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

==== - If you want a worst‑case guarantee over all graphs/instances of the kind stated in your open problem—i.e., a universal δ\deltaδ such that plain FISTA/linear coupling updates ≤δ/ρ\le \delta/\rho≤δ/ρ coordinates each iteration—then I don’t see a path via “standard acceleration analysis” alone, because the whole difficulty is controlling the extrapolated iterates’ active set, and generic proximal identification theory tells you you cannot control that uniformly without a nondegeneracy margin. ====
* The only provable acceleration-with-sparsity result I’m aware of in this line is essentially active-set acceleration (Rubio et al.), which introduces exactly the kind of “subproblem” / restricted-space solves you want to avoid. Proceedings of Machine Learning Research<ref>{{cite web|title=Proceedings of Machine Learning Research|url=https://proceedings.mlr.press/v195/martinez-rubio23b/martinez-rubio23b.pdf|publisher=Proceedings of Machine Learning Research|access-date=2025-12-19}}</ref>

===== The most plausible compromise (theorem-wise) is: =====
# prove a finite identification result for your PageRank instance class (needs a margin condition or a probabilistic model), then
# run a single accelerated method on the identified support.

This avoids repeated expensive restricted solves, but it won’t give a clean worst-case O~((αρ)−1)\tilde O((\sqrt{\alpha}\rho)^{-1})O~((αρ)−1) unless you can lower-bound the margin in terms of ρ\rhoρ for all graphs—which is exactly the kind of statement that seems false in worst case (symmetries/degeneracy).

If you tell me which “standard accelerated algorithm” you have in mind (plain FISTA, strongly-convex APG with fixed β\betaβ, linear coupling variant, etc.), I can be more concrete about which invariants one would need to establish (e.g., order-bounds 0≤yk≤c x\''0\le y_k\le c\,x^\''0≤yk≤cx\*, or a per-iteration certificate that prevents activation outside a small frontier), and where the known theory breaks.