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In typical deployments, visual data captured at edge devices (clients) is transmitted to the server for VLM inference. However, transmitting full-resolution images incurs high communication cost. Conversely, aggressive downsizing or excessive compression to mitigate communication overhead can discard fine-grained details, leading to accuracy degradation. To overcome this limitation, we design a communication-efficient two-stage framework. In the first stage, the server performs inference on the downsized thumbnail (global image) and quantifies the min-entropy of the output tokens. 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To overcome this limitation, we design a communication-efficient two-stage framework. In the first stage, the server performs inference on the downsized thumbnail (global image) and quantifies the min-entropy of the output tokens. 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