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Learning-based methods in robotics hold the promise of generalization, but what can be done if a learned policy does not generalize to a new situation? In principle, if an agent can at least evaluate its own success (i.e., with a reward classifier that generalizes well even when the policy does not), it could actively practice the task and finetune the policy in this situation. We study this problem in the setting of industrial insertion tasks, such as inserting connectors in sockets and setting screws. Existing algorithms rely on precise localization of the connector or socket and carefully managed physical setups, such as assembly lines, to succeed at the task. But in unstructured environments such as homes or even some industrial settings, robots cannot rely on precise localization and may be tasked with previously unseen connectors. Offline reinforcement learning on a variety of connector insertion tasks is a potential solution, but what if the robot is tasked with inserting previously unseen connector? In such a scenario, we will still need methods that can robustly solve such tasks with online practice. One of the main observations we make in this work is that, with a suitable representation learning and domain generalization approach, it can be significantly easier for the reward function to generalize to a new but structurally similar task (e.g., inserting a new type of connector) than for the policy. This means that a learned reward function can be used to facilitate the finetuning of the robot's policy in situations where the policy fails to generalize in zero shot, but the reward function generalizes successfully. We show that such an approach can be instantiated in the real world, pretrained on 50 different connectors, and successfully finetuned to new connectors via the learned reward function. Videos can be viewed at https://sites.google.com/view/learningonthejob