发表机构
Samsung SAIL Montreal（三星SAIL蒙特利尔）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出ABrA-GD算法，利用可计算的原始-对偶证书自适应相对光滑凸优化，仅需相对强凸常数，实现对凸目标加速$O(1/k^2)$收敛，对强凸目标线性收敛。
AI 中文摘要
我们提出了ABrA-GD（自适应Bregman加速梯度下降），一种用于相对光滑凸优化的自适应算法。该算法源自一个可计算的原始-对偶证书，通过将目标值与正则化目标最小值下界进行比较来保证进展。该证书使得算法能够自适应于光滑性和几何结构，仅需相对强凸常数作为问题相关输入。我们引入了对偶Bregman长度畸变因子（BLDF），用于衡量在对偶Bregman长度在锚点移动或缩放下的变化。在局部有界对偶BLDF条件下，ABrA-GD对凸目标实现加速的$O(1/k^2)$收敛速率，对相对强凸目标实现加速线性收敛速率。
英文摘要
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.
