Box-Constrained Softmax Function and Its Application for Post-Hoc Calibration
This work addresses the need for trustworthy models in machine learning by enabling post-hoc calibration to mitigate underconfidence and overconfidence, though it is incremental as it builds on existing softmax methods.
The paper tackles the problem of enforcing hard constraints on output probabilities in softmax-based models, which is critical for reliable applications, by proposing the box-constrained softmax (BCSoftmax) function and demonstrating its effectiveness in improving calibration metrics on datasets like TinyImageNet, CIFAR-100, and 20NewsGroups.
Controlling the output probabilities of softmax-based models is a common problem in modern machine learning. Although the $\mathrm{Softmax}$ function provides soft control via its temperature parameter, it lacks the ability to enforce hard constraints, such as box constraints, on output probabilities, which can be critical in certain applications requiring reliable and trustworthy models. In this work, we propose the box-constrained softmax ($\mathrm{BCSoftmax}$) function, a novel generalization of the $\mathrm{Softmax}$ function that explicitly enforces lower and upper bounds on output probabilities. While $\mathrm{BCSoftmax}$ is formulated as the solution to a box-constrained optimization problem, we develop an exact and efficient computation algorithm for $\mathrm{BCSoftmax}$. As a key application, we introduce two post-hoc calibration methods based on $\mathrm{BCSoftmax}$. The proposed methods mitigate underconfidence and overconfidence in predictive models by learning the lower and upper bounds of the output probabilities or logits after model training, thereby enhancing reliability in downstream decision-making tasks. We demonstrate the effectiveness of our methods experimentally using the TinyImageNet, CIFAR-100, and 20NewsGroups datasets, achieving improvements in calibration metrics.