AI 中文总结
研究在\(f\)光滑但求值昂贵、\(g\)非光滑的\(\min_x f(x)+g(x)\)问题中，提出曲率循环DRS方法，通过传输残差、重用模型等操作，实现全局线性收敛与局部超线性减少，实验表明该方法能大幅减少昂贵梯度调用。
AI 中文摘要
我们考虑\(\min_x f(x)+g(x)\)，其中\(f\)光滑但其函数值和梯度计算代价高昂，而\(g\)非光滑且近端映射计算代价低。道格拉斯 - 拉赫福德分裂（DRS）需要一系列光滑近端求解，这些求解中心不同但曲率相同。我们引入曲率循环DRS（CR - DRS），该方法将旧的精确残差传输到新的近端中心，重用拟牛顿模型，并在评估第一个新梯度之前移动旧的近端状态。我们给出了直接和逆BFGS以及保护对称秩一的实现方式。一个经过认证的变体仅在通过下降和残差测试时才接受传输的拟牛顿步，否则使用固定数量的安全梯度步。在强凸性和平滑性条件下，一个小增益条件给出全局线性收敛，每次外层迭代有固定数量的新梯度。一个传输的丹尼斯 - 莫雷条件给出新近端误差的局部超线性减少。我们表明该条件并非自动成立，推导了全模型和活动子空间的充分条件，并将活动子空间与非光滑近端映射诱导的局部DRS动力学相关联。在\(\ell_1\)和总变差正则化问题上的实验表明，相对于重新启动和仅曲率拟牛顿近端求解，昂贵梯度调用大幅减少。对于直接近端\(\ell_1\)模型，我们还与FISTA进行了比较。
英文摘要
We consider \[ \min_x f(x)+g(x), \] where $f$ is smooth but its value and gradient are expensive, while $g$ is nonsmooth and has a cheap proximal map. Douglas--Rachford splitting (DRS) then requires a sequence of smooth proximal solves. These solves have different centers but share the same nonlinear curvature. We introduce curvature-recycling DRS (CR-DRS). The method transports an old exact residual to the new proximal center, reuses a quasi-Newton model, and moves the old proximal state before the first new gradient is evaluated. We give paired direct and inverse BFGS and safeguarded symmetric-rank-one realizations. A certified variant accepts a transported quasi-Newton step only when it passes descent and residual tests; otherwise it uses a fixed number of safe gradient steps. Under strong convexity and smoothness, a small-gain condition gives global linear convergence with a constant number of new gradients per outer iteration. A transported Dennis--Moré condition gives local superlinear reduction of the new proximal error. We show that this condition is not automatic, derive full-model and active-subspace sufficient conditions, and relate the active subspace to the local DRS dynamics induced by the nonsmooth proximal map. Experiments on $\ell_1$- and total-variation-regularized problems show large reductions in expensive-gradient calls relative to restarted and curvature-only quasi-Newton proximal solves. For directly proximal $\ell_1$ models, we also compare with FISTA.
Comments36 pages, 2 figures
