AI 中文总结
针对仅知次梯度且未知强凸参数的有界域在线凸优化，提出尺度不变算法，耦合投影自适应梯度下降与单一二次领导者，实现无对数因子的遗憾界及强凸改进。
AI 中文摘要
我们研究有界欧几里得域上的在线凸优化问题，其中学习者在每次预测时仅接收一个次梯度，且不知道时间范围、梯度界或损失函数的强凸参数。我们提出一种尺度不变算法，该算法通过一个正的财富过程将投影自适应梯度下降与一个锥约束二次领导者耦合。该算法保持$O(d)$状态，无需学习率网格或重启。对于直径至多为$D$的域，它保证遗憾至多为$\t{√}2D(\t{∑}_t\tt{‖}\t{widehat g}_t\t{‖}^2)^{1/2}+2D\t{max}_t\tt{‖}\t{widehat g}_t\t{‖}$，且不含对数因子。一个同时成立的二次距离界为未知强凸性提供了显式改进，包括由观测梯度尺度跳跃所控制的细化。对于每轮曲率变化的情况，该界取决于相对于事后选择的参考水平的总体不足量。证明使用了兼容裁剪的替代函数以及涉及二次最小值、对数财富和两个标量曲率统计量的势函数。所得到的强凸界在经典对数速率下并非对所有曲率与梯度之比均匀成立；文中给出了精确的参数依赖关系。贡献在于单一领导者构造及其分析，属于已有的通用和尺度无关在线学习文献。
英文摘要
We study online convex optimization on a bounded Euclidean domain when the learner receives only one subgradient at its prediction and knows neither the horizon, a gradient bound, nor the losses' strong-convexity parameters. We present a scale-invariant algorithm that couples projected adaptive gradient descent to one cone-constrained quadratic leader through a positive wealth process. The algorithm maintains $O(d)$ state, without a learning-rate grid or restarts. For a domain of diameter at most $D$, it guarantees regret at most $\sqrt{2}D(\sum_t\|\widehat g_t\|^2)^{1/2}+2D\max_t\|\widehat g_t\|$, with no logarithmic factor. A simultaneous quadratic-distance bound yields explicit improvements for unknown strong convexity, including a refinement governed by jumps in the observed gradient scale. For varying per-round curvature, the bound depends on the total shortfall below a reference level selected in hindsight. The proof uses a clipping-compatible surrogate and a potential involving the quadratic minimum, log wealth, and two scalar curvature statistics. The resulting strongly convex bound is not uniform at the classical logarithmic rate over all curvature-to-gradient ratios; the exact parameter dependence is stated. The contribution is a single-leader construction and its analysis, within an established literature on universal and scale-free online learning.
