregation is performed using split conformal predictions or black-box methodologies.
3. Finite-Sample Valid Guarantees:
   • CCR is demonstrated to provide finite-sample valid coverage guarantees, both in black-box settings and under split conformal predictions, which offer improved guarantees.
4. Application to Linear Models:
   • For linear models, the confidence region derived is represented as the feasible set of a Mixed-Integer Linear Program (MILP). This facilitates the deduction of confidence intervals for individual parameters and enables robust optimization.
5. Empirical Validation:
   • The proposed method is empirically validated in challenging settings, including heteroskedastic and non-Gaussian noise, comparing favorably to recent methodologies.

Methodological Insights

Confidence Sets for Noise-Free Outputs

The foundation of CCR lies in constructing prediction intervals over the noise-free outputs $f_{\theta_\star}(X)$. This construction leverages the finite-sample guarantees of conformal prediction but adapts it to handle noise-free outputs, which can yield more robust and realistic confidence regions. Assumption 1 establishes that CP prediction sets are intervals, while Assumption 2 ensures finite-sample validity by supposing a non-zero probability density condition $b$ on the noise.

Confidence Sets for Model Parameters

Building on the noise-free confidence regions, the CCR approach focuses on creating confidence regions in the parameter space. This is done by defining a set $\Theta(X)$ for each input such that the feasible parameter values yield outputs within conformal intervals. The confidence region for parameters, $\Theta_k$, is obtained through an aggregation method that counts how many individual confidence sets $\Theta(X_i)$ include the true parameter.

Several strategies for aggregating these confidence sets are:

• Fully Black-Box Approach: In the absence of information about the conformal prediction method, conservative bounds using Markov's inequality or Worst-Case Dependency are employed.
• Split Conformal Prediction: Leveraging the structure of split conformal prediction, tighter bounds are obtained, conditional on the distribution of calibration scores.

Applications and Practical Implications

MILP Formulation for Linear Models

In the linear model case, the feasible set $\Theta_k$ can be represented through a MILP, enabling the optimization of linear objectives and robust model parameter estimation. The authors demonstrate how MILP can be used to assess model linearity and provide bounds on specific parameter coordinates, aiding feature selection and interpretability.

Regression with Conformal Abstention

Another practical application of CCR is in regression with a rejection option. By rejecting predictions when the conformal prediction set is too large, the method provides a mechanism to avoid erroneous predictions, ensuring robust predictive models, particularly in high-uncertainty scenarios. This finite-sample valid approach stands out for its theoretical guarantees and practical robustness across diverse predictive tasks.

Empirical Validation

The empirical comparison of CCR against other methods, such as SPS and RII, highlights its robust performance under various noise conditions. Tables and figures in the paper showcase the consistent coverage of CCR, validating its practical efficacy and reliability. Specifically, even under challenging noise conditions, the CCR method reliably maintains the desired coverage probability, a testament to its robustness and accuracy.

Theoretical and Practical Implications

The theoretical implications of this work lie in providing a finite-sample valid methodology for constructing confidence regions with minimal assumptions. This capability is crucial for deploying reliable machine learning models in real-world scenarios where data is often limited and noise properties are not well-known.

Practically, the development of CCR paves the way for more robust, interpretable, and safe predictive models. Future developments in AI could build on this foundation to further refine parameter estimation techniques, enhance model reliability under uncertainty, and extend these methodologies to more complex, non-linear models.

Given these advancements, future research may delve into exploring more scalable optimization techniques for large-scale MILP problems, enhancing interpretability in non-linear settings, and extending the application of CCR to a broader array of machine learning tasks, bolstering confidence in AI-driven decision-making processes.