Complexity Bounds for Smooth Multiobjective Optimization

Abstract

We study the oracle complexity of finding $\varepsilon$-Pareto stationary points in smooth multiobjective optimization with $m$ objectives. Progress is measured by the Pareto stationarity gap $\mathcal{G}(x)$, the norm of the best convex combination of objective gradients. Our analysis relies on a non-degenerate lifting that embeds hard single-objective instances into MOO instances with distinct objectives and non-singleton Pareto fronts while preserving lower bounds on $\mathcal{G}$. We establish: (i) in the $\mu$-strongly convex case, any span first-order method has worst-case linear convergence no faster than $\exp(-\Theta(T/\sqrt{\kappa}))$ after $T$ oracle calls, yielding $\Theta(\sqrt{\kappa}\log(1/\varepsilon))$ iterations and matching accelerated upper bounds; (ii) in the convex case, an $\Omega(1/T)$ min-iterate lower bound for oblivious one-step methods and a universal last-iterate lower bound $\Omega(1/T^2)$ for oblivious span methods via polynomial-degree arguments, and we further show this latter bound is loose (for general adaptive methods) by importing geometric lower bounds to obtain an $\Omega(1/T)$ min-iterate lower bound for general adaptive first-order methods; (iii) in the nonconvex case with $L$-Lipschitz gradients, an $\Omega(\sqrt{L}/(T+1))$-type lower bound on $\mathcal{G}$ (tight in order), implying $\Omega(1/\varepsilon^2)$ iterations to reach $\mathcal{G}(x)\le\varepsilon$ up to natural scaling.