QAM: Quadratic-Accurate Checkpoint Merging via Sequential Consistency

Shihao Wang, Rui Kong, Xinran Chen, Hui Wu, Qipeng Qian +5

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…