Optimal Local Convergence Rates of Stochastic First-Order Methods under Local α-PL

We study the local convergence rate of stochastic first-order methods under a local α-Polyak-Lojasiewicz (α-PL) condition in a neighborhood of a target connected component M of the local minimizer set. The parameter α ∈ [1,2] is the exponent of the gradient norm in the α-PL inequality: α=2 recovers the classical PL case, α=1 corresponds to Holder-type error bounds, and intermediate values interpolate between these regimes. Our performance criterion is the number of oracle queries required to output hatx with F(hatx)-l ≤ varepsilon, where l := F(y) for any y ∈ M. We work in a local regime where the algorithm is initialized near M and, with high probability, its iterates remain in that neighborhood. We establish a lower bound Omega(varepsilon-2/α) for all stochastic first-order methods in this regime, and we obtain a matching upper bound O(varepsilon-2/α) for 1 ≤ α < 2 via a SARAH-type variance-reduced method with time-varying batch sizes and step sizes. In the convex setting, assuming a local α-PL condition on the varepsilon-sublevel set, we further show a complexity lower bound widetildeOmega(varepsilon-2/α) for reaching an varepsilon-global optimum, matching the varepsilon-dependence of known accelerated stochastic subgradient methods.