Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent

• URL: http://arxiv.org/abs/2606.04031v1
• Date: Mon, 01 Jun 2026 20:42:04 GMT
• Title: Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent
• Abstract summary: We develop a pseudospectral theory for block-triangular Jacobians.<n>We show that the Kreiss constant satisfies $K(J) leq 2/ (1-) + |C|/(4 (1-))$ when the diagonal blocks are symmetric with spectral radii at most $ 1$.<n>Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Coupled gradient descent--where the update of one parameter block depends on another--underlies bilevel optimization, two-time-scale stochastic approximation, and adversarial training. When the coupled Jacobian is block-triangular, asymptotic stability is governed by the spectral radii of the diagonal blocks, yet transient amplification before convergence can be arbitrarily large due to non-normality. We develop a sharp pseudospectral theory for such block-triangular Jacobians, proving that the Kreiss constant satisfies $K(J) \leq 2/(1-γ) + \|C\|/(4(1-γ))$ when the diagonal blocks are symmetric with spectral radii at most $γ< 1$, and we establish matching minimax lower bounds. We characterize the critical coupling threshold for spectral instability and extend the analysis to nearly self-referential systems via a Neumann-series perturbation framework. As a consequence, we obtain a finite-horizon iteration-complexity bound of $O(K(J)^2 \log(1/δ))$ for stochastic coupled descent. Framed as scaling laws for non-stationary two-time-scale optimization, our results expose a non-asymptotic, instance-dependent regime of high-dimensional learning dynamics that is invisible to spectral-radius analysis. Experiments on linear-quadratic problems, IQC-based comparisons, and neural-network training confirm the theory.

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