# Stochastic single-call last-iterate rate

## Author claim mapped

For the stated simplified single-call variant, the paper gives an `O(1/sqrt(k))` last-iterate expected squared-residual rate.

## Scope and source route

- Assumptions: monotone, `L`-Lipschitz `G`; an unbiased stochastic oracle; and `E[||Ghat(x, xi)||^2 | x] <= sigma^2 + kappa ||G(x)||^2` with `kappa >= 1`.
- Update: `eq:sto_updates`.
- Schedule: `beta_k = 1/(k+2)` and `eta_k = 1/(L sqrt(kappa) (k+2)^(3/4))`.
- Statement: `thm:last_iter_stoch_rho0`.
- Appendix route: `appx:last_iter_stoch_rho0`, joining `lem:det-reference` (`appx:lem_det_reference`) and `lem:stoch-stability` (`appx:lem_stoch_stability`).
- Evidence tier: source/theorem crosswalk only.

## Pin

Use the TeX archive SHA-256 in [`../hash_manifest.json`](../hash_manifest.json), then inspect `example_paper.tex` and `appendix.tex`.