arXiv:2607. 22927v1 Announce Type: new Abstract: Weights and biases are normally optimized as separate parameter tensors, yet they do not represent separate functions when the input to an affine layer has nonzero mean.
arXiv:2607. 13380v1 Announce Type: new Abstract: Predictive Coding (PC) offers a biologically motivated alternative to backpropagation via local weight updates, yet routing error between layers still relies on an autograd Jacobian-transpose ($J^\top$) product - the last non-local operation in PC.
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 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:2605.07815v2 Announce Type: replace Abstract: Muon fixes the \emph{direction} of every matrix-valued update at the polar factor of its momentum, while each layer's step \emph{magnitude} is addr...
arXiv:2608. 02991v1 Announce Type: new Abstract: Matrix spectral optimizers reshape weight-update spectra but usually delegate vector-valued biases to a separate optimizer.
arXiv:2608.08322v2 Announce Type: replace-cross Abstract: Adaptive loss-balancing schemes for physics-informed neural networks rest on a premise that every residual should be driven to zero. For leve...
arXiv:2608. 05136v1 Announce Type: new Abstract: Gradient descent on a factored model $W = UV^\top$ is implicitly biased toward low-rank solutions, while Adam, starting from the same small initialization, is not.
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. 29176v1 Announce Type: new Abstract: A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation.
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.
The paper investigates why adaptive optimizers like Adam outperform SGD when fine‑tuning Transformers. It introduces gradient heterogeneity—the variation in gradient norms across parameter blocks—and shows, both theoretically and experimentally, that this heterogeneity, together with Hessian heterogeneity, hampers SGD convergence while sign‑based methods such as SignSGD are less affected. The study links the source of gradient heterogeneity to layer‑normalization placement, finding that Post‑LN architectures exhibit the strongest effect, and uses SignSGD as a tractable proxy to analyze Adam‑like behavior and learning‑rate scaling.
A deep network's loss is invariant to continuous symmetries of its parameters: the logit shift, the ReLU rescaling, the LayerNorm scale, the per-head attention rotation. Adam's per-coordinate preconditioner drifts along each symmetry orbit, which pulls the trajectory off the symmetry quotient where the optimization lives and blurs the singular-learning rate the quotient makes readable.