Subword tokenization is a key part of most NLP pipelines. However, little is known about why some tokenizer and hyperparameter combinations lead to improved downstream model performance over others. We propose that good tokenizers lead to efficient channel usage, where the channel is the means by which some input is conveyed to the model and efficiency can be quantified in information-theoretic terms as the ratio of the Shannon entropy to the maximum entropy of the subword distribution. Nevertheless, an optimal encoding according to Shannon entropy assigns extremely long codes to low-frequency subwords and very short codes to high-frequency subwords.Defining efficiency in terms of Rényi entropy, on the other hand, penalizes distributions with either very high or very low-frequency subwords.We posit that (1) extremely high-frequency subwords are problematic because their meaning is not distinct and (2) that low-frequency subwords may not appear frequently enough for their meaning to be learned properly; encodings that induce unigram distributions with either can harm model performance. In machine translation, we find that across multiple tokenizers, the Rényi entropy has a very strong correlation with BLEU: 0.82 in comparison to just -0.30 for compressed length.

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/sigma*(1-e(-sigma))-approximation of an optimal merge sequence, where sigma is the total backward curvature with respect to the optimal merge sequence. Empirically the lower bound of the approximation is approx0.37.We provide a faster implementation of BPE which improves the runtime complexity from O(NM) 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.