Geometric Moment Contraction for Stochastic Nesterov Acceleration

Practical Muon maintains momentum and performs a small, fixed number of Newton--Schulz iterations separately for each parameter matrix, often with a Nesterov correction. We analyze these layer-wise finite-step updates jointly on a coupled nonconvex objective, rather than replacing them by exact polar factors or one global orthogonalization. Under gradient-dependent $(\mathcal L_0,\mathcal L_1,q)$-smoothness and conditionally unbiased stochastic gradients with bounded layer-wise variance, we establish an $\mathcal O(T^{-1/4})$ bound on the expected average Frobenius gradient norm. The analysis retains the Nesterov recursion and requires neither bounded stochastic gradients, symmetric noise, nor a uniform positive lower bound on the nonzero output singular values. Its constants contain no explicit matrix-dimension or rank factors when the number of blocks and problem constants are fixed. The proof follows a descent inequality and a decomposition of the momentum tracking error into initialization, noise, and drift. For the original five-step quintic, we verify the required scalar-map bounds analytically; the result also allows step-dependent coefficients satisfying the same bounds. A complementary nuclear-norm result quantifies rank dependence under a stronger spectral condition. The vanishing rate uses coupled learning-rate and momentum schedules, including the standard single-coefficient Nesterov rule.

We survey Lyapunov-based techniques for the finite-time analysis of stochastic iterative algorithms, also known as stochastic approximation (SA) algorithms, for solving fixed-point equations $\bar{F}(x)=x$, where the operator $\bar{F}(\cdot)$ can only be accessed through a noisy oracle. We first focus on the standard setting in which $\bar{F}(\cdot)$ is contractive with respect to some norm and the noise is i.i.d., and explain how generalized Moreau envelopes serve as universal Lyapunov functions, regardless of the underlying norm. We then show how this framework yields mean-square convergence guarantees and applies to stochastic gradient descent, linear SA, and value-based reinforcement learning algorithms such as Q-learning and temporal-difference learning. Finally, we discuss extensions to Markovian noise, seminorm-contractive operators, dissipative operators, and high-probability bounds, and conclude with open problems. The goal is to present a unified and self-contained roadmap for the finite-time analysis of SA and its applications, especially in reinforcement learning.

Large neural networks are commonly trained across many workers, and communication between them is a major cost. Workers therefore take several local optimization steps before sending their net parameter displacements, or pseudo-gradients, to an outer optimizer, which often applies Nesterov momentum. We study how the outer momentum, the number of local steps and the averaging over workers control the training dynamics, and in particular their stability. For random quadratics, we derive an exact mean-square stability criterion in any dimension, which reveals that the inner batch size determines whether increasing the number of local steps eventually stabilizes or destabilizes the outer iteration. Moreover, while averaging the pseudo-gradients of independent workers can make larger outer momentum values admissible, the number of workers required for stability can grow exponentially with the number of local steps. Finally, we show that the mean-square dynamics governing stability can be probed directly at intermediate neural-network checkpoints using shared-minibatch perturbations. We test our theoretical findings across networks, and measure their consequences for training loss.