Individuals and businesses have been significantly benefited by Large Language Models (LLMs) including PaLM, Gemini and ChatGPT in various ways. For example, LLMs enhance productivity, reduce costs, and enable us to focus on more valuable tasks. Furthermore, LLMs possess the capacity to sift through extensive datasets, uncover underlying patterns, and furnish critical insights that propel the frontiers of technology and science. However, LLMs also pose privacy concerns. Users' interactions with LLMs may expose their sensitive personal or company information. A lack of robust privacy safeguards and legal frameworks could permit the unwarranted intrusion or improper handling of individual data, thereby risking infringements of privacy and the theft of personal identities. To ensure privacy, it is essential to minimize the dependency between shared prompts and private information. Various randomization approaches have been proposed to protect prompts' privacy, but they may incur utility loss compared to unprotected LLMs prompting. Therefore, it is essential to evaluate the balance between the risk of privacy leakage and loss of utility when conducting effective protection mechanisms. The current study develops a framework for inferring privacy-protected Large Language Models (LLMs) and lays down a solid theoretical basis for examining the interplay between privacy preservation and utility. The core insight is encapsulated within a theorem that is called as the NFL (abbreviation of the word No-Free-Lunch) Theorem.

翻译：大型语言模型（LLMs），包括PaLM、Gemini和ChatGPT，已在诸多方面为个人和企业带来显著效益。例如，LLMs能够提升生产效率、降低成本，并使我们得以专注于更具价值的任务。此外，LLMs具备筛选海量数据集、揭示潜在模式并提供关键洞见的能力，从而推动技术与科学的前沿发展。然而，LLMs也引发了隐私方面的担忧。用户与LLMs的交互可能暴露其敏感的个人或公司信息。若缺乏健全的隐私保护措施与法律框架，可能导致对个人数据的不当侵入或处理，从而带来隐私侵犯与身份盗用的风险。为确保隐私，必须最小化共享提示词与私人信息之间的关联性。已有多种随机化方法被提出以保护提示词的隐私，但与未受保护的LLMs提示相比，这些方法可能导致效用损失。因此，在实施有效的保护机制时，评估隐私泄露风险与效用损失之间的平衡至关重要。本研究构建了一个用于推理隐私保护大型语言模型（LLMs）的框架，并为审视隐私保护与效用之间的相互作用奠定了坚实的理论基础。其核心见解被概括为一个定理，称为NFL（即“无免费午餐”的缩写）定理。