Sharp First-Order Lower Bounds under Sublevel $α$-Polyak-Lojasiewicz Conditions

Abstract

We study the optimal complexity of first-order methods under the $α$-Polyak-Lojasiewicz condition with $α\in[1,2)$. This condition bounds the suboptimality gap by a power $α$ of the gradient norm; $α=2$ recovers the classical Polyak-Lojasiewicz condition, $α=1$ corresponds to a Holder error-bound regime, and intermediate exponents arise near degenerate minima in local Kurdyka-Lojasiewicz geometry. We first prove a structural obstruction: if global smoothness and a global $α$-Polyak-Lojasiewicz inequality are imposed on $\mathbb{R}^d$, then every such function is constant for $α<2$. This motivates the globally smooth, sublevel-$α$-Polyak-Lojasiewicz class, where the inequality is required only on the initial sublevel set. On this class, we prove sharp minimax lower bounds for first-order methods. In the deterministic oracle model, any first-order method requires $Ω(Lτ^{2/α}ε^{-(2-α)/α})$ queries to reach accuracy $ε$, matching gradient descent. In the bounded-variance stochastic-gradient oracle model, any stochastic first-order method requires $Ω(Lσ^2τ^{4/α}ε^{-(4-α)/α})$ queries in the noise-dominated regime, matching known SGD upper rates under trajectory-containment assumptions.