Cautious Deep Learning

TL;DR

This paper replaces single-label decisions with conformal prediction sets C(x) based on class-conditional densities p(x|y), achieving distribution-free coverage guarantees while allowing empty sets to signal out-of-distribution inputs. By estimating p(x|y) per class and thresholding via empirical quantiles, the method yields cautious, adaptable predictions that gracefully handle outliers and adversarial inputs. The authors demonstrate scalability to large, high-dimensional data (ImageNet, CelebA, IMDB-Wiki) and show advantages in robustness, class adaptivity, and interpretability over traditional p(y|x)-based approaches. Limitations include computational overhead for density estimation in high dimensions, motivating future work on ordering-consistent approximations and GPU-accelerated implementations. Overall, the approach provides a principled framework for uncertainty-aware multiclass classification with controllable coverage and flexible class management.

Abstract

Most classifiers operate by selecting the maximum of an estimate of the conditional distribution $p(y|x)$ where $x$ stands for the features of the instance to be classified and $y$ denotes its label. This often results in a {\em hubristic bias}: overconfidence in the assignment of a definite label. Usually, the observations are concentrated on a small volume but the classifier provides definite predictions for the entire space. We propose constructing conformal prediction sets which contain a set of labels rather than a single label. These conformal prediction sets contain the true label with probability $1-α$. Our construction is based on $p(x|y)$ rather than $p(y|x)$ which results in a classifier that is very cautious: it outputs the null set --- meaning "I don't know" --- when the object does not resemble the training examples. An important property of our approach is that adversarial attacks are likely to be predicted as the null set or would also include the true label. We demonstrate the performance on the ImageNet ILSVRC dataset and the CelebA and IMDB-Wiki facial datasets using high dimensional features obtained from state of the art convolutional neural networks.

Figures (6)

• Figure 1: Classification boundaries for different methods for the Iris dataset. For the conformal prediction method (c) (with $\alpha=0.05$) the overlapping areas are classified as multiple classes and white areas are classified as the null set. For the standard methods (a-b), the decision boundaries can change significantly with small changes in some of the data points and the prediction cannot be justified in most of the space. Online version in color.
• Figure 2: An example of more complex estimation function of $p(x|y)$ providing $0.95$ coverage on the Iris dataset. $p(x|y)$ is higher for points which has high density within the class and low density in the other classes. The decision boundary closely resemble standard methods, while still providing cautious prediction and robustness to out of sample predictions.
• Figure 3: (a) Performance plot for the conformal method. Accuracy is empirically linear as a function of $\alpha$ but affect the number of classes predicted per sample. (b-c) are illustrative examples. When $\alpha=0.6$ both black and red classes are predicted. When $\alpha=0.8$ the red classes remain.
• Figure 4: Classification results for (a) random noise; (b) Jackson Pollock "Rabit Hole"; (c) Muhammad Ali towering over Sonny Liston (1965 rematch). These pictures are outliers for the Imagenet categories. The left labels of each picture are provided by our method and the right are the results of the Inception-v4 model.
• Figure 5: A collage of the first $20$ images in the ImageNet validation set with $\alpha=0.7$. TL denotes the image true label and Prediction is the method output. By design only $0.3$ accuracy is expected, yet both the true and the false predictions are reasonable. Online version in color.

Theorems & Definitions (2)

• Proposition 1
• Proposition 2