A Symplectic Analysis of Alternating Mirror Descent

TL;DR

This paper analyzes the Alternating Mirror Descent algorithm for bilinear zero-sum games using symplectic Euler methods, revealing new conserved quantities and deriving improved regret and duality gap bounds.

Contribution

It introduces a symplectic analysis framework for AMD, computes the modified Hamiltonian explicitly, and establishes tighter error bounds and convergence guarantees.

Findings

Derived closed-form modified Hamiltonian for quadratic Hamiltonians.

Established an improved total regret bound of O(K^{1/5}).

Achieved an O(K^{-4/5}) duality gap for average iterates.

Abstract

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 $\mathcal{O}(K^{1/5})$ total regret bound and an $\mathcal{O}(K^{-4/5})$ 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 $\mathcal{O}\left(K^{\varepsilon}\right)$ and the duality gap of the average iterates as $\mathcal{O}\left(K^{-1+\varepsilon}\right)$ for any $\varepsilon>0$, and we can take $\varepsilon=0$ upon certain convergence conditions for the MH.