针对光滑凸集的线性收敛无投影算法
A Linearly Convergent Projection-Free Algorithm for Smooth Convex Sets
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中文总结 AI 辅助
针对光滑凸集上光滑强凸函数的极小化问题，提出用梯度计算和支撑切线计算替代昂贵的投影，实现线性收敛，且切线可通过\tilde{O}(d)个成员查询近似，突破了此前仅多面体等集能实现无投影线性收敛的局限。
中文摘要 AI 辅助
我们考虑在凸集上极小化一个光滑强凸函数。在该场景下，投影梯度下降算法已知会线性收敛，但每次迭代都需要向可行集做投影，这在计算上可能十分昂贵。我们证明，当可行集是光滑的，投影可被替换为每次迭代的一次梯度计算和一次支撑切线计算，同时保持线性收敛性。此外，所需的切线可使用\tilde{O}(d)个成员查询以足够精度近似，其中d是环境维度。此前，无投影线性收敛仅针对多面体集或既光滑又强凸的集已知存在。
英文摘要
We consider minimizing a smooth, strongly convex function over a convex set. Projected gradient descent is known to converge linearly in this setting, but each iteration requires a projection onto the feasible set, which may be computationally expensive. We show that when the feasible set is smooth, projection can be replaced by one gradient computation and a single supporting-tangent computation per iteration, while preserving linear convergence. Moreover, the required tangent can be approximated to sufficient accuracy using $\widetilde O(d)$ membership-oracle queries, where $d$ is the ambient dimension. Previously, projection-free linear convergence was known only for polyhedral sets or for sets that are both smooth and strongly convex.
