We study Online Convex Optimization (OCO) over a convex set $K\subseteq \mathbb R^d$, where in each round $t$ the learner selects $x_t\in K$ and then observes a convex loss $f_t:K\to[0,1]$, with the goal of minimizing regret to the best fixed decision in hindsight. We introduce a unified probing model that generalizes two recent lines of work: sublinear best-expert queries in the experts setting, and pairwise (comparison-based) feedback available every round in OCO. In our framework, the learner has a budget of $k\le T$ pairwise probes; on a probed round it may query two points and learn which one has smaller loss. Our main result shows that even a sublinear and noisy probe budget can provably improve worst-case regret in the full feedback OCO regime. With $k$ $δ$-noisy pairwise probes, we obtain: $ \text{Reg}_T \le O\left(\min\left\{\sqrt{dT\ln T},\; \frac{dT\ln T}{k|1-2δ|}\right\}\right) $, which is tight (up to logarithmic factors in $T$) across $T$, $k$ and $δ$. Specifically regarding the noise parameter $δ\in [0,1]$, the regret guarantee smoothly degrades as the oracle response approaches a coin flip, i.e., $δ$ is close to $\frac{1}{2}$. When applying the same techniques to a finite $K$ for the prediction with $d$ experts setting, the resulting rates are instead completely tight in all parameters, including $d$. Our analysis gives a streamlined treatment of pairwise probing in OCO by quantifying the benefit of probing via a variance reduction effect, combined with a second-order (variance-based) analysis of Continuous Exponential Weights.

翻译：我们研究凸集 $K\subseteq \mathbb R^d$ 上的在线凸优化（OCO），其中每轮 $t$ 中学习器选择 $x_t\in K$ 并观测到凸损失函数 $f_t:K\to[0,1]$，目标是最小化相对于事后最优固定决策的遗憾值。我们提出一个统一探测模型，该模型概括了近期两个研究方向：专家设置中亚线性最优专家查询，以及 OCO 中每轮可用的成对（基于比较）反馈。在此框架中，学习器拥有至多 $k\le T$ 次成对探测的预算；在被探测的轮次中，它可以查询两个点并获知哪个点的损失较小。我们的主要结果表明，即使是在完全反馈 OCO 机制下，亚线性且含噪声的探测预算也能确保改进最坏情况遗憾值。利用 $k$ 次 $\delta$ 噪声成对探测，我们得到：$ \text{Reg}_T \le O\left(\min\left\{\sqrt{dT\ln T},\; \frac{dT\ln T}{k|1-2\delta|}\right\}\right) $，该界在 $T$、$k$ 和 $\delta$ 上（忽略 $T$ 的对数因子）是紧的。具体而言，对于噪声参数 $\delta\in [0,1]$，当预言机响应趋近于抛硬币（即 $\delta$ 接近 $\frac{1}{2}$）时，遗憾保证平滑退化。当将相同技术应用于具有 $d$ 个专家的有限集 $K$ 上的预测问题时，所得到的速率在所有参数（包括 $d$）上完全紧。我们的分析通过量化探测带来的方差缩减效应，并结合连续指数加权法的二阶（基于方差）分析，为 OCO 中的成对探测提供了简洁的处理方法。