The Benefit of Being Bayesian in Online Conformal Prediction (2410.02561v2)

Abstract: Based on the framework of Conformal Prediction (CP), we study the online construction of confidence sets given a black-box machine learning model. By converting the target confidence levels into quantile levels, the problem can be reduced to predicting the quantiles (in hindsight) of a sequentially revealed data sequence. Two very different approaches have been studied previously: (i) Assuming the data sequence is iid or exchangeable, one could maintain the empirical distribution of the observed data as an algorithmic belief, and directly predict its quantiles. (ii) Due to the fragility of statistical assumptions, a recent trend is to consider the non-distributional, adversarial setting and apply first-order online optimization algorithms to moving quantile losses. However, it requires the oracle knowledge of the target quantile level, and suffers from a previously overlooked monotonicity issue due to the associated loss linearization. This paper presents an adaptive CP algorithm that combines their strengths. Without any statistical assumption, it is able to answer multiple arbitrary confidence level queries with low regret, while also overcoming the monotonicity issue suffered by first-order optimization baselines. Furthermore, if the data sequence is actually iid, then the same algorithm is automatically equipped with the "correct" coverage probability guarantee. To achieve such strengths, our key technical innovation is to regularize the aforementioned algorithmic belief (the empirical distribution) by a Bayesian prior, which robustifies it by simulating a non-linearized Follow the Regularized Leader (FTRL) algorithm on the output. Such a belief update backbone is shared by prediction heads targeting different confidence levels, bringing practical benefits analogous to the recently proposed concept of U-calibration (Kleinberg et al., 2023).

• The paper introduces a novel Bayesian framework that integrates regularization with online conformal prediction to robustly improve the accuracy of confidence sets.
• It demonstrates that the algorithm supports multiple confidence levels while achieving an optimal frequentist regret bound under both adversarial and iid conditions.
• The research offers practical solutions for adaptivity to distribution shifts and memory reduction, paving the way for real-time predictive applications in various domains.

The Benefit of Being Bayesian in Online Conformal Prediction

The paper presented by Zhang et al. explores the integration of Bayesian methods within the framework of Online Conformal Prediction (OCP). The authors address the problem of constructing valid confidence sets for sequentially revealed data using a black-box ML model. This paper links Bayesian regularization with conformal prediction, providing new insights into how Bayesian methods can enhance the accuracy and reliability of confidence sets in an online setting.

The core innovation of this research is a novel Bayesian framework for conformal prediction that combines the strengths of both direct and indirect approaches previously studied. The direct approach assumes an iid sequence, maintaining an empirical distribution of observed data and predicting its quantiles. The indirect approach, on the other hand, adopts an adversarial perspective using first-order online optimization. This Bayesian approach uniquely offers a low regret bound without any statistical assumptions, effectively addressing the validity issues commonly faced by first-order optimization.

Key Contributions

1. Bayesian Framework Integration: The algorithm uses a Bayesian prior to regularize the empirical distribution, effectively robustifying predictions by simulating a non-linearized Follow the Regularized Leader (FTRL) algorithm. This approach mitigates the validity issues of first-order traditional algorithms.
2. Support for Multiple Confidence Levels: The presented algorithm can handle multiple arbitrary confidence level queries online, ensuring provable low regret across these levels.
3. Regret Minimization: The paper demonstrates that the algorithm achieves an optimal frequentist regret bound of $O(R\sqrt{T})$ under a uniform prior. This robustness is achieved without prior knowledge of quantile level $\alpha$.
4. Adaptivity to IID: The Bayesian algorithm shows adaptability to an iid environment, securing similar guarantees to Empirical Risk Minimization (ERM) concerning coverage probability and excess quantile risk.
5. Quantization and Reduced Memory Usage: A quantized version of the proposed algorithm demonstrates that significant memory reductions are possible with only minimal impact on performance.
6. Handling Continual Distribution Shifts: The algorithm extends to scenarios involving continuous distribution shifts through a discounted method, maintaining performance with sliding-window assurances.

Implications and Future Work

The implications of this work are noteworthy for online predictive modeling and decision-making, where model uncertainty and data non-stationarity are critical concerns. The Bayesian approach holds potential for applications requiring real-time adaptive prediction mechanisms such as finance, healthcare monitoring, and dynamic risk management.

Future research could focus on exploring group-conditional guarantees and addressing broader classes of performance metrics. Further empirical testing across varied datasets could deepen the understanding of its practical efficacy and robustness. Exploring methodological enhancements for handling extreme adversarial conditions would also be valuable, as well as integrating this Bayesian conformal framework with other predictive methodologies in ensemble models.

In summary, Zhang et al. provide a compelling argument for the employment of Bayesian methods in OCP, presenting a conceptually robust yet computationally feasible framework that stands to significantly enrich both theoretical insights and pragmatic applications in AI and ML domains.