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该论文发现RL后训练中fp32主权重经bf16四舍五入导出时，大多数小于半个bf16间隔的策略更新被静默丢弃，使采样器落后优化器50-150步；作者提出仅需一行代码的Fair Cast（随机舍入/误差反馈），在Qwen3-0.6B到4B的25组配对实验中平均提升6.1分。
The paper derives Bellman-certified sensitivity envelopes for rounding continuous row-mixture policies into sparse binary policies in finite MDPs, proving unavoidable linear dimension dependence while showing candidate-specific bounds and local curvature integration substantially tighten pre-deployment return guarantees.
This paper analyzes generalization error, uniform stability, and uniform argument stability of gradient descent (GD) and stochastic gradient descent (SGD) over discrete parameter spaces with deterministic or stochastic rounding, showing that rounding degrades generalization for GD and introduces dimension-dependent errors for stochastic rounding.
Explores the behavior of floor and ceil functions when applied to denormalized floating-point numbers, highlighting differences between CPU and GPU implementations and potential pitfalls.