Title: On Position Embeddings in BERT. Abstract: Various Position Embeddings (PEs) have been proposed in Transformer based architectures~(e.g. BERT) to model word order. These are empirically-driven and perform well, but no formal framework exists to systematically study them. To address this, we present three properties of PEs that capture word distance in vector space:  translation invariance, monotonicity, and  symmetry. These properties formally capture the behaviour of PEs and allow us to reinterpret sinusoidal PEs in a principled way.
Moreover, we propose a new probing test (called `identical word probing') and  mathematical  indicators to quantitatively detect the general  attention patterns with respect to the above properties. An empirical evaluation of seven PEs (and their combinations) for classification (GLUE) and span prediction (SQuAD) shows that: (1) both  classification and span prediction benefit from  translation invariance and local monotonicity, while symmetry slightly decreases performance;
(2) The fully-learnable absolute PE performs better in classification, while relative PEs perform better in span prediction.  We contribute the first formal and quantitative analysis of desiderata for PEs, and  a principled discussion about their correlation to the performance of typical downstream tasks.