Research map / Optimization / Nonconvex and smooth optimization
Local Convergence of Gradient Methods for Min-Max Games: Partial Curvature Generically Suffices
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Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Allow nonzero partial curvature S, with interaction eigenvectors in general position relative to ker S | Gradient methods for smooth two-player zero-sum games · Local convergence to a local Nash equilibrium | local convergence | Helped |
| Use a small symmetric component αS when interaction dominates | Gradient Flow for min-max games · Games with Jacobian Mα = A + αS | normalized local convergence rate: 1/2 α (min_{1≤j≤n} u_jᵀQ u_j + v_jᵀR v_j) + O(α³) | Mixed |
| Use interaction directions that are uniformly spread relative to a well-spread curvature spectrum | Gradient Flow for min-max games · S ≪ A; well-spread spectrum | local convergence rate: trace(S)/n | Helped |
| Optimize over support locations as well as mixed-strategy weights | Conic particle methods · Continuous games with sparse mixed Nash equilibria | local convergence rate | Helped |
| Use EG rather than Mirror Prox | Conic particle methods · Continuous games with sparse mixed Nash equilibria | convergence-rate scaling with η: EG: η; MP: η² | Helped |
| Reduce the support-location step-size through γ | Conic particle methods · Continuous games with sparse mixed Nash equilibria | local convergence rate: γ² | Hurt |
| Use simultaneous GDA instead of Extra-Gradient when interaction dominates | Gradient methods for min-max games · Games with small symmetric part | local convergence rate: Sim-GDA: +η²σ_j²; EG: -η²σ_j² | Mixed |
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