Provable Uncertainty Decomposition via Higher-Order Calibration (2412.18808v1)

Abstract: We give a principled method for decomposing the predictive uncertainty of a model into aleatoric and epistemic components with explicit semantics relating them to the real-world data distribution. While many works in the literature have proposed such decompositions, they lack the type of formal guarantees we provide. Our method is based on the new notion of higher-order calibration, which generalizes ordinary calibration to the setting of higher-order predictors that predict mixtures over label distributions at every point. We show how to measure as well as achieve higher-order calibration using access to $k$-snapshots, namely examples where each point has $k$ independent conditional labels. Under higher-order calibration, the estimated aleatoric uncertainty at a point is guaranteed to match the real-world aleatoric uncertainty averaged over all points where the prediction is made. To our knowledge, this is the first formal guarantee of this type that places no assumptions whatsoever on the real-world data distribution. Importantly, higher-order calibration is also applicable to existing higher-order predictors such as Bayesian and ensemble models and provides a natural evaluation metric for such models. We demonstrate through experiments that our method produces meaningful uncertainty decompositions for image classification.

• The paper introduces higher-order calibration based on k-snapshots to provably decompose predictive uncertainty into aleatoric and epistemic components with formal guarantees.
• This approach uses mutual information under higher-order calibration to ensure the estimated aleatoric uncertainty matches the true average uncertainty, facilitating meaningful separation.
• By providing reliable uncertainty quantification without data distribution assumptions, this method enhances model interpretability and is applicable to critical areas like medical imaging and autonomous systems.

Analyzing Provable Uncertainty Decomposition via Higher-Order Calibration

The paper introduces a novel approach to decomposing predictive uncertainty in machine learning models, focusing on the distinction between aleatoric and epistemic uncertainties. This decomposition is vital in many practical machine learning applications because it helps practitioners ascertain whether uncertainty in predictions arises from the inherent variability in data (aleatoric) or from a model's insufficient knowledge about the data-generating process (epistemic). Traditional methods often conflate these two sources of uncertainty or provide decompositions without formal guarantees. The authors present a method based on the concept of higher-order calibration to address this gap, offering rigorous semantic underpinnings to the decomposition.

Higher-Order Calibration and Snapshots

The paper introduces "higher-order calibration," an extension of standard calibration that applies to higher-order predictors—those predicting mixtures over label distributions rather than single distributions. The innovation here lies in measuring and achieving this calibration using "k-snapshots," which are sets of k independent observations per data point, thus allowing a model to capture additional information about the distribution of possible outcomes. The significant promise of this calibration approach is that it allows the aleatoric uncertainty estimated by the predictor to match the actual real-world aleatoric uncertainty averaged over a given set of predictions, independent of the distributional assumptions about the data.

Key Contributions

1. Higher-Order Calibration Definition: The researchers propose a formal definition of higher-order calibration, requiring that whenever a model predicts a particular mixture of distributions, one observes the same mixture consistently on average. They also introduce the notion of $k$-th order calibration as a practical relaxation, which becomes increasingly accurate as k, the snapshot size, increases.
2. Uncertainty Decomposition: The paper provides a framework for decomposing uncertainty using mutual information. Under higher-order calibration, the aleatoric uncertainty estimated by the model corresponds to the true aleatoric uncertainty averaged over the points in a given predictive set, while epistemic uncertainty represents the divergence within these distributions.
3. Practical Implementation and Evaluation: The authors demonstrate that higher-order calibration can be practically achieved using post-hoc calibration algorithms that operate on $k$-snapshots. Experiments highlight that this method leads to meaningful uncertainty decompositions in complex tasks like image classification, emphasizing its applicability to practical machine learning problems.
4. Implications for Machine Learning: From a theoretical standpoint, higher-order calibration offers formal guarantees about uncertainty decomposition without reliance on specific data distribution assumptions, making it an enticing approach for applications like active learning and model performance diagnosis.

Theoretical and Practical Implications

The work has wide-ranging implications for both theory and practice:

• Theoretical Impact: By providing rigorous definitions and proving that the epistemic and aleatoric uncertainties can be accurately separated, this research lays a robust foundation for future studies and applications requiring trustworthy uncertainty quantifications.
• Practical Utility: In contexts like medical imaging or autonomous systems, where understanding the source of prediction uncertainty is critical, this method could become a standard part of the machine learning toolkit, enabling more reliable and interpretable AI systems.

Future Directions

Future research could explore the application of higher-order calibration in diverse domains such as natural language processing or drug discovery, where uncertainty quantification is critical. Furthermore, techniques to scale this approach to large snapshot sizes efficiently, thereby reducing computational overhead, would enhance its practicality in real-world applications. Additionally, investigating the relationship between $k$-snapshots and active learning strategies could provide insights into optimizing data collection and model improvements, effectively balancing between model enhancement and computational cost.

Overall, the introduction of higher-order calibration provides a formal, practical framework for understanding and utilizing uncertainty decompositions in machine learning, promising enhancements in both the interpretability and performance of predictive models.