Sharp First-Order Lower Bounds under $\alpha$-Polyak-Lojasiewicz Conditions
We study first-order oracle complexity under the $\alpha$-Polyak-Lojasiewicz condition $f(x)-f_\star\leq \tau\|\nabla f(x)\|^\alpha$ for $\alpha\in[1,2]$. For $\alpha<2$, we first show that global $L$-smoothness together with a global $\alpha$-Polyak-Lojasiewicz inequality forces the objective to be constant. This motivates a nontrivial model in which smoothness remains global but the inequality is required only on the initial sublevel set. On this class, we establish sharp minimax lower bounds for every $\alpha\in[1,2)$. Deterministic first-order methods require $\Omega(L\tau^{2/\alpha}\varepsilon^{-(2-\alpha)/\alpha})$ oracle calls, matching gradient descent. With unbiased stochastic gradients of conditional variance at most $\sigma^2$, randomized first-order methods require $\Omega(L\tau^{2/\alpha}\varepsilon^{-(2-\alpha)/\alpha}+L\sigma^2\tau^{4/\alpha}\varepsilon^{-(4-\alpha)/\alpha})$ calls, matching the corresponding SGD dependence when the inequality holds along the stochastic trajectory. At the classical endpoint $\alpha=2$, a separate construction yields the variance-dependent lower bound $\Omega(L\tau^2\sigma^2/\varepsilon)$ even for globally smooth objectives satisfying the Polyak-Lojasiewicz inequality globally. In contrast, the sharp variance-dependent complexity for smooth $\mu$-strongly convex objectives is $\Theta(\sigma^2/(\mu\varepsilon))$; with $\tau=(2\mu)^{-1}$, the worst-case global Polyak-Lojasiewicz class exhibits an additional factor $L\tau=L/(2\mu)$.