Speculative decoding accelerates large language model inference using a smaller draft model. In this paper, we establish a surprising connection between speculative sampling and the concept of channel simulation from information theory, which aims at simulating a noisy channel using as few bits as possible. This connection allows us to provide an information-theoretic analysis of the speed up that can be achieved by speculative sampling. Leveraging this link, we derive an explicit relation between generation speed-up and the number of tokens k generated by the draft model for large k, which serves as an upper bound for all k. We also propose a novel speculative sampling method via exponential races called ERSS that matches state-of-the-art performance.

LLM training is resource-intensive. Quantized training improves computational and memory efficiency but introduces quantization noise, which can hinder convergence and degrade model accuracy. Stochastic Rounding (SR) has emerged as a theoretically attractive alternative to deterministic rounding, offering unbiased gradient estimates. However, its interaction with other training factors—especially batch size—remains underexplored. In this paper, we present a theoretical and empirical study of mini-batch stochastic gradient descent (SGD) with SR, showing that increased batch sizes can compensate for reduced precision during backpropagation. Furthermore, we show that quantizing weights and activations impacts gradient variance in distinct ways. Our experiments validate these theoretical insights. Our experiments validate these theoretical insights.