DEL: Digit Entropy Loss for Numerical Learning of Large Language Models

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. 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 conditional probability and binary cross-entropy to guide the entropy optimization into a supervised manner; deprecating the distance term to bypass the issue of numerical distance; and generalizing the integer-based numerical learning to floating-point number optimization, enabling more accurate number prediction. Our DEL formulation can incorporate integers, decimals, and decimal points, expanding the learning objective from a single digit to the floating-point number domain. Experiments conducted on seven mathematical reasoning benchmarks with four representative LLMs, including CodeLlama, Mistral, DeepSeek, and Qwen-2.5, demonstrate that DEL consistently outperforms its counterparts in both overall prediction accuracy and numerical distance. Source codes and trained models will be publicly available.