From Lyapunov Analysis to Algorithm Design in two-sided PL Minimax Optimization

We derive algorithms for smooth nonconvex nonconcave minimax optimization and establish linear convergence rates for problems that satisfy the two-sided Polyak-Lojasiewicz (PL) inequality. At the core of our approach is the observation that Lyapunov functions can be used not only to certify convergence a posteriori, but also to design algorithms. By replacing an idealized, intractable Lyapunov function with a computable surrogate based on gradient information, we derive TALDA (Tri-Action Lyapunov Descent Ascent), a single-loop algorithm that enforces Lyapunov descent by construction. TALDA guarantees linear convergence under the two-sided PL condition, with a rate that depends explicitly on the cross-smoothness constant. This recovers existing worst-case guarantees while yielding sharper convergence rates in weakly coupled min–max problems.