Border identification (BI) was previously proposed to help learning systems focus on the most relevant portion of the training set so as to improve learning accuracy. This paper argues that the traditional BI implementation suffers from a serious limitation: it is only able to identify partial borders. This paper proposes a new BI method called Progressive Border Sampling (PBS), which addresses this limitation by borrowing ideas from recent research on Progressive Sampling. PBS progressively learns optimal borders from the entire training sets by, first, identifying a full border, thus, avoiding the limitation of the traditional BI method, and, second, by incrementing the size of that border until it converges to an optimal sample, which is smaller than the original training set. Since PBS identifies the full border, it is expected to discover more optimal samples than traditional BI. Our experimental results on the selected 30 benchmark datasets from the UCI repository show that, indeed, in the context of classification, PBS is more successful than traditional BI at reducing the size of the training sets and optimizing the accuracy results. [ABSTRACT FROM AUTHOR]