Close fixed-dimensional complexity gaps and characterize surface-entropy necessity

Determine the optimal complexity of smooth Polyak–Łojasiewicz optimization in fixed dimensions, including the condition-number-dependent deterministic gaps for dimensions d≥2 and the stochastic complexity in the complementary regime α/(α−1)<d−1, and establish whether the metric-entropy factor arising from reconstructing (d−1)-dimensional separating surfaces is information-theoretically necessary or an artifact of gradient-flow trapping.

Background

The paper establishes matching or near-matching complexity results for several regimes of smooth PL optimization with conditionally unbiased stochastic gradients having a finite centered α-moment. In fixed dimensions one, two, and three, the stochastic term is characterized up to logarithmic factors, while for fixed d>3 the same stochastic characterization is obtained when α/(α−1)≥d−1. The upper bound in the complementary regime contains an additional condition-number-dependent surface-entropy factor.

The conclusion identifies unresolved gaps beyond these results. It asks whether the remaining deterministic condition-number dependence in dimensions d≥2 can be removed, what the correct stochastic complexity is when α/(α−1)<d−1, and whether the metric-entropy factor caused by dyadic reconstruction of separating surfaces reflects a fundamental information-theoretic obstruction or merely the limitations of the gradient-flow-trapping method.