发表机构
The University of Tokyo; RIKEN（东京大学; 理化学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一族几何加速Bregman近端梯度算法，无需三角形缩放等额外条件，通过局部回溯自适应步长，实现凸目标O(k^-2)、相对强凸线性、非凸O(k^-1)收敛，并在多个数值实验中验证了加速效果。
AI 中文摘要
我们研究了在相对光滑性条件下，针对凸、相对强凸和非凸目标函数的Bregman近端梯度（BPG）算法。现有的针对凸目标的加速BPG算法通常需要对Bregman散度施加额外假设，尤其是三角形缩放条件，这可能导致收敛速度较慢。我们提出了一族几何加速BPG算法，该算法在近端梯度和镜像空间更新中利用Bregman几何，而不施加诸如三角形缩放条件之类的额外几何条件。我们的方法通过局部回溯和可计算的接受准则来调整步长和镜像空间更新，无需输入全局相对光滑常数。我们根据迭代过程中接受的参数推导出收敛界。当几何加速参数保持一致有界时，这些界对凸目标产生$Ø(k^{-2})$的收敛速率，对相对强凸目标产生线性速率。对于非凸目标，我们在不要求下Bregman界或全定义域Bregman散度的情况下，建立了关于平稳性度量的$Ø(k^{-1})$速率。在反问题、熵正则化最小二乘、D-最优设计和非负矩阵分解上的数值实验表明，与已建立的Bregman基线相比，所提方法在实际中收敛更快。
英文摘要
We study Bregman proximal gradient (BPG) algorithms under relative smoothness for convex, relatively strongly convex, and nonconvex objectives. Existing accelerated BPG algorithms for convex objectives typically require additional assumptions on Bregman divergences, most notably triangle-scaling conditions, which can lead to slower convergence rates. We propose a family of geometry-accelerated BPG algorithms that exploit Bregman geometry in both proximal-gradient and mirror-space updates, without imposing additional geometric conditions such as triangle-scaling conditions. Our methods adapt the stepsizes and mirror-space updates through local backtracking and computable acceptance criteria, without a global relative-smoothness constant as input. We derive convergence bounds in terms of the parameters accepted during the iterations. These bounds yield an $Ø(k^{-2})$ rate for convex objectives and a linear rate for relatively strongly convex objectives when the geometry-acceleration parameters remain uniformly bounded. For nonconvex objectives, we establish an $Ø(k^{-1})$ rate for a stationarity measure without requiring a lower Bregman bound or a full-domain Bregman divergence. Numerical experiments on inverse problems, entropy-regularized least squares, D-optimal design, and nonnegative matrix factorization demonstrate faster practical convergence than established Bregman baselines.
Comments31pages. 6 figures
