Large Language models (LLMs) are trained on large amounts of data, which can include sensitive information that may compromise personal privacy. LLMs showed to memorize parts of the training data and emit those data verbatim when an adversary prompts appropriately. Previous research has primarily focused on data preprocessing and differential privacy techniques to address memorization or prevent verbatim memorization exclusively, which can give a false sense of privacy. However, these methods rely on explicit and implicit assumptions about the structure of the data to be protected, which often results in an incomplete solution to the problem. To address this, we propose a novel framework that utilizes a reinforcement learning approach (PPO) to fine-tune LLMs to mitigate approximate memorization. Our approach utilizes a negative similarity score, such as BERTScore or SacreBLEU, as a reward signal to learn a dissimilarity policy. Our results demonstrate that this framework effectively mitigates approximate memorization while maintaining high levels of coherence and fluency in the generated samples. Furthermore, our framework is robust in mitigating approximate memorization across various circumstances, including longer context, which is known to increase memorization in LLMs.