Fine-Grained Sketch-Based Image Retrieval (FG-SBIR) aims at finding a specific image from a large gallery given a query sketch. Despite the widespread applicability of FG-SBIR in many critical domains (e.g., crime activity tracking), existing approaches still suffer from a low accuracy while being sensitive to external noises such as unnecessary strokes in the sketch. The retrieval performance will further deteriorate under a more practical on-the-fly setting, where only a partially complete sketch with only a few (noisy) strokes are available to retrieve corresponding images. We propose a novel framework that leverages a uniquely designed deep reinforcement learning model that performs a dual-level exploration to deal with partial sketch training and attention region selection. By enforcing the model's attention on the important regions of the original sketches, it remains robust to unnecessary stroke noises and improve the retrieval accuracy by a large margin. To sufficiently explore partial sketches and locate the important regions to attend, the model performs bootstrapped policy gradient for global exploration while adjusting a standard deviation term that governs a locator network for local exploration. The training process is guided by a hybrid loss that integrates a reinforcement loss and a supervised loss. A dynamic ranking reward is developed to fit the on-the-fly image retrieval process using partial sketches. The extensive experimentation performed on three public datasets shows that our proposed approach achieves the state-of-the-art performance on partial sketch based image retrieval.