Restless Multi-Process Multi-Armed Bandits with Applications to Self-Driving Microscopies
High-content screening microscopy generates large amounts of live-cell imaging data, yet its potential remains constrained by the inability to determine when and where to image most effectively. Optimally balancing acquisition time, computational capacity, and photobleaching budgets across thousands of dynamically evolving regions of interest remains an open challenge, further complicated by limited field-of-view adjustments and sensor sensitivity. Existing approaches either rely on static sampling or heuristics that neglect the dynamic evolution of biological processes, leading to inefficiencies and missed events. Here, we introduce the restless multi-process multi-armed bandit (RMPMAB), a new decision-theoretic framework in which each experimental region is modeled not as a single process but as an ensemble of Markov chains, thereby capturing the inherent heterogeneity of biological systems such as asynchronous cell cycles and heterogeneous drug responses. Building upon this foundation, we derive closed-form expressions for transient and asymptotic behaviors of aggregated processes, and design scalable Whittle index policies with sub-linear complexity in the number of imaging regions. Through both simulations and a real biological live-cell imaging dataset, we show that our approach achieves substantial improvements in throughput under resource constraints. Notably, our algorithm outperforms Thomson Sampling, Bayesian UCB, epsilon-Greedy, and Round Robin by reducing cumulative regret by more than 37% in simulations and capturing 93% more biologically relevant events in live imaging experiments, underscoring its potential for transformative smart microscopy. Beyond improving experimental efficiency, the RMPMAB framework unifies stochastic decision theory with optimal autonomous microscopy control, offering a principled approach to accelerate discovery across multidisciplinary sciences.