[Paper Review] PL conditions do not guarantee convergence of gradient descent-ascent dynamics

Jean-Christophe Mourrat|arXiv (Cornell University)|Feb 18, 2026

Stochastic processes and financial applications0 citations

TL;DR

The paper constructs a smooth function on a compact domain that satisfies a two-sided PL condition yet yields a gradient descent-ascent flow that is periodic, proving that two-sided PL does not guarantee global convergence to a saddle point. It also shows local convergence under the same condition.

ABSTRACT

We give an example of a function satisfying a two-sided Polyak-Lojasiewicz condition but for which a gradient descent-ascent flow line fails to converge to the saddle point, circling around it instead.

Motivation & Objective

• Motivate understanding of convergence under PL-type conditions in saddle-point problems.
• Construct a concrete C-infinity function on a compact domain with two-sided PL property that induces periodic GDA trajectories.
• Distinguish between local convergence and global convergence under two-sided PL conditions.
• Provide auxiliary results clarifying when PL implies local convergence for GDA in low dimensions.

Proposed method

• Define and analyze a two-variable function f on [-1,1]^2 with a unique critical point at the origin.
• Show f satisfies a two-sided PL condition via criteria for PL on I^2 and local Hessian constraints.
• Construct f by prescribing level lines as flow lines of a carefully designed vector field v, ensuring PL properties.
• Demonstrate that GDA dynamics remain periodic away from the origin by introducing a conserved quantity along trajectories.
• Prove local convergence near the origin using linearization and stability of the Jacobian of the GDA vector field.

Figure 1. The flow lines with a color scale from dark blue to yellow are the level lines of the function $f$ we build for Theorem 1.1 (the color scale indicates the magnitude of $\mathbf{v}$ ). The value of $f$ is not shown and is prescribed along the two orange lines according to ( 3.2 ). These two

Experimental results

Research questions

• RQ1 Does a two-sided Polyak-Łojasiewicz condition ensure global convergence of gradient descent-ascent dynamics to a saddle point?
• RQ2 Can a function satisfy two-sided PL while still yielding non-convergent (periodic) GDA trajectories?
• RQ3 Under what circumstances does PL imply local convergence for GDA in low dimensions?

Key findings

• There exists a C-infinity function f on [-1,1]^2 with a unique critical point at the origin that satisfies both f(x,y)-inf_x f(x,y) <= C|∂_x f|^2 and sup_y' f(x,y')-f(x,y) <= C|∂_y f|^2 for all (x,y) in [-1,1]^2.
• The gradient descent-ascent flow for this f is periodic for a family of initial conditions, despite the two-sided PL property.
• Locally near the origin, the GDA flow converges to the saddle point, illustrating a mismatch between local and global behavior under two-sided PL.
• Away from the origin, the construction yields an integral of motion (an L^4-norm after a π/8 rotation), which prevents convergence along certain trajectories.
• The paper provides a general criterion (Propositions 2.1 and 2.2) clarifying when a function on I^2 satisfies the two-sided PL condition based on Hessian signs at zero-gradient points.
• Proposition 1.2 confirms local convergence under the two-sided PL condition in 2D for initial points in a small ball.