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The widely adopted maximum likelihood estimation (MLE) for LLM training is not tailored to number prediction. Recently, penalty-driven approaches, e.g., Number Token Loss and Discretized Distance Loss, introduce an inductive bias of numerical distance but induce over-sharpened and over-flattened digit distributions, respectively. In this paper, we make an in-depth analysis on LLM numerical learning, and show that existing numerical learning methods conceptually follow a criterion-distance formulation, where the criterion term represents optimization pattern and the distance term instills geometric prior. Consequently, we present Digit Entropy Loss (DEL) for auto-regressive numerical learning, which reformulates the conventional unsupervised entropy optimization in three key designs: leveraging digit ","title":"DEL: Digit Entropy Loss for Numerical Learning of Large Language Models","url":"https://arxiv.org/abs/2605.20369","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.20369v1 Announce Type: cross \nAbstract: Number prediction stands as a fundamental capability of large language models (LLMs) in mathematical problem-solving and code generation. The widely adopted maximum likelihood estimation (MLE) for LLM training is not tailored to number prediction. Recently, penalty-driven approaches, e.g., Number Token Loss and Discretized Distance Loss, introduce an inductive bias of numerical distance but induce over-sharpened and over-flattened digit distributions, respectively. 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