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Pre-trained large language models, such as GPT\nobreakdash-2 and BERT, are often fine-tuned to achieve state-of-the-art performance on a downstream task. One natural example is the ``Smart Reply'' application where a pre-trained model is tuned to provide suggested responses for a given query message. Since the tuning data is often sensitive data such as emails or chat transcripts, it is important to understand and mitigate the risk that the model leaks its tuning data. We investigate potential information leakage vulnerabilities in a typical Smart Reply pipeline. We consider a realistic setting where the adversary can only interact with the underlying model through a front-end interface that constrains what types of queries can be sent to the model. Previous attacks do not work in these settings, but require the ability to send unconstrained queries directly to the model. Even when there are no constraints on the queries, previous attacks typically require thousands, or even millions, of queries to extract useful information, while our attacks can extract sensitive data in just a handful of queries. We introduce a new type of active extraction attack that exploits canonical patterns in text containing sensitive data. We show experimentally that it is possible for an adversary to extract sensitive user information present in the training data, even in realistic settings where all interactions with the model must go through a front-end that limits the types of queries. We explore potential mitigation strategies and demonstrate empirically how differential privacy appears to be a reasonably effective defense mechanism to such pattern extraction attacks.