Multi-stage Bayesian Optimisation under Unknown Stopping Constraints

We consider Bayesian optimisation over multi-stage decision processes with unknown stage-wise transitions and stopping constraints, where a single violation aborts the current run and prevents all downstream observations. Such problems are common in sequential experimental and manufacturing settings, where failed runs are acceptable during learning but remain costly. To explore uncertain but potentially high-value trajectories under this tolerance while limiting the cumulative number of aborted runs, we formulate the problem as a feasibility-indicated Markov decision process and propose \fiovi, a value iteration method for optimistic lookahead planning that uses an annealed feasibility margin to provide fine-grained control over the exploration-failure trade off. Under standard RKHS regularity and feasibility assumptions, we derive sublinear rates for both cumulative abort count and cumulative value regret. We demonstrate competitive performance against existing baselines on synthetic benchmarks and real-world problems.