Construct a strongly monotone example exhibiting β²/α² tracking-error dependence

Construct a nonlinear two-time-scale stochastic approximation instance satisfying the Lipschitzness and one-point strong-monotonicity assumptions on the averaged slow dynamics such that the asymptotic MSE of the fast iterate depends on (β/α)^2.

Background

The paper gives an example in which the fast iterate tracks the slow iterate with asymptotic MSE proportional to (β/α){2/γ}, approaching the β²/α² dependence as γ approaches one. However, that example violates the slow-dynamics assumption requiring Lipschitzness and strong monotonicity. The authors explicitly state that they have not found an example satisfying that assumption, leaving open whether the β²/α² dependence persists under the full theorem hypotheses.

References

It is an open question if one can construct an example that satisfies A.\ref{assumption: Lip and Mono of g}, and whose asymptotic MSE for $x$ has a dependence on $(\beta/\alpha)$.

— The Bias of Nonlinear Two-Time-scale Stochastic Approximation under Constant Step-Sizes  (2609.20409 - Lamouri et al., 17 Sep 2026) in Appendix, Section "Illustrations," subsection "Example showing a lower bound in Ω(β²/α²)," paragraph "Discussion on the assumptions"