Efficient estimation of the strong convexity parameter without restarts

Determine whether the strong convexity parameter m of a Lipschitz-smooth, strongly convex objective f in the class F(L,m) can be efficiently estimated online without using restart schemes. The goal is to develop an estimator that operates during optimization, does not rely on prior knowledge of m, and avoids restarts.

Background

Accelerated first-order methods typically require knowledge of both the smoothness parameter L and the strong convexity parameter m. While L can be estimated via backtracking, estimating m has been widely regarded as difficult, leading to the prevailing use of restart schemes to cope with unknown m.

The paper positions this challenge as a central open problem motivating their method (NAG-free). They highlight that existing approaches either rely on heuristics, continuous-time motivations, or exhaustive sweeps over candidate m values, and lack guarantees that avoid restarts while keeping worst-case performance reasonable.

References

While the Lipschitz smoothness parameter L can be easily estimated by backtracking, directly estimating (i.e., without restarts) the strong convexity parameter m remains an open problem.

— Adaptive Acceleration Without Strong Convexity Priors Or Restarts  (2506.13033 - Cavalcanti et al., 16 Jun 2025) in Abstract (page 1)

Estimating the $L$-smoothness and $\mu$-strong convexity parameters without prior problem information remains challenging . It is worth investigating whether adaptive iteration-dependent parameters $L_k$ and $\mu_k$ can be incorporated into the UGM framework ugm to achieve accelerated convergence, following the parameter strategy of Nesterov’s FGM .

— 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 (ii)