The paper introduces Activation-Keyed Momentum (AK‑Momentum), a momentum update that uses the input activation of a linear layer as a key to apply a delta‑rule update, allowing each direction to decay at a rate proportional to its frequency of appearance. AK‑Momentum is proven to be a valid momentum, incorporates input‑side curvature correction without matrix inversion, and clears stale directions faster than traditional exponential moving average (EMA) under both fixed and drifting optima. It can replace the momentum buffer of any optimizer, scales with width under μP, adds only 22–25% extra compute, and demonstrates significant step‑count reductions in FineWeb‑Edu pretraining and other benchmarks.
whyItMatters":"AK‑Momentum offers a principled, efficient way to adapt momentum decay to anisotropic training dynamics, improving convergence speed and stability across a range of models and datasets."
By Euijin Hong, Guannan Qu
The paper introduces a physical response-and-memory model for the Muon optimizer, explaining its semi‑orthogonalized momentum update as the maximally dissipative direction under an output‑side safety budget. It treats the weight matrix as a responsive medium with internal stress, showing that momentum corresponds to accumulated stress whose relaxation occurs over multiple timescales—fast and slow. Based on this, the authors propose the Bi‑Maxwell optimizer, which uses a two‑timescale memory kernel and achieves target loss in fewer steps on a public large‑language‑model benchmark.
By Yinze Hu, Hongjun Xiang, Xingao Gong, Hongyu Yu
The paper identifies a problem called Gradient-Update Mismatch (GUM), where optimizers can alter conflict-free gradient directions produced by gradient surgery, leading to conflicts between physics residual and boundary condition losses in Physics-Informed Neural Networks (PINNs). To address this, the authors propose Gradient-Update Alignment (GUA), which projects the optimizer’s update onto the conflict-free cone and adjusts internal optimizer state accordingly. Experiments show GUM is common across many optimizers, and GUA consistently eliminates conflicts and significantly reduces error in PINN training.
By Jing Xiao, Xinhai Chen, Qinglin Wang, Menghan Jia, Zhiquan Lai, Dongsheng Li, Jie Liu, Tiejun Li
The paper introduces the Drift Contract, a spectral update geometry for local learning that improves depth robustness and hyperparameter stability. By applying momentum orthogonalization with spectral step scaling to per‑layer updates, the authors achieve consistent performance across a wide range of widths and depths on CIFAR‑10 MLPs, outperforming local Adam and providing a per‑layer, input‑conditioned drift bound. The study also shows that the spectral geometry itself, rather than step‑size rules, drives the observed depth robustness, while a negative result indicates that the stability benefit is limited to non‑normalized layers.
By Fabien Polly
arXiv:2609.07666v1 Announce Type: new
Abstract: Full-parameter fine-tuning of large language models has substantial memory costs because backpropagation stores activations and gradients. Zeroth-order...
By Yuyang Wang, Haoyu Yao, Pengcheng Xie
The paper argues that the policy governing how tensors are rounded and reused during the backward pass—termed the backward‑state policy—is an integral part of the learning algorithm, not merely a memory or precision detail. Experiments with 390 M‑parameter models show that whether the backward pass reuses a forward’s rounded output or generates a new rounding can decisively affect training success, even when copy accuracy is high. The authors propose a method to determine, for each operator, which value should be read or substituted to preserve the correct gradient, and validate these predictions on PyTorch and Transformer Engine.
By Shuxiao Xie, Shuyang Xie, Dezhi Ran, Wei Yang, Tao Xie
arXiv:2606. 22932v2 Announce Type: replace Abstract: Reverse-mode differentiation computes every weight gradient, writes it to memory, and only then lets the optimizer read it back.
By Dikshant Kukreja, Kritarth Prasad, Avinash Anand, Zhengkui Wang, Erik Cambria, Timothy Liu, Aik Beng Ng, Simon See, Bapi Chatterjee
arXiv:2608. 04407v1 Announce Type: cross Abstract: Memory-efficient matrix optimizers such as Sinkhorn gradient descent remove most AdamW optimizer state for dense Transformer matrices, but direct application to Mixture-of-Experts (MoE) training is unreliable.
By Masato Fujitake
arXiv:2606. 17526v1 Announce Type: new Abstract: Efficient optimization is essential for training large language models.
By Da Chang, Ganzhao Yuan
arXiv:2610. 03306v1 Announce Type: cross Abstract: Existing convergence analyses of Muon either assume exact orthogonalization or analyze classical Newton--Schulz polynomials, and guarantee only stationarity, so it is unresolved what Muon's five tuned Newton--Schulz steps preserve and whether that suffices to reach a prescribed neural-network training loss.
By Amartya Roy, Souvik Chakraborty
arXiv:2502. 11034v3 Announce Type: replace Abstract: Loss spikes remain a persistent obstacle in large-scale language model pretraining.
By Guoxia Wang, Shuai Li, Congliang Chen, Jinle Zeng, Jiabin Yang, Dianhai Yu, Yanjun Ma, Li Shen
The paper investigates how different optimizers acquire distinct generalizing relations in single‑hidden‑layer ReLU networks. By manipulating nuisance‑connected weights and performing a strict‑parity sweep, the authors show that training conditions reshape both shared failures and relative optimizer advantages, with specific interventions improving accuracy on hard stages. The study also compares optimizer behavior across tasks and an external architecture (CIFAR‑10).
By Gongyue Zhang, Honghai Liu