The paper presents a theoretical study of Adam in non‑stationary stochastic optimization, distinguishing two regimes: Euclidean tracking under adaptive strong monotonicity and high‑probability projected stationarity for general smooth objectives. It derives finite‑time bounds that decompose into initialization, objective drift, first‑moment tracking error (β₁), and preconditioner perturbation (β₂), and characterizes burn‑in times for constant and step‑decay schedules. The analysis reveals a noise–drift tradeoff, showing that in noise‑dominated settings Adam’s adaptive mechanisms can improve guarantees, while in drift‑dominated settings they may worsen tracking, potentially making vanilla SGD preferable.
arXiv:2603. 09923v4 Announce Type: replace Abstract: Exponential moving averages (EMAs) are a central component of widely used adaptive optimizers such as Adam.
arXiv:2601. 12238v5 Announce Type: replace-cross Abstract: In this paper, we provide a comprehensive theoretical analysis of Stochastic Gradient Descent (SGD) and its momentum variants (Polyak Heavy-Ball and Nesterov) for tracking time-varying optima under strong convexity and smoothness.
arXiv:2606. 28879v1 Announce Type: new Abstract: The adaptive moment estimation algorithm, known as Adam, is widely used in modern machine learning, owing to its low per-iteration complexity and strong empirical performance.
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:2602.13960v2 Announce Type: replace Abstract: Constant-stepsize stochastic approximation (SA) is widely used in learning for computational efficiency, yet the distribution of the iterates is ty...
arXiv:2605. 18694v2 Announce Type: replace-cross Abstract: Many tasks in modern machine learning are observed to involve heavy-tailed gradient noise during the optimization process.
arXiv:2609. 12785v1 Announce Type: new Abstract: Classical convergence guarantees for stochastic gradient methods typically assume Lipschitz-smooth objectives and finite-variance gradient noise, both frequently violated in practice.
arXiv:2608.30382v1 Announce Type: new Abstract: Popular adaptive stochastic gradient descent (SGD) methods to train artificial intelligence (AI) systems include the RMSprop, the Adam, and the AdamW o...
arXiv:2608. 15824v1 Announce Type: new Abstract: Adam retains a moving average of past squared gradients in its denominator, but the optimization cost of this memory is not well understood.
arXiv:2310. 15976v4 Announce Type: replace Abstract: signSGD is attractive in nonconvex optimization because it communicates sign-valued rather than full-precision gradients.
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:2608. 25551v1 Announce Type: new Abstract: Stochastic gradient descent (SGD) is typically analyzed at a deterministic horizon chosen before the algorithm is run, even though practical stopping decisions are made adaptively by inspecting the evolving trajectory.