Adaptive Discretization of Bregman Gradient Flows: A Dynamical Systems Perspective on Mirror Descent Convergence Geometry

Abstract

The limited convergence efficiency of classical first-order methods in high-dimensional, non-Euclidean geometries is addressed by analyzing the continuous-time limit of the Bregman gradient method and its connection to Mirror Descent. Focusing on convex optimization with emphasis on ℓ1-regularized sparse regression and convex classification, the Bregman gradient flow is derived via a variational formulation, yielding the governing ODEs. A Lyapunov energy is constructed to characterize decay; its time derivative induces a principled, Lyapunov-guided adaptive step-size rule. During discretization, a stabilization term is introduced to ensure that the numerical scheme tracks the continuous flow under curvature-dependent mirror geometries.The proposed continuum-guided Mirror Descent (CG-MD) adapts step size to local geometry and demonstrates improved efficiency and stability. On synthetic sparse-regression benchmarks, CG-MD reduces mean error by ≈30% relative to standard Mirror Descent. On real-world sparse regression, CG-MD reaches an error threshold of 0.01 in 140 iterations versus >200 for the baseline. In convex classification, CG-MD matches the speed of accelerated Mirror Descent while achieving lower terminal error. A sensitivity study across common mirror maps (entropy, Tsallis, log-barrier) and step-size policies indicates consistent gains for CG-MD. Assumptions and limits (convexity, smoothness, discretization overhead) are detailed, and potential extensions to stochastic and nonconvex settings are outlined.