In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods.

Combining machine learning and formal methods (FMs) provides a possible solution to overcome the safety issue of autonomous driving (AD) vehicles. However, there are gaps to be bridged before this combination becomes practically applicable and useful. In an attempt to facilitate researchers in both FMs and AD areas, this paper proposes a framework that combines two well-known tools, namely CommonRoad and UPPAAL. On the one hand, CommonRoad can be enhanced by the rigorous semantics of models in UPPAAL, which enables a systematic and comprehensive understanding of the AD system's behaviour and thus strengthens the safety of the system. On the other hand, controllers synthesised by UPPAAL can be visualised by CommonRoad in real-world road networks, which facilitates AD vehicle designers greatly adopting formal models in system design. In this framework, we provide automatic model conversions between CommonRoad and UPPAAL. Therefore, users only need to program in Python and the framework takes care of the formal models, learning, and verification in the backend. We perform experiments to demonstrate the applicability of our framework in various AD scenarios, discuss the advantages of solving motion planning in our framework, and show the scalability limit and possible solutions.

In this work, we provide a fundamental unified convergence theorem used for deriving expected and almost sure convergence results for a series of stochastic optimization methods. Our unified theorem only requires to verify several representative conditions and is not tailored to any specific algorithm. As a direct application, we recover expected and almost sure convergence results of the stochastic gradient method (SGD) and random reshuffling (RR) under more general settings. Moreover, we establish new expected and almost sure convergence results for the stochastic proximal gradient method (prox-SGD) and stochastic model-based methods (SMM) for nonsmooth nonconvex optimization problems. These applications reveal that our unified theorem provides a plugin-type convergence analysis and strong convergence guarantees for a wide class of stochastic optimization methods.