Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent
This paper presents a sharp pseudospectral analysis of transient amplification in coupled gradient descent algorithms, which are common in bilevel optimization, two-time-scale stochastic approximation, and adversarial training. The authors prove bounds on the Kreiss constant for block-triangular Jacobians and provide a finite-horizon iteration-complexity bound. Their theory reveals a non-asymptotic regime invisible to traditional spectral analysis, confirmed by experiments on linear-quadratic problems and neural network training.
[2606.04031] Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent
[Submitted on 1 Jun 2026]
Title:Pseudospectral Bounds for Transient Amplification in Coupled Gradient Descent
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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 $K(J) \leq 2/(1-\gamma) + \|C\|/(4(1-\gamma))$ when the diagonal blocks are symmetric with spectral radii at most $\gamma
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