Online Convex Optimization with Sublinear Noisy Probes

• URL: http://arxiv.org/abs/2606.14640v1
• Date: Fri, 12 Jun 2026 17:05:08 GMT
• Title: Online Convex Optimization with Sublinear Noisy Probes
• Abstract summary: We study Online Convex Optimization (OCO) over a convex set $Ksubseteq mathbb Rd$.<n>We introduce a unified probing model that generalizes two recent lines of work: sublinear best-expert queries in the experts setting, and pairwise feedback available every round in OCO.<n>Our main result shows that even a sublinear and noisy probe budget can provably improve worst-case regret in the full feedback OCO regime.
• Score: 6.998574163422056
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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