Myopically Verifiable Probabilistic Certificate for Long-term Safety

Physics-Informed Representation and Learning: Control and Risk Quantification

“…Risk quantification is the key enabler for many long-term stochastic safe control methods (Wang et al 2021;Jing and Nakahira 2022). Existing methods often use rare event simulation through Monte Carlo (MC) and importance sampling to estimate the long-term risk in stochastic systems (Botev, L'Ecuyer, and Tuffin 2013;Hanna, Niekum, and Stone 2021;Stadie et al 2018;Madhushani et al 2021).…”

“…Comparison Theorem Autoencoder-like NN Feynman-Kac Formula Physics-informed Learning The goal is to find the safety probability F over the state space for a long-term horizon T . Once the safety probability is acquired, existing safe control methods can be used to guarantee safety of the system (Wang et al 2021;Jing and Nakahira 2022). One standard approach to acquire such safety probability is to run Monte Carlo simulation with the nominal controller N multiple times and calculate the empirical probability of the system being safe, i.e.,…”

“…) where N safe is the number of safe trajectories over N trajectories. However, such estimation has a sample complexity that scales linearly with horizon T and exponentially with system dimension n (Rubino and Tuffin 2009;Wang et al 2021), and is thus inefficient for high-dimensional systems with long-term safety requirements. We aim to overcome these issues and efficiently estimate long-term safety probabilities of high-dimensional systems with the proposed framework.…”

Myopically Verifiable Probabilistic Certificate for Long-term Safety

“…For example, for safe control, chance-constrained predictive control takes probabilistic safety requirements as the constraints in an optimization-based controller, and solves a minimal distance control to a nominal controller to find its safe counterpart [38]- [40]. For safe learning, probabilistic safety certificate can ensure long-term safety of neural network-based controller through forward invariance on the probability space [41]. For safe reinforcement learning, the probabilistic safety certificate can safe guard reinforcement learning agents during and after training [42]- [45] (e.g., policy gradient and Q-learning agents [44]), and similar frameworks can be potentially used in meta learning settings as well [46].…”

“…The distribution of (11) and (12) will allow us to study the property of recovery from a variety of perspectives, such as the average recovery time, tail distribution of recovery, and the mean and tail distribution of recovery vs. crashes. The efficient calculation of ( 7), ( 8), (11), and (12) will allow us to design online stochastic safe control methods with probabilistic guarantees [41].…”