Compared to traditional machine learning models, recent large language models (LLMs) can exhibit multi-task-solving capabilities through multi-modal data sources and multi-turn conversations. These unique characteristics of LLMs, together with their large model size, make their deployment more challenging. Specifically, (i) deploying LLMs on devices faces computational, memory, and energy resource issues, while (ii) deploying them in the cloud cannot guarantee real-time service and incurs communication/usage costs. In this paper, we design TMO, a device-cloud LLM inference system with Three-M Offloading: Multi-modal, Multi-task, and Multi-turn. TMO incorporates (i) a lightweight on-device LLM that can process simple tasks at high speed and (ii) a large-scale cloud LLM that can handle multi-modal data sources. We develop a resource-constrained reinforcement learning (RCRL) strategy for TMO that optimizes the inference location (i.e., device vs. cloud) and multi-modal data sources to use for each task in multi-turn conversations, aiming to maximize the long-term reward (response quality, latency, and usage cost) while adhering to resource constraints. We also contribute M4A1, a new dataset we curated across multiple modalities, tasks, conversation turns, and LLM configurations, enabling evaluation of offloading decisions. We demonstrate the effectiveness of TMO compared to several exploration-decision and LLM-as-Router baselines, showing significant improvements in latency, cost, and response quality. Our code and dataset are available at https://github.com/liangqiyuan/TMO.

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