QAM: Quadratic-Accurate Checkpoint Merging via Sequential Consistency
Shihao Wang, Rui Kong, Xinran Chen, Hui Wu, Qipeng Qian, Jinman Zhao, Jiashu Zhao, Yuchen Li, Jimmy Huang, Dawei Yin
Abstract
Saved checkpoints record states along a training trajectory, but generally do not determine the updates at states that would be visited under a different schedule. We study how accurately these checkpoints can reconstruct the endpoint of a sequential reference with prescribed update strengths. Under a common local transition model, two checkpoint-index moment conditions characterize all convex merges that agree with this reference through second order. We then prove an information limit that for nondegenerate profiles, no algorithm using only a fixed-length gradient-descent (GD) history with step size $h$ can achieve $o(h^3)$ endpoint error uniformly over a fixed class of smooth, strongly convex losses. The lower bound follows from two losses with identical GD checkpoint histories but sequential reference endpoints separated by $Ω(h^3)$. \textbf{Quadratic-Accurate Merging} (QAM) achieves a matching uniform $O(h^3)$ endpoint error bound. Its explicit coefficients also define the unique profile-dependent merge that exactly matches the sequential GD reference across all fixed quadratic objectives. Across two public Adam checkpoint trajectories (SmolLM3-3B and OpenEuroLLM-Prelude-9B), three windows and three profiles per model, and 15 tasks, QAM shows mixed results for short windows and broader advantages over \textbf{Warmup-Stable and Merge} (WSM) for longer windows. Matched-moment GSM8K diagnostics further show that local consistency alone does not fully determine downstream scores. These results characterize the reconstruction limits of saved histories, provide a coefficient rule that attains the optimal rate, and assess its practical utility.