A Formal Perspective on Byte-Pair Encoding

Byte-Pair Encoding (BPE) is a popular algorithm used for tokenizing data in NLP, despite being devised initially as a compression method. BPE appears to be a greedy algorithm at face value, but the underlying optimization problem that BPE seeks to solve has not yet been laid down. We formalize BPE as a combinatorial optimization problem. Via submodular functions, we prove that the iterative greedy version is a

)-approximation of an optimal merge sequence, where σ(µ ⋆ ) is the total backward curvature with respect to the optimal merge sequence µ ⋆ . Empirically the lower bound of the approximation is ≈ 0.37.

We provide a faster implementation of BPE which improves the runtime complexity from O (N M ) to O (N log M ), where N is the sequence length and M is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization.