A Formal Perspective on Byte-Pair Encoding
This work addresses the lack of formal understanding and efficiency in BPE, a widely used tokenization method in NLP, offering theoretical insights and practical improvements.
The paper formalizes Byte-Pair Encoding (BPE) as a combinatorial optimization problem, proving that its greedy version approximates an optimal merge sequence with a theoretical bound of about 0.37 and providing a faster implementation that reduces runtime from O(NM) to O(N log M).
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 $\frac{1}{σ(\boldsymbolμ^\star)}(1-e^{-{σ(\boldsymbolμ^\star)}})$-approximation of an optimal merge sequence, where ${σ(\boldsymbolμ^\star)}$ is the total backward curvature with respect to the optimal merge sequence $\boldsymbolμ^\star$. Empirically the lower bound of the approximation is $\approx 0.37$. We provide a faster implementation of BPE which improves the runtime complexity from $\mathcal{O}\left(N M\right)$ to $\mathcal{O}\left(N \log M\right)$, where $N$ is the sequence length and $M$ is the merge count. Finally, we optimize the brute-force algorithm for optimal BPE using memoization.