[Paper Review] Post-hoc Calibration of Neural Networks.

Amir Rahimi, Kartik Gupta+4 more|arXiv (Cornell University)|Jun 23, 2020

Adversarial Robustness in Machine Learning22 references4 citations

TL;DR

This paper provides a theoretical foundation for post-hoc calibration of neural networks by proving that even when a base network is not optimal, adding a learnable calibration head (g) and minimizing Negative Log-Likelihood (NLL) on the output of the base network (f) can yield a calibrated final model (g∘f). Experiments on image classification benchmarks confirm the theoretical findings.

ABSTRACT

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.

Motivation & Objective

• To understand the theoretical underpinnings of post-hoc calibration methods for neural networks.
• To investigate whether learning an additional function after training a base network can lead to theoretically calibrated predictions.
• To relax the requirement that the base network must achieve the global optimum of NLL for calibration to be achieved.
• To provide theoretical justification for the empirical success of post-hoc calibration techniques.

Proposed method

• The paper introduces a two-stage model: a pre-trained base network f and a learnable calibration head g, forming the composition g∘f.
• It proves that minimizing NLL on the output of f via optimization of g's parameters leads to a calibrated model, even if f is not globally optimal.
• The theoretical analysis relies on the known property that minimizing NLL yields calibration on the training set when the global optimum is reached.
• The method treats the base network f as fixed and optimizes only the parameters of g to calibrate the outputs.
• The approach is applicable to any base network and calibration head architecture, as long as g can model the necessary transformation.

Experimental results

Research questions

• RQ1 Can post-hoc calibration via a learnable head g achieve theoretical calibration even when the base network f is not globally optimal?
• RQ2 What conditions ensure that the composition g∘f is calibrated after optimizing g's parameters?
• RQ3 How does minimizing NLL on the output of f through g relate to the calibration of the final model?
• RQ4 Does the theoretical justification hold in practice across diverse image classification benchmarks?

Key findings

• Theoretical analysis proves that minimizing NLL on the output of a pre-trained base network f via a learnable head g results in a calibrated final model g∘f, even if f is not globally optimal.
• The method provides a less stringent condition than requiring the base network to achieve the global optimum of NLL for calibration.
• The theoretical justification supports the empirical success of post-hoc calibration methods in practice.
• Experiments on multiple image classification benchmarks validate the theoretical claims, showing effective calibration.