Dynamics of Nesterov's Accelerated Gradient Descent in Quadratic Games

Abstract

We analyze Nesterov's accelerated gradient descent (NAGD) for Nash equilibrium seeking in $N$-player quadratic games. While the continuous-time NAGD dynamics, specifically the Su-Boyd-Candès ODE, are well understood for convex optimization, their behavior with non-symmetric pseudo-gradient matrices arising in games has not been previously studied. We establish sharp spectral characterizations: stability holds if and only if all eigenvalues of the pseudo-gradient matrix $G$ lie in $\mathbb{R}_{\geq 0}$, with the convergence direction additionally requiring diagonalizability of $G$. Remarkably, complex eigenvalues with positive real parts, which ensure stability for first-order gradient dynamics, induce exponential instability in NAGD. This reveals that the momentum mechanism enabling $O(1/t^2)$ convergence in optimization can be detrimental for equilibrium seeking in non-potential games, providing theoretical guidance for algorithm selection.

Figures (5)

• Figure 1: Symmetric positive definite $G$ with eigenvalues $0.32, 0.88$: (a) phase portrait showing convergence to the origin; (b) norm evolution confirming the $O(t^{-3/2})$ decay rate from Theorem \ref{['thm:symmetric']}(a).
• Figure 2: Normal $G$ with complex eigenvalues $6 \pm 1.5i$: (a) first-order dynamics $\dot{x} = -Gx$ converge; (b) NAGD dynamics diverge; (c) norm evolution showing exponential decay $\sim e^{-6t}$ for first-order versus exponential growth $\sim e^{|\mathrm{Im}(\sqrt{\lambda})|t}/t^{3/2}$ with $|\mathrm{Im}(\sqrt{\lambda})| \approx 0.30$ for NAGD. This demonstrates the surprising instability from Corollary \ref{['cor:complex']}.
• Figure 3: Diagonal $G$ with eigenvalues $1$ and $-0.5$: (a) phase portrait showing unbounded growth in the $x_2$ direction; (b) component evolution confirming $x_1 \to 0$ (stable mode) while $|x_2|$ grows as $e^{\sqrt{0.5}t}$ (unstable mode), validating Theorems \ref{['thm:symmetric']}(b) and \ref{['thm:chetaev_negative']}.
• Figure 4: Positive semidefinite $G$ with eigenvalues $0, 0.5$: (a) trajectory converges to the null space $\mathcal{N}(G)$ (dashed line); (b) distance to $\mathcal{N}(G)$ decays at rate $O(t^{-3/2})$, confirming Theorem \ref{['thm:symmetric']}(a) and Lemma \ref{['lem:nullspace']}.
• Figure 5: Multiplayer potential games with symmetric positive definite $G$: (a) 3-player trajectory in $\mathbb{R}^3$ converging to the origin; (b) 3-player norm decay matching the $O(t^{-3/2})$ rate; (c) 4-player component evolution showing damped oscillations; (d) 4-player norm decay confirming Theorem \ref{['thm:symmetric']}(a) for $N > 2$.

Theorems & Definitions (41)

• Definition 1: Pseudo-gradient rosen1965
• Remark 1: Translation invariance
• Remark 2: Symmetry and potential games
• Remark 3: Role of the damping coefficient
• Lemma 1: Modal projection
• proof
• Lemma 2: General solution
• proof
• Lemma 3: Modal asymptotics
• proof