We study interactive learning of LLM-based language agents based on user edits made to the agent's output. In a typical setting such as writing assistants, the user interacts with a language agent to generate a response given a context, and may optionally edit the agent response to personalize it based on their latent preference, in addition to improving the correctness. The edit feedback is naturally generated, making it a suitable candidate for improving the agent's alignment with the user's preference, and for reducing the cost of user edits over time. We propose a learning framework, PRELUDE that infers a description of the user's latent preference based on historic edit data. The inferred user preference descriptions are used to define prompts for generating responses in the future. This avoids fine-tuning the agent, which is costly, challenging to scale with the number of users, and may even degrade its performance on other tasks. Furthermore, learning descriptive preference improves interpretability, allowing the user to view and modify the learned preference. However, user preference can be complex, subtle, and vary based on context, making it challenging to learn. To address this, we propose a simple yet effective algorithm named CIPHER that leverages the LLM to infer the user preference for a given context based on user edits. In the future, CIPHER retrieves inferred preferences from the k-closest contexts in the history, and forms an aggregate preference for response generation. We introduce two interactive environments -- summarization and email writing, and use a GPT-4 simulated user for evaluation. On both tasks, CIPHER outperforms several baselines by achieving the lowest edit distance cost while only having a small overhead in LLM query cost. Our analysis reports that user preferences learned by CIPHER show significant similarity to the ground truth latent preferences.