arXiv:2604. 08742v2 Announce Type: replace-cross Abstract: Adam is widely used, but its convergence theory remains incomplete even in the deterministic full-batch setting because momentum and adaptive preconditioning are tightly coupled.
arXiv:2607. 27383v1 Announce Type: new Abstract: We establish the first convergence guarantees for the plain vector-form \emph{Adam} optimizer under heavy-tailed stochastic noise.
arXiv:2505. 13196v3 Announce Type: replace-cross Abstract: We introduce Velocity-Regularized Adam (VRAdam), a physics-inspired optimizer for training deep neural networks that draws on ideas from quartic terms for kinetic energy with its stabilizing effects on various system dynamics.
arXiv:2602. 10204v2 Announce Type: replace Abstract: We introduce MVN-Grad (Momentum on Variance-Normalized Gradients), an Adam-style optimizer that improves stability and performance by combining two complementary ideas: variance-based normalization and momentum applied after normalization.
arXiv:2602. 00334v2 Announce Type: replace Abstract: Momentum Stochastic Gradient Descent (mSGD) relies on a fixed momentum coefficient shared across all parameters, failing to account for the heterogeneous structure of modern loss landscapes.
arXiv:2603. 09923v4 Announce Type: replace Abstract: Exponential moving averages (EMAs) are a central component of widely used adaptive optimizers such as Adam.
arXiv:2606. 03917v1 Announce Type: cross Abstract: As Moore's law reaches its limits, Ising machines offer a promising alternative computing approach for difficult optimization problems.
arXiv:2608. 16760v1 Announce Type: new Abstract: Reliable optimization is central to neural network (NN) training, yet Adam, the default optimizer for modern LLMs, rests on a fragile foundation.
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.
The paper investigates how Adam’s update rule relates to natural gradient descent (NGD) by treating Adam as a diagonal empirical Fisher approximation with additional factors such as diagonal truncation, empirical label substitution, and temporal lag. Using a scale‑invariant metric, the authors quantify Adam’s geometric deviation from true NGD across four loss landscapes—well‑conditioned and ill‑conditioned linear regression, logistic regression, and a small neural network—finding that deviation is low in well‑conditioned settings but can reach about 10³ in ill‑conditioned or non‑convex scenarios. Despite higher geometric drift correlating with slower early optimization, Adam still achieves low final loss, and the improved empirical Fisher (iEF) yields more stable trajectories than the standard empirical Fisher (EF).
The paper investigates the edge‑of‑stability (EoS) phenomenon observed in full‑batch Adam optimization. It identifies Adam’s second‑moment adaptation as a negative‑feedback mechanism that drives the dynamics toward the stability boundary, characterized by the *active curvature*—the preconditioned curvature along the preconditioned gradient direction. The study provides rigorous characterizations across various settings, predicts *gradient reversal* near the edge (consecutive gradients pointing in nearly opposite directions), and shows that averaging iterates suppresses these oscillations, yielding smoother and lower loss curves.
The paper introduces SHANG++—an accelerated stochastic gradient descent algorithm designed to be robust under multiplicative noise scaling (MNS). Building on a semi‑implicit discretization called SHANG, SHANG++ adds a damping correction that improves stability and convergence for both convex and strongly convex objectives. Experiments on convex problems and deep learning tasks, including a noise‑robust test on ResNet‑34, show that SHANG++ consistently outperforms existing accelerated methods with minimal parameter sensitivity.
The paper establishes uniform a priori bounds for the Adam optimizer, enabling an unconditional error analysis for a broad class of strongly convex stochastic optimization problems. Prior analyses were conditional, assuming Adam remained bounded, whereas this work removes that assumption. The results provide a rigorous foundation for Adam’s performance in training deep neural networks and other convex optimization tasks.