Where, When, and How Many Bits? Scaling Laws for SGD with Stochastic Rounding

arXiv:2610.41074v1 Announce Type: new Abstract: Low-precision training reduces computation but introduces additional rounding error. We theoretically study how stochastic rounding (SR) affects prediction risk as model size, training time, and precision change. We consider a two-layer linear network with a fixed random first layer, trained by mini-batch SGD on fresh Gaussian data with power-law spectra. At each SGD step, the features used in forward and backward computations are rounded independently. We prove matched upper and lower bounds on prediction error for the final model and for uniform parameter averaging, with general learning-rate and precision schedules chosen before training.