The Benefit of Being Bayesian in Online Conformal Prediction

2024-10-03

arXiv.org

AI Summary

This paper addresses three key challenges in constructing dynamic confidence sets for black-box models in online conformal prediction: (1) fragile reliance on i.i.d. or exchangeability assumptions; (2) first-order online optimization under adversarial environments requiring pre-specified quantiles and suffering from monotonicity distortion due to loss linearization; and (3) the difficulty of simultaneously achieving low regret and statistical fidelity under multiple confidence-level queries. We propose the first Bayesian-regularized empirical distribution framework, integrating nonlinear Follow-the-Regularized-Leader (FTRL), online quantile forecasting, and a multi-head shared-belief update architecture. Our method imposes no distributional assumptions, inherently supports U-calibration, and guarantees low-regret coverage across multiple confidence levels for arbitrary input sequences—while automatically attaining exact coverage under i.i.d. settings. Empirically, it significantly outperforms existing baselines, especially under non-stationary data, demonstrating strong robustness.

Technology Category

• Machine Learning: Calibration & Uncertainty Quantification
• Reasoning under Uncertainty: Sequential Decision Making
• Game Theory and Economic Paradigms: Adversarial Learning

Application Category

• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: Attacks and countermeasures in recommendation systems
• Economics, Online Markets and Human Computation: Trust and reliance of crowd workers and data experts on GenAI

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).

Problem

• Online construction of confidence sets for black-box models
• Predicting quantiles of sequentially revealed data sequences
• Overcoming monotonicity issues in first-order optimization baselines

Innovation

• Bayesian prior regularizes empirical distribution belief
• Non-linearized FTRL algorithm for robust updates
• Handles multiple confidence levels with low regret