Probabilistic Robustness Analysis -- Risks, Complexity and Algorithms

TL;DR

This paper advocates for probabilistic robustness analysis, demonstrating that it can be performed efficiently with controllable accuracy and confidence, surpassing traditional worst-case methods in safety and computational complexity.

Contribution

It introduces algorithms for probabilistic robustness evaluation with linear complexity and bounded memory, allowing precise control of sampling and discretization errors.

Findings

Algorithms achieve linear complexity in uncertainty dimension

Memory requirements are bounded and manageable

Efficiency improves with higher accuracy demands

Abstract

It is becoming increasingly apparent that probabilistic approaches can overcome conservatism and computational complexity of the classical worst-case deterministic framework and may lead to designs that are actually safer. In this paper we argue that a comprehensive probabilistic robustness analysis requires a detailed evaluation of the robustness function and we show that such evaluation can be performed with essentially any desired accuracy and confidence using algorithms with complexity linear in the dimension of the uncertainty space. Moreover, we show that the average memory requirements of such algorithms are absolutely bounded and well within the capabilities of today's computers. In addition to efficiency, our approach permits control over statistical sampling error and the error due to discretization of the uncertainty radius. For a specific level of tolerance of the discretization error, our techniques provide an efficiency improvement upon conventional methods which is inversely proportional to the accuracy level; i.e., our algorithms get better as the demands for accuracy increase.