On the Spectral Dependence of Decentralized Stochastic Gradient Tracking

Abstract

We study how the spectral geometry of the mixing matrix enters the convergence of decentralized stochastic gradient tracking (GT) for smooth strongly convex problems. General bounds contain the factor ${\Delta }_{\lambda }^{-3}$ in the higher-order stochastic term, where ${\Delta }_{\lambda }$ is the absolute spectral gap. We show that, even for two agents with identical quadratic objectives, the exact steady-state mean-square disagreement scales as ${\alpha }^2{\sigma }^2{\Delta }_{\lambda }^{-3}$ as the nontrivial eigenvalue approaches $-1$, provided the stepsize $\alpha$ is sufficiently small relative to the gap. Here ${\sigma }^2$ is the gradient-noise variance. By unrolling the disagreement recursion, we establish a stochastic bound that averages all nontrivial eigenvalue contributions for diagonalizable mixing matrices, up to an eigenvector conditioning factor. As a real or complex eigenvalue approaches the unit circle away from $1$, its contribution grows as the inverse cube of its distance to the circle; growth can be slower near $1$. For symmetric mixing with gaps $p$ at $1$ and $\nu$ at $-1$, the weighted sum ${\theta }_1/p+{\theta }_2/{\nu }^3$ replaces the prior factor $({\Delta }_{\lambda }{\nu }^2{)}^{-1}$. For $N$ agents, $N{\theta }_1$ and $N{\theta }_2$ count the nonnegative and negative nontrivial eigenvalues, respectively. On a symmetric family, a finite-horizon lower bound for the average iterate matches the worst-case factor $p^{-1}+{\nu }^{-3}$ up to universal constants. Numerical experiments confirm the predicted scaling.