Editing Openai/693e3ce6-229c-8008-97dc-ab720cb1f95a (section)

==== Not by itself. ====
* The bound is quadratic in 1/γ1/\gamma1/γ, not linear in 1/ρ1/\rho1/ρ.
* To turn it into a “local algorithm” guarantee, you would need to relate γ\gammaγ to ρ\rhoρ (and control ∥yk−x⋆∥\|y_k-x^\star\|∥yk−x⋆∥ by something that doesn’t hide an nnn-dependence). In worst case, γ\gammaγ can be arbitrarily small even when ∣S⋆∣|S^\star|∣S⋆∣ is O(1/ρ)O(1/\rho)O(1/ρ), so the lemma allows very large transient supports.

So the best honest summary is:
* Yes, you can bound transient spurious activations using a margin parameter, but the natural bound you can prove from standard smoothness + margin is of the form ∣Ak∣≲∥yk−x⋆∥2/γ2,|A_k| \lesssim \|y_k-x^\star\|^2/\gamma^2,∣Ak∣≲∥yk−x⋆∥2/γ2, which is not the O~(1/ρ)\tilde O(1/\rho)O~(1/ρ) sparsity you ultimately want.
* Getting a uniform O~(1/ρ)\tilde O(1/\rho)O~(1/ρ) per-iteration support bound for acceleration typically needs extra invariants/controls (active-set control, monotonicity/restarter tricks, etc.), which is exactly where the “nonstandard” machinery comes in—consistent with why the identification papers emphasize oscillations/inertial effects. arXiv<ref>{{cite web|title=arXiv|url=https://arxiv.org/pdf/1503.03703|publisher=arxiv.org|access-date=2025-12-19}}</ref>

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