Muon has demonstrated strong empirical performance in large-scale language-model training, motivating a deeper theoretical understanding of its practical implementation. Unlike the idealized polar-factor update, practical Muon employs a fixed number of quintic Newton-Schulz iterations. Existing analyses primarily establish average- or best-iterate stationarity guarantees, while last-iterate convergence remains less understood. In this work, we investigate last-iterate gradient convergence of practical finite-step Muon on smooth, lower-bounded nonconvex objectives. In the deterministic setting, we show that Muon without input stabilization can exhibit cyclic behavior even on a smooth convex quadratic. In contrast, with positive stabilization, we establish last-iterate gradient convergence and square-summability of gradients for any fixed momentum coefficient below one and a sufficiently small constant learning rate. In the stochastic setting, we first demonstrate that persistent bounded, unbiased noise can prevent convergence under fixed learning rates and momentum, even with positive stabilization. We then establish almost-sure and mean-square last-iterate gradient convergence under bounded-variance stochastic gradients by combining diminishing learning rates with increasing momentum. Finally, we construct a smooth counterexample demonstrating that neither convergence theorem implies a pointwise polynomial convergence rate. Nevertheless, we establish density-one convergence rates, guaranteeing polynomial gradient decay for all but an asymptotically vanishing fraction of iterations.
This paper characterizes the second-order coding region of the scalar Gaussian broadcast channel along the curved boundary of its capacity region. The characterization holds under both average and maximal error criteria and allows correlation between the receiver noises. At each interior point of this boundary, the second-order region is a halfspace, as in the work of Scarlett and the author on Gaussian multiple-access channels with degraded message sets. Achievability follows from spherical superposition coding with a blocklength-dependent power allocation. The main innovation lies in the converse. Using Ornstein--Uhlenbeck entropy dissipation and Gaussian functional inequalities, we construct an auxiliary observation through an additive Gaussian noise channel and establish quantitative relative entropy bounds for the channel output distributions conditioned on this observation. The strong functional representation lemma of Li and El Gamal then allows us to select a realization of its independent random seed to obtain a deterministic partition of the retained receiver-1 message set. The component label plays a role analogous to the auxiliary random variable in the capacity characterization, while providing the distributional control needed at the second-order scale. These properties guide the construction of auxiliary output measures whose likelihood ratios admit the cancellations and normal approximation needed to evaluate an information spectrum converse.
We study \textit{best-of-both-worlds} (BOBW) algorithms for linear bandits, which aim to simultaneously guarantee $\gO(\sqrt T)$ regret in the adversarial regime and $\gO(\log T)$ regret in the stochastic regime over $T$ rounds. For finite action sets, we introduce \emph{information-acquisition follow-the-regularized-leader} (\IAFTRL{}), the first BOBW algorithm based purely on the follow-the-regularized-leader (FTRL) framework that attains an $\gO(\cstar\log T)$-type regret in the stochastic regime, where $\cstar$ is the information-acquisition complexity that characterizes instance-dependent optimality in linear bandits, while retaining the $\gO(\sqrt{T})$ regret guarantee in the adversarial regime. This resolves the open problem posed in~\cite{LeeEtAl2021,KongZ023,ItoT23,ZhaoItoLi2025}.