A Low-Rank Symplectic Gradient Adjustment Method for Computing Nash Equilibria

Abstract

This work presents a theoretical and numerical investigation of the symplectic gradient adjustment (SGA) method and of a low-rank SGA (LRSGA) method for efficiently solving revviolet optimization problems arising from two-player Nash games. The SGA method outperforms the gradient method by including second-order mixed derivatives computed at each iterate, which requires considerably larger computational effort. For this reason, an LRSGA method is proposed where the approximation to second-order mixed derivatives is obtained by rank-one updates. The theoretical analysis presented in this work focuses on novel convergence estimates for the SGA and LRSGA methods, including parameter bounds. The numerical experiments complement the theory by studying the behavior of LRSGA on explicit deterministic games with known equilibria and by evaluating its computational advantage over exact SGA on a CLIP-inspired neural-network training task, where LRSGA achieves comparable loss values lower CPU time than SGA with explicitly assembled mixed-derivative blocks.