On Continuity of Robust and Accurate Classifiers
Ramin Barati Reza Safabakhsh Mohammad Rahmati
The reliability of a learning model is key to the successful deployment of
machine learning in various applications. Creating a robust model, particularly
one unaffected by adversarial attacks, requires a comprehensive understanding
of the adversarial examples phenomenon. However, it is difficult to describe
the phenomenon due to the complicated nature of the problems in machine
learning. It has been shown that adversarial training can improve the
robustness of the hypothesis. However, this improvement comes at the cost of
decreased performance on natural samples. Hence, it has been suggested that
robustness and accuracy of a hypothesis are at odds with each other. In this
paper, we put forth the alternative proposal that it is the continuity of a
hypothesis that is incompatible with its robustness and accuracy. In other
words, a continuous function cannot effectively learn the optimal robust
hypothesis. To this end, we will introduce a framework for a rigorous study of
harmonic and holomorphic hypothesis in learning theory terms and provide
empirical evidence that continuous hypotheses does not perform as well as
discontinuous hypotheses in some common machine learning tasks. From a
practical point of view, our results suggests that a robust and accurate
learning rule would train different continuous hypotheses for different regions
of the domain. From a theoretical perspective, our analysis explains the
adversarial examples phenomenon as a conflict between the continuity of a
sequence of functions and its uniform convergence to a discontinuous function.