ABrA-GD: Adaptive Bregman Accelerated Gradient Descent for Relatively Smooth and (Strongly-)Convex Optimization

We propose ABrA-GD (Adaptive Bregman Accelerated Gradient Descent), an adaptive algorithm for relatively smooth convex optimization. The algorithm is derived from a computable primal–dual certificate that guarantees progress by comparing the objective value with a lower bound on the minimum of a regularized objective. This certificate enables adaptation to both smoothness and geometry, using only the relative strong convexity constant as a problem-dependent input. We introduce the dual Bregman Length Distortion Factor (BLDF), which measures how dual Bregman lengths change under an anchor shift or rescaling. Under bounded local dual BLDF, ABrA-GD achieves an accelerated $O(1/k^2)$ rate for convex objectives and an accelerated linear rate for relatively strongly convex objectives.