The paper investigates how momentum influences optimization in a river‑valley loss landscape, where a low‑loss manifold is surrounded by steep orthogonal directions. It shows that momentum stabilizes large learning rates that vanilla gradient descent cannot tolerate, enabling faster progress along the river. The study also finds that in very flat, slowly spinning rivers, momentum itself does not directly accelerate tracking, but the larger permissible learning rate does.
arXiv:2605. 22432v2 Announce Type: replace Abstract: Modern deep learning commonly relies on AdamW with prescribed learning rate schedules, but recent works challenge both components: Schedule-Free optimization removes explicit schedules via iterate averaging, and Muon improves the update geometry by orthogonalizing momentum for matrix parameters.
The paper investigates how the local landscape geometry of language model pre‑training evolves, identifying two distinct phases. In Phase I, the landscape starts sharp, causing instability and loss plateaus at high learning rates, which explains the need for learning‑rate warmup and suggests longer warmups for larger peak rates. In Phase II, the geometry is governed by gradient noise scale, revealing a depth‑flatness trade‑off that motivates a dynamic batch‑size scheduler that starts small and grows later in training.
arXiv:2510. 14717v2 Announce Type: replace-cross Abstract: Increasing the batch size during training -- a ''batch ramp'' -- is a promising strategy to accelerate large language model pretraining.
arXiv:2505. 01423v2 Announce Type: replace-cross Abstract: Efficient computation of min-max problems is a central question in optimization, learning, games, and control.
arXiv:2606. 09012v1 Announce Type: cross Abstract: Post-training quantization (PTQ) converts a trained full-precision model into low-bit weights without task-level retraining, while quantization-aware training (QAT) incorporates quantization into the training loop.
arXiv:2601. 10962v2 Announce Type: replace Abstract: Stochastic gradient descent (SGD) is central to deep learning, yet the dynamical origin of its preference for flatter, more generalizable solutions remains unclear.
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."
The paper derives an exact discrete‑time law that captures how learning‑rate schedules and weight decay interact in scale‑invariant neural networks, showing that a single scalar quantity governs the effective step size. It demonstrates that the balance point between contraction and expansion is intrinsically unstable, leading to recurrent dynamics when using constant learning rates with weight decay. The authors extend this analysis to various optimizers and datasets, confirming the law’s precision and showing that performance peaks sharply at the predicted boundary.
arXiv:2606. 03938v1 Announce Type: cross Abstract: Multi-epoch training is becoming the standard now that compute is growing faster than the supply of high-quality text.
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.
arXiv:2605. 29547v2 Announce Type: replace-cross Abstract: Deep learning optimization relies heavily on the assumption of smooth loss landscapes, a condition systematically violated by modern architectures due to non-smooth components such as ReLU activations and quantization operators.
arXiv:2607. 10959v1 Announce Type: new Abstract: Standard learning rate schedules such as cosine annealing are tied to a fixed training horizon, limiting their ability to accommodate post hoc horizon extension.