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."
arXiv:2608. 19491v1 Announce Type: new Abstract: Most modern optimizers form their momentum as an exponential moving average (EMA) of past gradients, forgetting every direction at one fixed rate.
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.
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.
arXiv:2606. 17526v1 Announce Type: new Abstract: Efficient optimization is essential for training large language models.
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.
The paper studies how different optimizers perform as training duration (overtraining) increases, focusing on matrix‑preconditioned methods (Muon, SOAP) and a momentum‑scheduled method (ADANA) compared to AdamW. Across models ranging from 51M to 253M parameters and overtraining factors up to 256×, the authors find that optimal learning‑rate schedules, weight‑decay coefficients, and memory settings shift with horizon, and that ADANA consistently outperforms AdamW, especially with log‑time weight decay and momentum cooldown. Muon and SOAP maintain roughly constant token‑efficiency advantages, with SOAP potentially improving at the highest overtraining levels.
arXiv:2607. 22444v1 Announce Type: new Abstract: For scale-invariant deep networks, Hyperball-style optimizers have shown strong performance in large-scale training by fixing the norms of matrix-valued parameters and normalizing updates.
arXiv:2608. 01997v1 Announce Type: new Abstract: Single-optimizer training is a poor fit for the distinct phases of deep network optimization: adaptive methods handle noisy early gradients well but overshoot flat minima, while SGD with momentum generalizes better in the late phase but converges slowly early on.
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.
arXiv:2606. 25971v1 Announce Type: new Abstract: Modern neural network training relies on optimizers such as Adam and Muon which act on each weight matrix as a single object.
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...
arXiv:2607. 19058v1 Announce Type: cross Abstract: Optimizer state is the largest single line item in the memory budget of mixture-of-experts (MoE) training: on a 6.