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A growing number of people are seeking healthcare advice online. Usually, they diagnose their medical conditions based on the symptoms they are experiencing, which is also known as self-diagnosis. From the machine learning perspective, online disease diagnosis is a sequential feature (symptom) selection and classification problem. Reinforcement learning (RL) methods are the standard approaches to this type of tasks. Generally, they perform well when the feature space is small, but frequently become inefficient in tasks with a large number of features, such as the self-diagnosis. To address the challenge, we propose a non-RL Bipartite Scalable framework for Online Disease diAgnosis, called BSODA. BSODA is composed of two cooperative branches that handle symptom-inquiry and disease-diagnosis, respectively. The inquiry branch determines which symptom to collect next by an information-theoretic reward. We employ a Product-of-Experts encoder to significantly improve the handling of partial observations of a large number of features. Besides, we propose several approximation methods to substantially reduce the computational cost of the reward to a level that is acceptable for online services. Additionally, we leverage the diagnosis model to estimate the reward more precisely. For the diagnosis branch, we use a knowledge-guided self-attention model to perform predictions. In particular, BSODA determines when to stop inquiry and output predictions using both the inquiry and diagnosis models. We demonstrate that BSODA outperforms the state-of-the-art methods on several public datasets. Moreover, we propose a novel evaluation method to test the transferability of symptom checking methods from synthetic to real-world tasks. Compared to existing RL baselines, BSODA is more effectively scalable to large search spaces.