Ordinary positive-loss empirical local minima

Determine whether positive-loss local minima of the empirical loss can be ruled out for finite teacher-generated datasets without assuming fixed-radius stability of the minima.

Background

The paper transfers the population benign-landscape result to empirical loss only for minima that minimize the empirical loss on a parameter ball of a prescribed fixed radius. The required sampling accuracy depends on that radius.

The authors explicitly leave unresolved whether the stronger conclusion holds for ordinary empirical local minima, which need not persist on a fixed-radius ball.

References

Moreover, our result on the empirical loss landscape concerns minima that persist on a prescribed parameter ball. Whether one can rule out ordinary positive-loss empirical local minima, without a fixed-radius stability assumption, remains open.

— Removing spurious minima for planar features by skip connections  (2610.01728 - Zimmermann et al., 1 Oct 2026) in Section 6, Conclusion and Limitations