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Video captioning has shown impressive progress in recent years. One key reason of the performance improvements made by existing methods lie in massive paired video-sentence data, but collecting such strong annotation, i.e., high-quality sentences, is time-consuming and laborious. It is the fact that there now exist an amazing number of videos with weak annotation that only contains semantic concepts such as actions and objects. In this paper, we investigate using weak annotation instead of strong annotation to train a video captioning model. To this end, we propose a progressive visual reasoning method that progressively generates fine sentences from weak annotations by inferring more semantic concepts and their dependency relationships for video captioning. To model concept relationships, we use dependency trees that are spanned by exploiting external knowledge from large sentence corpora. Through traversing the dependency trees, the sentences are generated to train the captioning model. Accordingly, we develop an iterative refinement algorithm that refines sentences via spanning dependency trees and fine-tunes the captioning model using the refined sentences in an alternative training manner. Experimental results demonstrate that our method using weak annotation is very competitive to the state-of-the-art methods using strong annotation.