In plain words: A system's predictability—how fast tiny state changes grow—decides whether its trajectory can be solved in parallel instead of step by step. Predictable ones finish in at worst the square of the log of the sequence length; chaotic ones are too slow.
Abstract · Predictability Enables Parallelization of Nonlinear State Space Models
The rise of parallel computing hardware has made it increasingly important to understand which nonlinear state space models can be efficiently parallelized. Recent advances like DEER (arXiv:2309.12252) and DeepPCR (arXiv:2309.16318) recast sequential evaluation as a parallelizable optimization problem, sometimes yielding dramatic speedups. However, the factors governing the difficulty of these optimization problems remained unclear, limiting broader adoption. In this work, we establish a precise relationship between a system's dynamics and the conditioning of its corresponding optimization problem, as measured by its Polyak-Lojasiewicz (PL) constant. We show that the predictability of a system, defined as the degree to which small perturbations in state influence future behavior and quantified by the largest Lyapunov exponent (LLE), impacts the number of optimization steps required for evaluation. For predictable systems, the state trajectory can be computed in at worst $O((\log T)^2)$ time, where $T$ is the sequence length: a major improvement over the conventional sequential approach. In contrast, chaotic or unpredictable systems exhibit poor conditioning, with the consequence that parallel evaluation converges too slowly to be useful. Importantly, our theoretical analysis shows that predictable systems always yield well-conditioned optimization problems, whereas unpredictable systems lead to severe conditioning degradation. We validate our claims through extensive experiments, providing practical guidance on when nonlinear dynamical systems can be efficiently parallelized. We highlight predictability as a key design principle for parallelizable models.
Xavier Gonzalez, Leo Kozachkov, David M. Zoltowski, Kenneth L. Clarkson, Scott W. Linderman
arXiv:2508.16817 · math.OC, cs.LG, eess.SY, math.DS, stat.ML · submitted Aug 22, 2025 · updated Feb 7, 2026
abstract · pdf · html · NeurIPS '25. XG and LK dual lead authors. Code: https://github.com/lindermanlab/predictability_enables_parallelization