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Fine-grained sketch-based image retrieval (FG-SBIR) addresses the problem of retrieving a particular photo instance given a user's query sketch. Its widespread applicability is however hindered by the fact that drawing a sketch takes time, and most people struggle to draw a complete and faithful sketch. In this paper, we reformulate the conventional FG-SBIR framework to tackle these challenges, with the ultimate goal of retrieving the target photo with the least number of strokes possible. We further propose an on-the-fly design that starts retrieving as soon as the user starts drawing. To accomplish this, we devise a reinforcement learning-based cross-modal retrieval framework that directly optimizes rank of the ground-truth photo over a complete sketch drawing episode. Additionally, we introduce a novel reward scheme that circumvents the problems related to irrelevant sketch strokes, and thus provides us with a more consistent rank list during the retrieval. We achieve superior early-retrieval efficiency over state-of-the-art methods and alternative baselines on two publicly available fine-grained sketch retrieval datasets.