Smoothing DiLoCo with Primal Averaging for Faster Training of LLMs

We propose Generalized Primal Averaging (GPA), an extension of Nesterov's method that unifies and generalizes recent averaging-based optimizers like single-worker DiLoCo and Schedule-Free, within a non-distributed setting. While DiLoCo relies on a memory-intensive two-loop structure to periodically aggregate pseudo-gradients using Nesterov momentum, GPA eliminates this complexity by decoupling Nesterov's interpolation constants to enable smooth iterate averaging at every step. Structurally, GPA resembles Schedule-Free but replaces uniform averaging with exponential moving averaging. Empirically, GPA consistently outperforms single-worker DiLoCo and AdamW with reduced memory overhead. GPA achieves speedups of 8.71%, 10.13%, and 9.58% over the AdamW baseline in terms of steps to reach target validation loss for Llama-160M, 1B, and 8B models, respectively. Similarly, on the ImageNet ViT workload, GPA achieves speedups of 7% and 25.5% in the small and large batch settings respectively. Furthermore, we prove that for any base optimizer with $O(\sqrt{T})$ regret, where $T$ is the number of iterations, GPA matches or exceeds the original convergence guarantees depending on the interpolation constants.