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

Abstract

We study first-order oracle complexity under the $α$-Polyak-Lojasiewicz condition $f(x)-f_\star\leq τ\|\nabla f(x)\|^α$ for $α\in[1,2]$. For $α<2$, we first show that global $L$-smoothness together with a global $α$-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 $α\in[1,2)$. Deterministic first-order methods require $Ω(Lτ^{2/α}\varepsilon^{-(2-α)/α})$ oracle calls, matching gradient descent. With unbiased stochastic gradients of conditional variance at most $σ^2$, randomized first-order methods require $Ω(Lτ^{2/α}\varepsilon^{-(2-α)/α}+Lσ^2τ^{4/α}\varepsilon^{-(4-α)/α})$ calls, matching the corresponding SGD dependence when the inequality holds along the stochastic trajectory. At the classical endpoint $α=2$, a separate construction yields the variance-dependent lower bound $Ω(Lτ^2σ^2/\varepsilon)$ even for globally smooth objectives satisfying the Polyak-Lojasiewicz inequality globally. In contrast, the sharp variance-dependent complexity for smooth $μ$-strongly convex objectives is $Θ(σ^2/(μ\varepsilon))$; with $τ=(2μ)^{-1}$, the worst-case global Polyak-Lojasiewicz class exhibits an additional factor $Lτ=L/(2μ)$.