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

We study the optimal complexity of first-order methods under the $\alpha$-Polyak-Lojasiewicz condition with $\alpha\in[1,2)$. This condition bounds the suboptimality gap by a power $\alpha$ of the gradient norm; $\alpha=2$ recovers the classical Polyak-Lojasiewicz condition, $\alpha=1$ 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 $\alpha$-Polyak-Lojasiewicz inequality are imposed on $\mathbb{R}^d$, then every such function is constant for $\alpha<2$. This motivates the globally smooth, sublevel-$\alpha$-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 $\Omega(L\tau^{2/\alpha}\epsilon^{-(2-\alpha)/\alpha})$ queries to reach accuracy $\epsilon$, matching gradient descent. In the bounded-variance stochastic-gradient oracle model, any stochastic first-order method requires $\Omega(L\sigma^2\tau^{4/\alpha}\epsilon^{-(4-\alpha)/\alpha})$ queries in the noise-dominated regime, matching known SGD upper rates under trajectory-containment assumptions.