凸优化的维度自适应精度证书
Dimension-Adaptive Accuracy Certificates for Convex Optimization
- HSE University（高等经济大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
针对凸优化中保留完整梯度成本高的问题，提出维度自适应精度证书，结合二次正则化的精确校正项，在核SVM等实验中减少了证书通信量和停止时间。
AI中文摘要：
许多凸优化方法使用仿射下界模型来证明候选解接近最优。当有效方向占据一个初始未知的小子空间时，保留完整梯度的成本很高。我们开发了精度证书，该证书存储投影仿射次梯度以及被丢弃分量的边界。当子空间、查询和证书权重被自适应选择时，证书仍然有效。对于二次正则化，我们推导了一个精确校正项，用于说明每个被丢弃分量的记录时间，以及一个用于选择证书权重的紧凑问题。我们还刻画了当环境方向未被使用时，压缩记录所支持的最强下界。该证书可作为提供有效仿射次梯度的方法的停止准则。我们还分析了一种自适应割平面构造，其神谕边界取决于已发现的维度。在核SVM和ν-SVR问题上的实验表明，在模拟带宽约束下，证书通信量和停止时间均减少。
英文摘要：
Many convex optimization methods use affine lower models to certify that a candidate solution is near optimal. Keeping full gradients can be costly when the informative directions occupy a small, initially unknown subspace. We develop accuracy certificates that store projected affine minorants together with bounds on the discarded components. The certificates remain valid as the subspace, queries, and certificate weights are chosen adaptively. For quadratic regularization, we derive an exact correction that accounts for when each discarded component was recorded, and a compact problem for selecting certificate weights. We also characterize the strongest lower bound supported by the compressed records when an ambient direction remains unused. The certificates can be used as stopping criteria for methods that provide valid affine minorants. We also analyze an adaptive cutting-plane construction whose oracle bound depends on the discovered dimension. Experiments on kernel SVM and $ν$-SVR problems show reduced certificate communication and stopping times under an emulated bandwidth constraint.
