Andreea-Alexandra Muşat

Andreea-Alexandra Muşat, Nicolas Boumal

Optimization algorithms are unlikely to converge to strict saddle points. Proofs to that effect rely on the Center-Stable Manifold Theorem (CSMT), casting algorithms as dynamical systems: $x_{k+1} = g_k(x_k)$. In its standard form, the CSMT is limited to autonomous systems (the maps $g_k$ are all the same). To study algorithms such as gradient descent with non-constant step-size schedules, we need a non-autonomous CSMT. There are a few, but they are unable to handle, for example, vanishing step sizes. To cover such scenarios, we establish a new Center-Stable Set Theorem (CSST) for non-autonomous systems. We use it to prove saddle avoidance for gradient descent (Euclidean and Riemannian) and for the proximal point method, without assuming Lipschitz gradients or isolated saddles, and allowing vanishing step sizes.

Antoine Gonon, Andreea-Alexandra Muşat, Nicolas Boumal

Muon updates weight matrices along (approximate) polar factors of the gradients and has shown strong empirical performance in large-scale training. Existing attempts at explaining its performance largely focus on single-step comparisons (on quadratic proxies) and worst-case guarantees that treat the inexactness of the polar-factor as a nuisance ``to be argued away''. We show that already on simple strongly convex functions such as $L(W)=\frac12\|W\|_{\text{F}}^2$, these perspectives are insufficient, suggesting that understanding Muon requires going beyond local proxies and pessimistic worst-case bounds. Instead, our analysis exposes two observations that already affect behavior on simple quadratics and are not well captured by prevailing abstractions: (i) approximation error in the polar step can qualitatively alter discrete-time dynamics and improve reachability and finite-time performance -- an effect practitioners exploit to tune Muon, but that existing theory largely treats as a pure accuracy compromise; and (ii) structural properties of the objective affect finite-budget constants beyond the prevailing conditioning-based explanations. Thus, any general theory covering these cases must either incorporate these ingredients explicitly or explain why they are irrelevant in the regimes of interest.