Mild cognitive impairment (MCI) is a prodromal phase in the progression from normal aging to dementia, especially Alzheimers disease. Even though there is mild cognitive decline in MCI patients, they have normal overall cognition and thus is challenging to distinguish from normal aging. Using transcribed data obtained from recorded conversational interactions between participants and trained interviewers, and applying supervised learning models to these data, a recent clinical trial has shown a promising result in differentiating MCI from normal aging. However, the substantial amount of interactions with medical staff can still incur significant medical care expenses in practice. In this paper, we propose a novel reinforcement learning (RL) framework to train an efficient dialogue agent on existing transcripts from clinical trials. Specifically, the agent is trained to sketch disease-specific lexical probability distribution, and thus to converse in a way that maximizes the diagnosis accuracy and minimizes the number of conversation turns. We evaluate the performance of the proposed reinforcement learning framework on the MCI diagnosis from a real clinical trial. The results show that while using only a few turns of conversation, our framework can significantly outperform state-of-the-art supervised learning approaches.