Extending LISO theory beyond local strong convexity

Establish theoretical guarantees for Laplace Importance Sampling Optimization when the objective function is not locally strongly convex around the global minimizer but satisfies a weaker geometric condition, for example the Polyak–Łojasiewicz inequality, thereby handling functions that exhibit only directional or PL-type strong convexity.

Background

The current theoretical results assume a unique global minimizer and local strong convexity near x*, conditions that enable Laplace-principle-based approximations and underpin the convergence analysis. The method relies on convex combinations, and without these assumptions both theory and practice may fail.

Many practical objectives satisfy weaker conditions, such as the Polyak–Łojasiewicz inequality, rather than full local strong convexity. Extending the analysis to these settings would broaden the applicability of LISO and address a stated limitation of the present work.

References

An important question for future work is to handle functions which are not locally strongly convex but still enjoy a weaker form of strong convexity when restricted to some directions, such as Polyak-Łojasiewicz functions .

— Importance Sampling Optimization with Laplace Principle  (2604.02882 - Dragomir et al., 3 Apr 2026) in Section Discussion