Directly determine the adaptive inner-product sign

Determine the sign of the inner product \(\langle \nabla f(x^{k+1}),x^{k+1}-z^{k+1}\rangle\) without relying on Lemma 3.3, thereby avoiding additional full-gradient evaluations or corresponding directional computations and reducing the computational overhead of the adaptive UGM parameter rules.

Background

The adaptive UGM-II and UGM-III schemes select the extrapolation parameter using information about the inner product ⟨∇f(xk+1),xk+1−zk+1⟩\langle \nabla f(x^{k+1}),x^{k+1}-z^{k+1}\rangle. Its sign determines whether the parameter should be adjusted in one direction or the other, but evaluating the sign through Lemma \ref{lem:est} requires either an additional full-gradient evaluation or an equivalent directional computation. A direct sign-determination technique would make the adaptive accelerated methods less costly while remaining compatible with the unified Lyapunov analysis.

References

One interesting direction is to estimate the sign of $\langle \nabla! f(x{k+1}),x{k+1}-z{k+1} \rangle$ without relying on Lemma~\ref{lem:est}, which requires either additional full-gradient evaluations or the corresponding directional computations.

— UGM: A Unified Framework and New Perspectives for Accelerated Gradient Methods in Smooth and Strongly Convex Optimization  (2608.27368 - Zhou et al., 27 Aug 2026) in Section 5, Concluding remarks, item (i)

Examining its behavior under the current algorithmic setting constitutes another interesting research direction.

— UGM: A Unified Framework and New Perspectives for Accelerated Gradient Methods in Smooth and Strongly Convex Optimization  (2608.27368 - Zhou et al., 27 Aug 2026) in Section 5, Concluding remarks, item (iii)