A Rate-Distortion View of Uncertainty Quantification

Abstract: In supervised learning, understanding an input's proximity to the training data can help a model decide whether it has sufficient evidence for reaching a reliable prediction. While powerful probabilistic models such as Gaussian Processes naturally have this property, deep neural networks often lack it. In this paper, we introduce Distance Aware Bottleneck (DAB), i.e., a new method for enriching deep neural networks with this property. Building on prior information bottleneck approaches, our method learns a codebook that stores a compressed representation of all inputs seen during training. The distance of a new example from this codebook can serve as an uncertainty estimate for the example. The resulting model is simple to train and provides deterministic uncertainty estimates by a single forward pass. Finally, our method achieves better out-of-distribution (OOD) detection and misclassification prediction than prior methods, including expensive ensemble methods, deep kernel Gaussian Processes, and approaches based on the standard information bottleneck.

• The paper introduces a novel method, Distance Aware Bottleneck (DAB), that formulates uncertainty quantification as a rate-distortion problem.
• It employs a codebook of latent encoders to estimate uncertainty with a single forward pass, outperforming ensemble methods in OOD detection and misclassification prediction.
• Experimental results on benchmarks like CIFAR-10 demonstrate high AUROC scores, underscoring the method's scalability and practical viability.

A Rate-Distortion View of Uncertainty Quantification

The paper "A Rate-Distortion View of Uncertainty Quantification" introduces a novel approach, termed Distance Aware Bottleneck (DAB), for enhancing deep neural networks (DNNs) with principled uncertainty quantification. The central tenet of this work is to leverage a rate-distortion framework to create a compressed representation of the training dataset in the form of a codebook of latent encoders. Crucially, the expected distance of a new input from this codebook is utilized as an uncertainty estimate.

This approach addresses the inherent weakness in many deep learning models, where conventional neural networks lack mechanisms to determine their confidence based on the proximity of new inputs to the training data. Unlike probabilistic models such as Gaussian Processes, where such uncertainty estimation is more straightforward, DNNs require additional architectural or training modifications to achieve similar capabilities.

Contributions

The key contributions of this research are multi-fold:

1. Formulating Uncertainty as a Rate-Distortion Problem: The paper frames uncertainty quantification within a rate-distortion perspective. It constructs a compressed representation of the training dataset via a codebook. This codebook comprises centroids that represent the training data minimally compressed by a statistical distance measure. The expectation of a new data point’s distance from these centroids serves as the uncertainty estimate.
2. Meta-Probabilistic Perspective: The authors adopt a meta-probabilistic viewpoint for the rate-distortion problem, utilizing the Information Bottleneck (IB) framework. They define the distortion function in terms of statistical distances between distributions of embeddings, thereby enforcing a regularization that renders the DNN distance-aware.
3. Practical Learning Algorithm: A practical deep learning algorithm is proposed, which involves successive estimates of the rate-distortion function to identify the centroids. This algorithm is designed to be simple to implement and ensures that distances from the codebook can be computed deterministically with a single forward pass.
4. Experimental Validation: The paper validates the proposed method using several benchmarks. The results indicate that DAB outperforms existing methods, including ensemble techniques, deep kernel Gaussian Processes, and standard IB methods, in tasks such as out-of-distribution (OOD) detection and misclassification prediction. Notably, DAB closes the performance gap between single forward pass methods and expensive ensemble methods in terms of calibration.
5. Scalability: DAB is shown to be effective and scalable when applied to large-scale datasets. It can be trained and applied post-hoc to large, pre-trained feature extractors, thus proving its utility in various practical scenarios.

Numerical Results

The authors present strong numerical evidence in support of DAB. For instance, on the CIFAR-10 dataset, DAB achieves an AUROC of 0.986 for detecting SVHN (a far OOD dataset) and 0.922 for CIFAR-100 (a near OOD dataset). These results surpass those of competitive baselines such as standard VIB, deep kernel methods, and even ensemble models. Furthermore, DAB demonstrates significant improvements in misclassification prediction tasks, with a Calibration AUROC of 0.930, closely approaching the performance of deep ensembles.

Implications

The theoretical and practical implications of this research are profound:

• Theoretical Implications:

The proposed method extends the conventional IB framework by integrating a rate-distortion perspective that operates directly on latent distributions. This provides a novel way to understand and quantify uncertainty in DNNs, offering a foundation for further exploration and development of probabilistic architectures that are both scalable and interpretable.

• Practical Implications:

From a practical standpoint, the ability to deterministically estimate the uncertainty of models in a single forward pass makes DAB highly attractive for deployment in real-world applications where computational resources and time are constrained. Additionally, the approach’s effectiveness in large-scale settings such as ImageNet demonstrates its versatility and scalability, paving the way for its adoption in various machine learning tasks, including reinforcement learning, natural language processing, and beyond.

Future Directions

The framework presented opens several avenues for future research:

1. Exploration of Different Statistical Distances: While the current work employs the Kullback-Leibler divergence, investigating alternative statistical distances could yield further insights and possibly enhanced performance.
2. Integration with Stochastic Decoders: Extending the method to include distance-aware stochastic decoders could make use of the uncertainty scores to adaptively modulate prediction uncertainty, potentially leading to improvements in tasks requiring nuanced confidence estimates.
3. Application to Diverse Modalities: Applying DAB to other data modalities, such as text and sequence data, would test its robustness and extend its applicability across broader domains.
4. Outlier Exposure and Data Augmentation: Integrating outlier exposure techniques and data augmentation methods with DAB could further bolster its OOD detection capabilities.

Conclusion

"A Rate-Distortion View of Uncertainty Quantification" sets forth a compelling case for incorporating a rate-distortion framework to achieve distance-aware uncertainty estimates in DNNs. The comprehensive experimental analysis underscores the method’s superiority over several established baselines, both in terms of OOD detection and misclassification prediction. The scalability and practical applicability of DAB make it a significant contribution to the field, with promising future extensions and applications.