Abstract: Computational intelligence is empowering many critical society applications in health, education, transportations, manufacturing, etc. The intelligent computing techniques are required to be socially responsible in terms of safety, fairness, sustainability, etc. To meet these requirements, the computing operators need to make context-aware decisions on computing resource allocation. Importantly, the decisions must be trustworthy in terms of the concerned performance metrics even in the worst case, but existing intelligent algorithms have limited capacity to guarantee the trustworthiness for any possible problem instance. In this talk, I will introduce our recent progress of algorithm foundations on trustworthy sequential decision-making for responsible computing. The talk will first give online learning algorithms with worst-case performance guarantee. Subsequently, the talk will introduce the trustworthy algorithms that utilize ML predictions to improve the average performance while guaranteeing the worst-case performance for adversarial instances. The talk will conclude with the results on sustainable AI inference for battery-powered devices and future perspectives.