Abstract:We analyze Nesterov's accelerated gradient descent (NAGD) for Nash equilibrium seeking in $N$-player quadratic games. While the continuous-time NAGD dynamics - 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 characterized precisely. We establish spectral characterizations via Bessel function modal analysis: the equilibrium is unstable whenever any eigenvalue of the pseudo-gradient matrix $G$ lies outside $\mathbb{R}_{\geq 0}$, and all trajectories converge when every eigenvalue lies in $\mathbb{R}_{\geq 0}$ and $G$ is diagonalizable. 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.
From: Jay Paek [view email]
[v1]
Thu, 19 Feb 2026 00:52:58 UTC (216 KB)
[v2]
Wed, 25 Mar 2026 18:15:31 UTC (195 KB)
[v3]
Mon, 22 Jun 2026 23:21:17 UTC (195 KB)
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