A Reduction Algorithm for Markovian Contextual Linear Bandits
arXiv:2603.12530v1 Announce Type: new Abstract: Recent work shows that when contexts are drawn i.i.d., linear contextual bandits can be reduced to single-context linear bandits. This ``contexts are cheap" perspective is highly advantageous, as it allows for sharper finite-time analyses and leverages mature techniques from the linear bandit literature, such as those for misspecification and adversarial corruption. Motivated by applications with temporally correlated availability, we extend this perspective to Markovian contextual linear bandits, where the action set evolves via an exogenous Markov chain. Our main contribution is a reduction that applies under uniform geometric ergodicity. We construct a stationary surrogate action set to solve the problem using a standard linear bandit oracle, employing a delayed-update scheme to control the bias induced by the nonstationary conditional context distributions. We further provide a phased algorithm for unknown transition distributions th
arXiv:2603.12530v1 Announce Type: new Abstract: Recent work shows that when contexts are drawn i.i.d., linear contextual bandits can be reduced to single-context linear bandits. This ``contexts are cheap" perspective is highly advantageous, as it allows for sharper finite-time analyses and leverages mature techniques from the linear bandit literature, such as those for misspecification and adversarial corruption. Motivated by applications with temporally correlated availability, we extend this perspective to Markovian contextual linear bandits, where the action set evolves via an exogenous Markov chain. Our main contribution is a reduction that applies under uniform geometric ergodicity. We construct a stationary surrogate action set to solve the problem using a standard linear bandit oracle, employing a delayed-update scheme to control the bias induced by the nonstationary conditional context distributions. We further provide a phased algorithm for unknown transition distributions that learns the surrogate mapping online. In both settings, we obtain a high-probability worst-case regret bound matching that of the underlying linear bandit oracle, with only lower-order dependence on the mixing time.
Executive Summary
This article presents a reduction algorithm for Markovian contextual linear bandits, extending the 'contexts are cheap' perspective to temporally correlated availability applications. The proposed algorithm constructs a stationary surrogate action set to solve the problem using a standard linear bandit oracle, employing a delayed-update scheme to control bias. The method provides a high-probability worst-case regret bound, with lower-order dependence on the mixing time. This innovation enables the application of mature linear bandit techniques to Markovian contextual linear bandits, leading to sharper finite-time analyses and more robust decision-making. The algorithm's phased version learns the surrogate mapping online for unknown transition distributions, demonstrating adaptability and flexibility. The reduction's uniform geometric ergodicity assumption facilitates analytical tractability and scalability.
Key Points
- ▸ Extension of the 'contexts are cheap' perspective to Markovian contextual linear bandits
- ▸ Construction of a stationary surrogate action set for linear bandit oracle solution
- ▸ Delayed-update scheme for bias control in nonstationary conditional context distributions
- ▸ Phased algorithm for unknown transition distributions with online surrogate mapping learning
Merits
Strength
The proposed algorithm provides a high-probability worst-case regret bound, matching that of the underlying linear bandit oracle, with lower-order dependence on the mixing time.
Strength
The reduction's assumption of uniform geometric ergodicity facilitates analytical tractability and scalability.
Strength
The phased algorithm's adaptability and flexibility enable learning the surrogate mapping online for unknown transition distributions.
Strength
The method leverages mature linear bandit techniques, enabling sharper finite-time analyses and more robust decision-making.
Demerits
Limitation
The algorithm's performance relies on the assumption of uniform geometric ergodicity, which might not be universally applicable.
Limitation
The delayed-update scheme may introduce additional computational complexity and latency in practice.
Limitation
The phased algorithm's adaptability may come at the cost of increased algorithmic overhead and memory requirements.
Expert Commentary
The article makes a significant contribution to the field of contextual bandits by extending the 'contexts are cheap' perspective to Markovian contextual linear bandits. The proposed algorithm's ability to learn the surrogate mapping online for unknown transition distributions demonstrates adaptability and flexibility. However, the assumption of uniform geometric ergodicity may limit the algorithm's applicability in practice. Furthermore, the delayed-update scheme and phased algorithm's overhead may introduce additional computational complexity and latency. Nevertheless, the article's findings have far-reaching implications for the development of more robust and adaptable decision-making policies in the presence of temporal correlations and nonstationarity.
Recommendations
- ✓ Recommendation 1: Future research should focus on relaxing the assumption of uniform geometric ergodicity to broaden the algorithm's applicability.
- ✓ Recommendation 2: The authors should investigate the impact of the delayed-update scheme and phased algorithm's overhead on computational complexity and latency in practice.