Radio Access Network Virtualization (vRAN) will spearhead the quest towards supple radio stacks that adapt to heterogeneous infrastructure: from energy-constrained platforms deploying cells-on-wheels (e.g., drones) or battery-powered cells to green edge clouds. We perform an in-depth experimental analysis of the energy consumption of virtualized Base Stations (vBSs) and render two conclusions: (i) characterizing performance and power consumption is intricate as it depends on human behavior such as network load or user mobility; and (ii) there are many control policies and some of them have non-linear and monotonic relations with power and throughput. Driven by our experimental insights, we argue that machine learning holds the key for vBS control. We formulate two problems and two algorithms: (i) BP-vRAN, which uses Bayesian online learning to balance performance and energy consumption, and (ii) SBP-vRAN, which augments our Bayesian optimization approach with safe controls that maximize performance while respecting hard power constraints. We show that our approaches are data-efficient and have provably performance, which is paramount for carrier-grade vRANs. We demonstrate the convergence and flexibility of our approach and assess its performance using an experimental prototype.