2024/05/06 by Jonas Katona, Katona, Jonas, Xiuyuan Wang +3 · 2 citations
Engineering · #37M15 #65P10 #Computer Science and Game Theory (cs.GT) #Dynamical Systems (math.DS) #F.2.1 #FOS: Computer and information sciences #FOS: Mathematics #G.1.0 #G.1.7 #Laser and Thermal Forming Techniques #Machine Learning (cs.LG) #Numerical Analysis (math.NA) #Optimization and Control (math.OC) #Vibration and Dynamic Analysis
paper · pdf · doi:10.48550/arxiv.2405.03472
openalex publication_date 2024/05/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by understanding the behavior of the Alternating Mirror Descent (AMD) algorithm for bilinear zero-sum games, we study the discretization of continuous-time Hamiltonian flow via the symplectic Euler method. We provide a framework for analysis using results from Hamiltonian dynamics, Lie algebra, and symplectic numerical integrators, with an emphasis on the existence and properties of a conserved quantity, the modified Hamiltonian (MH), for the symplectic Euler method. We compute the MH in closed-form when the original Hamiltonian is a quadratic function, and show that it generally differs from the other conserved quantity known previously in that case. We derive new error bounds on the MH when truncated at orders in the stepsize in terms of the number of iterations, K, and use these bounds to show an improved O(K1/5) total regret bound and an O(K-4/5) duality gap of the average iterates for AMD. Finally, we propose a conjecture which, if true, would imply that the total regret for AMD scales as O(Kε) and the duality gap of the average iterates as O(K-1+ε) for any ε>0, and we can take ε=0 upon certain convergence conditions for the MH.