Synchronization on circles and spheres with nonlinear interactions

We consider the dynamics of $n$ points on a sphere in $\mathbb{R}^d$ ($d \geq 2$) which attract each other according to a function $\varphi$ of their inner products. When $\varphi$ is linear ($\varphi(t) = t$), the points converge to a common value (i.e., synchronize) in various connectivity scenarios: this is part of classical work on Kuramoto oscillator networks. When $\varphi$ is exponential ($\varphi(t) = e^{\beta t}$), these dynamics correspond to a limit of how idealized transformers process data, as described by Geshkovski et al. (2024). Accordingly, they ask whether synchronization occurs for exponential $\varphi$. In the context of consensus for multi-agent control, Markdahl et al. (2018) show that for $d \geq 3$ (spheres), if the interaction graph is connected and $\varphi$ is increasing and convex, then the system synchronizes. What is the situation on circles ($d=2$)? First, we show that $\varphi$ being increasing and convex is no longer sufficient. Then we identify a new condition (that the Taylor coefficients of $\varphi'$ are decreasing) under which we do have synchronization on the circle. In so doing, we provide some answers to the open problems posed by Geshkovski et al. (2024).

Accelerated Methods for Riemannian Min-Max Optimization Ensuring Bounded Geometric Penalties

In this work, we study optimization problems of the form $\min_x \max_y f(x, y)$, where $f(x, y)$ is defined on a product Riemannian manifold $\mathcal{M} \times \mathcal{N}$ and is $\mu_x$-strongly geodesically convex (g-convex) in $x$ and $\mu_y$-strongly g-concave in $y$, for $\mu_x, \mu_y \geq 0$. We design accelerated methods when $f$ is $(L_x, L_y, L_{xy})$-smooth and $\mathcal{M}$, $\mathcal{N}$ are Hadamard. To that aim we introduce new g-convex optimization results, of independent interest: we show global linear convergence for metric-projected Riemannian gradient descent and improve existing accelerated methods by reducing geometric constants. Additionally, we complete the analysis of two previous works applying to the Riemannian min-max case by removing an assumption about iterates staying in a pre-specified compact set.