Probabilistic verification problems of neural networks are concerned with formally analysing the output distribution of a neural network under a probability distribution of the inputs. Examples of probabilistic verification problems include verifying the demographic parity fairness notion or quantifying the safety of a neural network. We present a new algorithm for solving probabilistic verification problems of neural networks based on an algorithm for computing and iteratively refining lower and upper bounds on probabilities over the outputs of a neural network. By applying state-of-the-art bound propagation and branch and bound techniques from non-probabilistic neural network verification, our algorithm significantly outpaces existing probabilistic verification algorithms, reducing solving times for various benchmarks from the literature from tens of minutes to tens of seconds. Furthermore, our algorithm compares favourably even to dedicated algorithms for restricted probabilistic verification problems. We complement our empirical evaluation with a theoretical analysis, proving that our algorithm is sound and, under mildly restrictive conditions, also complete when using a suitable set of heuristics.

Counterexample-guided repair aims at creating neural networks with mathematical safety guarantees, facilitating the application of neural networks in safety-critical domains. However, whether counterexample-guided repair is guaranteed to terminate remains an open question. We approach this question by showing that counterexample-guided repair can be viewed as a robust optimisation algorithm. While termination guarantees for neural network repair itself remain beyond our reach, we prove termination for more restrained machine learning models and disprove termination in a general setting. We empirically study the practical implications of our theoretical results, demonstrating the suitability of common verifiers and falsifiers for repair despite a disadvantageous theoretical result. Additionally, we use our theoretical insights to devise a novel algorithm for repairing linear regression models based on quadratic programming, surpassing existing approaches.