Objective-Gap Convergence Rate of Goldstein Subgradient Descent

Establish the convergence rate of the objective-value gap \(J_\infty(K(n))-J_\infty^\ast\) for Goldstein's subgradient method applied to state-feedback \(\mathcal{H}_\infty\) policy optimization with step sizes proportional to \(1/n\).

Background

Prior work by Guo et al. established that Goldstein's subgradient method for state-feedback H∞\mathcal{H}_\infty policy optimization achieves asymptotic convergence of the objective value under step sizes proportional to $1/n$. However, that work did not provide a quantitative convergence rate for the global objective-value gap. Its alternative complexity guarantee concerns finding an approximate Goldstein-stationary point, which the paper notes does not necessarily imply a small objective-value gap.

References

While the authors showed that by setting $\eta(n)\propto 1/n$ one has $J_\infty(K(n))- J_\infty\ast\rightarrow 0$, the rate of convergence for the objective value gap is not known.

— Zeroth-Order Nonsmooth Nonconvex Optimization with Convex Liftings and Its Application to State-Feedback $H_\infty$ Policy Optimization  (2608.23178 - Wang et al., 24 Aug 2026) in Section 3.2, comparison with Guo et al.