Second Order Bounds for Contextual Bandits with Function Approximation

Aldo Pacchiano

Keywords: theory, contextual bandits, variance aware, second order, bandits

TL;DR: We derive second order bounds for contextual bandits with adversarial contexts and function approximation

Abstract: Many works have developed no-regret algorithms for contextual bandits with function approximation, where the mean rewards over context-action pairs belong to a function class $\mathcal{F}$. Although there are many approaches to this problem, algorithms based on the principle of optimism, such as optimistic least squares have gained in importance. It can be shown the regret of this algorithm scales as $\widetilde{\mathcal{O}}\left(\sqrt{d_{\mathrm{eluder}}(\mathcal{F}) \log(\mathcal{F}) T }\right)$ where $d_{\mathrm{eluder}}(\mathcal{F})$ is a statistical measure of the complexity of the function class $\mathcal{F}$ known as eluder dimension. Unfortunately, even if the variance of the measurement noise of the rewards at time $t$ equals $\sigma_t^2$ and these are close to zero, the optimistic least squares algorithm’s regret scales with $\sqrt{T}$. In this work we are the first to develop algorithms that satisfy regret bounds for contextual bandits with function approximation of the form $\widetilde{\mathcal{O}}\left( \sigma \sqrt{\log(\mathcal{F})d_{\mathrm{eluder}}(\mathcal{F}) T } + d_{\mathrm{eluder}}(\mathcal{F}) \cdot \log(|\mathcal{F}|)\right) $ when the variances are unknown and satisfy $\sigma_t^2 = \sigma$ for all $t$ and $\widetilde{\mathcal{O}}\left( d_{\mathrm{eluder}}(\mathcal{F})\sqrt{\log(\mathcal{F})\sum_{t=1}^T \sigma_t^2 } + d_{\mathrm{eluder}}(\mathcal{F}) \cdot \log(|\mathcal{F}|)\right) $ when the variances change every time-step. These bounds generalize existing techniques for deriving second order bounds in contextual linear problems.

Primary Area: learning theory

Code Of Ethics: I acknowledge that I and all co-authors of this work have read and commit to adhering to the ICLR Code of Ethics.

Submission Guidelines: I certify that this submission complies with the submission instructions as described on https://iclr.cc/Conferences/2025/AuthorGuide.

Anonymous Url: I certify that there is no URL (e.g., github page) that could be used to find authors’ identity.

No Acknowledgement Section: I certify that there is no acknowledgement section in this submission for double blind review.

Submission Number: 10700