PASO: Step Parallel Stochastic Optimization

This paper approaches the fundamental challenge of accelerating the inherently autoregressive nature of gradient descent (GD) like SGD and Adam through a dynamic system perspective. Specifically, we introduce a unified framework that recasts the autoregressive GD process as solving a system of triangular nonlinear equations (TNEs), thereby enabling step-parallel training, where gradients for different GD steps are computed concurrently without sequential dependencies. Within this generic framework, we establish that: (1) the TNE system admits a unique solution corresponding precisely to the autoregressive GD iterative trajectory; (2) solving the TNEs system guarantees convergence to the GD iterative trajectory in at most the equal iterations. Building on these insights, we present PASO, the first step-parallel optimizer for accelerating a broad class of GD-based optimizers like SGD and Adam. Extensive experiments (e.g., Llama-3.2-1B and diffusion model) validate that PASO achieves up to 21$\times$ reduction in GD steps and 4.5$\times$ speedup in wall-clock time, with no model quality loss. Source code is available at: https://github.com/Jianrong-Lu/PASO.git.