Stochastic extensions of mirror Polyak

Establish non-asymptotic convergence guarantees for stochastic extensions of mirror Polyak, including settings without interpolation and potentially using the primal-dual lifting technique to remove knowledge of the optimal value.

Background

The paper develops mirror Polyak, a Bregman-projection generalization of the Euclidean Polyak step-size, and a primal-dual lifting construction that gives the lifted objective a known optimal value of zero. The concluding remarks identify the stochastic extension of these ideas as unresolved, particularly because applying Polyak-type step-sizes without interpolation is difficult.

The authors note that existing interpolation-based results and analyses of an idealized stochastic variant imply asymptotic convergence, but do not establish non-asymptotic behavior. They suggest that the primal-dual lifting construction may help improve existing stochastic schemes, leaving the development and analysis of such methods open.

References

Extensions of these ideas to the stochastic setting remain open and of interest for applications in machine learning. The Polyak step-size has been applied successfully for finite-sum problems under interpolation, when all functions share a minimizer and an optimal value of 0, but is difficult to apply without interpolation. Existing results assuming interpolation or using the idealized variant of~\citet{gower2025analysis} would show mirror Polyak converges asymptotically as it leads to a contraction~\citep[][Prop.~4.13]{bauschke2003bregman}, but its non-asymptotic behavior is unknown. Our lifting technique might help improve existing schemes in the stochastic case~\citep{garrigos2023functionvaluelearning,jiang2023adaptive}.

— Mirror Polyak and a Primal-Dual Lifting  (2608.17252 - Kunstner et al., 18 Aug 2026) in Section 4, “Concluding remarks,” paragraph “Stochastic extensions”

We explore extensions of {PolyakEG} to stochastic settings and their limitations, and they raise concrete questions for future work. Can the deterministic algorithm's adaptivity to H"older continuity and $(L_0,L_1)$-Lipschitzness be extended to stochastic operators?

— Polyak-Type Extragradient Methods for Monotone Root-Finding Problems  (2609.26581 - Yoon et al., 22 Sep 2026) in Section Conclusion

Which stability mechanisms can be incorporated into {DecPolyakSEG} to guarantee convergence for any stochastic monotone root-finding problems with adequate regularity assumptions?

— Polyak-Type Extragradient Methods for Monotone Root-Finding Problems  (2609.26581 - Yoon et al., 22 Sep 2026) in Section Conclusion