Post-hoc Calibration of Neural Networks by g-Layers

Calibration of neural networks is a critical aspect to consider when incorporating machine learning models in real-world decision-making systems where the confidence of decisions are equally important as the decisions themselves. In recent years, there is a surge of research on neural network calibration and the majority of the works can be categorized into post-hoc calibration methods, defined as methods that learn an additional function to calibrate an already trained base network. In this work, we intend to understand the post-hoc calibration methods from a theoretical point of view. Especially, it is known that minimizing Negative Log-Likelihood (NLL) will lead to a calibrated network on the training set if the global optimum is attained (Bishop, 1994). Nevertheless, it is not clear learning an additional function in a post-hoc manner would lead to calibration in the theoretical sense. To this end, we prove that even though the base network (f) does not lead to the global optimum of NLL, by adding additional layers (g) and minimizing NLL by optimizing the parameters of g one can obtain a calibrated network g circ f. This not only provides a less stringent condition to obtain a calibrated network but also provides a theoretical justification of post-hoc calibration methods. Our experiments on various image classification benchmarks confirm the theory.