Second-Order, First-Class: A Composable Stack for Curvature-Aware Training

Second-order methods make fast progress per iteration, but forming and factorizing the Hessian often dominates their wall-clock time. We study Split-Client, a task-parallel design in which a curvature client factorizes Hessians in the background while a gradient client takes cubic-regularized steps with the most recently published matrix, so that curvature cost appears as a delay rather than as blocking time. For delayed exact Hessians, we prove a first-order stationarity bound that depends on the average age of the curvature rather than its maximum, using an overlap count and a weighted Hardy inequality that allows any bounded-error initial matrix; the analysis extends to subsampled and uniformly inexact curvature. Under a fixed-cost model, the resulting wall-clock bound matches the scaling of optimally tuned Lazy Hessian without choosing a reuse period. In experiments on two LIBSVM datasets, an implementation decreases the objective earlier in wall-clock time than Lazy and Vanilla cubic Newton, with the largest gains at intermediate accuracy.

In this paper, we present CT-AGD (Curvature-Tuned Accelerated Gradient Descent), an optimization method for non-convex optimization problems in deep learning training tasks. CT-AGD is a general boosting procedure that accelerates first-order methods by explicitly capturing the local curvature using finite-difference quotients, and the development of heuristics aimed at mitigating noise and bias introduced by stochastic mini-batch training. CT-AGD has a comparable storage and computational overhead as adaptive gradient methods such as Adam. Our extensive experiments demonstrate that CT-AGD achieves the same level of accuracy as the baseline first-order methods, yet reduces the required training epochs by 33% on average.

Muon improves training efficiency over Adam in large language-model training by about two times, but the local geometric source of this advantage remains unclear. Our work takes a first step toward demystifying Muon's superiority over Adam from a curvature perspective. First, we apply a second-order Taylor approximation to the training landscape and show that Muon achieves a larger one-step loss decrease than Adam at matched validation loss. The two optimizers have comparable first-order gains, but Muon consistently incurs a smaller second-order curvature penalty. Second, we decompose this curvature penalty into the squared update norm and Normalized Directional Sharpness (NDS). We find that Muon and Adam have comparable update norms, so Muon's smaller curvature penalty is driven by lower NDS, not update scale. Third, we study how training data and model structure shape Muon's NDS advantage. Using Zipf-Probabilistic Context-Free Grammar (PCFG) data with controlled imbalance, we show that data imbalance amplifies Muon's NDS advantage over Adam. A within-/cross-layer decomposition further shows that, in the middle and late stages of training, Muon's lower NDS is mainly sustained by smaller within-layer curvature. Beyond empirical evidence, we analyze stylized quadratic problems with heterogeneous curvature and gradient alignment toward high-curvature modes. We prove that Muon attains a smaller average NDS than GD by balancing update energy across curvature groups; when curvature heterogeneity is sufficiently strong, this also yields lower local quadratic loss after the same number of steps.