random matrix products

This work addresses the lack of theoretical guarantees for the Adam algorithm in time-varying non-stationary systems, where existing analyses rely on the restrictive i.i.d. assumption. To bridge this gap, the authors develop a general theoretical framework tailored to dynamic environments by coupling the recursive dynamics of first- and second-order moments and introducing a novel stochastic Lyapunov function. They further establish analytical techniques for products of non-stationary dependent random matrices. Within this framework, they derive the first explicit bounds on both parameter tracking error and output prediction error for Adam, quantitatively characterizing the influence of step size, momentum parameters, gradient noise, and parameter drift. The theoretical findings are validated through experiments on both synthetic and real-world datasets, offering practical guidance for hyperparameter tuning.

This work addresses a critical gap in the theoretical understanding of the Adam optimizer in strongly convex stochastic optimization, where existing convergence analyses rely on the restrictive assumption that the solution sequence remains bounded and lack unconditional error bounds. For the first time, this study establishes a uniform priori boundedness of the Adam iterates, thereby removing the need for such an assumption. By leveraging tools from stochastic optimization theory, recursive inequalities, and uniform estimation techniques, the authors rigorously prove the unconditional convergence of Adam for a broad class of strongly convex stochastic problems and derive explicit upper bounds on the optimization error. These results provide a solid theoretical foundation that significantly strengthens the justification for Adam’s widespread practical use.

Existing adaptive optimizers (e.g., Adam) neglect the gradient covariance structure, resulting in poor directional adaptivity and slow convergence. This work proposes an implicit diagonalization method based on invertible linear coordinate transformations, which maps the preconditioning matrix into an approximately diagonal space without explicit low-rank or sparsity approximations—thereby balancing computational efficiency and directional modeling fidelity. The approach integrates seamlessly into memory-efficient optimizers (e.g., Adafactor) while preserving full compatibility with standard training pipelines. Evaluated on large language models including LLaMA, it achieves a 2× speedup in convergence and significantly accelerates training across diverse deep models, without increasing GPU memory overhead. The core contribution is the first use of invertible transformations to enable implicit diagonalization of preconditioning matrices, breaking the traditional trade-off between computational cost and accuracy in covariance modeling.

Existing adaptive optimization algorithms—such as AdaGrad, RMSProp, and Adam—lack a unified continuous-time characterization, hindering rigorous theoretical analysis and principled design. Method: The authors formulate these algorithms as first-order integro-differential equations, establishing the first unified continuous-time dynamical model that captures their implicit gradient-based dynamics and adaptive step-size mechanisms. Contribution/Results: Through rigorous numerical simulations, the proposed continuous model is shown to faithfully reproduce the discrete algorithms’ optimization trajectories, convergence rates, and adaptive behavior. This framework enables novel convergence analysis techniques and provides a theoretically grounded foundation for designing improved adaptive optimizers. By bridging discrete iterative updates with continuous dynamics, the work significantly extends the scope and rigor of continuous-time optimization theory.

This work addresses the convergence of RMSProp and Adam for generalized smooth nonconvex optimization. Under the weakest known assumptions—coordinate-wise generalized smoothness and affine noise variance—we establish the first tight theoretical guarantees. We introduce a novel descent lemma that overcomes critical challenges: adaptive step-size dependence, unbounded gradient estimates, and mismatched Lipschitz constants. Rigorously, we prove that both algorithms converge to an ε-stationary point in O(ε⁻⁴) iterations—the optimal rate matching the fundamental lower bound for nonconvex stochastic optimization. Our analysis operates under strictly weaker assumptions and yields tighter bounds than all prior works on RMSProp and Adam, thereby advancing the foundational understanding and reliability certification of adaptive optimization methods.

This study investigates whether Adam and its variants can guarantee convergence with vanishing average regret under arbitrary momentum decay parameters. By constructing a specific three-period linear loss sequence—with slopes \(c\), \(-1\), and \(-1\), where \(c\) is slightly greater than 2—and leveraging tools from online convex optimization together with the projected Adam algorithm, the analysis overcomes the conventional restriction requiring \(\beta_1 < \sqrt{\beta_2}\). The results demonstrate that even under more permissive parameter settings, several widely used optimizers—including Adam, AdamW, RMSProp, and NAdam—can still incur non-vanishing average regret. In fact, their average regret is shown to be bounded below by a positive constant, thereby exposing a fundamental convergence limitation of these methods in worst-case scenarios.

We provide a theoretical analysis of Adam under non-stationary stochastic objectives, separating two regimes: Euclidean tracking under adaptive strong monotonicity of the Adam-preconditioned mean-gradient operator, and high-probability projected stationarity guarantees under general $L$-smooth objectives. In the tracking regime, we derive finite-time expected and high-probability bounds that decompose sharply into four components: initialization, objective drift, a first-moment tracking error governed by $β_1$, and a preconditioner perturbation governed by $β_2$. We characterize the burn-in time to reach Adam's irreducible tracking floor under constant and step-decay schedules. We also prove a high-probability bound on the average projected stationarity gap for Adam under distribution shift. Across both analyses, our bounds reveal a noise--drift tradeoff: in noise-dominated regimes, first-moment averaging and adaptive preconditioning can improve the high-probability error, whereas in drift-dominated regimes, stale first-moment information and preconditioner perturbations can compound the cost of nonstationarity, allowing vanilla SGD to achieve a smaller tracking floor. Our explicit $(β_1,β_2,ε)$-dependent bounds delineate when adaptive step-sizing is beneficial versus harmful, and provide a theoretical mechanism for Adam's empirical instability and stabilization under distribution shift.

This work addresses the finite-time convergence of stochastic iterative algorithms for fixed-point equations accessible only through a noisy oracle. The authors propose a norm-independent, unified Lyapunov function framework constructed via a generalized Moreau envelope, which integrates Lyapunov stability theory with stochastic approximation analysis. This framework accommodates complex settings such as Markovian noise, seminorm contractive operators, and dissipative operators, yielding sharp non-asymptotic convergence bounds in both high-probability and mean-square senses. As a result, it provides a unified and refined finite-time convergence guarantee for a broad class of algorithms, including stochastic gradient descent, linear stochastic approximation, Q-learning, and temporal difference learning.

This work investigates the statistical behavior and convergence rate of averaged Adam as it approaches the attracting zero of its associated vector field. Leveraging stochastic approximation theory and dynamical systems analysis, the study establishes, for the first time, a central limit theorem for averaged Adam, rigorously demonstrating its classical \(n^{-1/2}\) convergence rate and explicitly characterizing the covariance structure of the limiting distribution. These results confirm the asymptotic efficiency equivalence between averaged Adam and classical stochastic approximation algorithms, thereby providing a solid theoretical foundation for understanding the statistical properties of adaptive optimization methods.