A Formal Perspective on Byte-Pair Encoding
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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 1/σ(μ^⋆)(1-e^-σ(μ^⋆))-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.
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