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Query optimization remains one of the most challenging problems in data management systems. Recent efforts to apply machine learning techniques to query optimization challenges have been promising, but have shown few practical gains due to substantive training overhead, inability to adapt to changes, and poor tail performance. Motivated by these difficulties and drawing upon a long history of research in multi-armed bandits, we introduce Bao (the BAndit Optimizer). Bao takes advantage of the wisdom built into existing query optimizers by providing per-query optimization hints. Bao combines modern tree convolutional neural networks with Thompson sampling, a decades-old and well-studied reinforcement learning algorithm. As a result, Bao automatically learns from its mistakes and adapts to changes in query workloads, data, and schema. Experimentally, we demonstrate that Bao can quickly (an order of magnitude faster than previous approaches) learn strategies that improve end-to-end query execution performance, including tail latency. In cloud environments, we show that Bao can offer both reduced costs and better performance compared with a sophisticated commercial system.