Bessel Function Analysis of Nesterov's ODE in $N$-Player Quadratic Games
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911544893243392 |
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| author | Paek, Jay |
| author_facet | Paek, Jay |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_16982 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Bessel Function Analysis of Nesterov's ODE in $N$-Player Quadratic Games Paek, Jay Optimization and Control 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. |
| title | Bessel Function Analysis of Nesterov's ODE in $N$-Player Quadratic Games |
| topic | Optimization and Control |
| url | https://arxiv.org/abs/2602.16982 |