The goal of decision-makers in contextual bandit problems is to minimize their loss by choosing actions that maximize their rewards. In this study, we explore linear models between losses and contexts, which allows us to analyze the problem using both optimistic and pessimistic perspectives. We aim to find a balance between these two approaches to achieve near-optimal performance in high-dimensional contextual bandits.
Contextual Bandits: A Primer
Contextual bandits are decision-making problems where the outcome of an action is dependent on both the chosen action and the context in which it was taken. In this study, we assume that each round, a decision-maker observes an independent and identically distributed (i.i.d.) context, draws an arm accordingly, and incurs a loss associated with the chosen arm. Our goal is to find the best arm selection strategy that minimizes the total loss over a sequence of rounds.
Linear Models: A Key Assumption
To analyze the linear contextual bandit problem, we assume that there are linear relationships between the losses and contexts. Specifically, we assume that each loss is a function of both the chosen arm and the context, with a parameter that captures the strength of this relationship. This assumption allows us to use tools from linear programming to analyze the problem.
Optimistic and Pessimistic Perspectives
In this study, we consider two different approaches to analyzing the linear contextual bandit problem: optimistic and pessimistic. The optimistic approach assumes that the best arm is always the one with the highest expected loss, while the pessimistic approach assumes that the best arm is always the one with the lowest expected loss. By combining these two approaches, we can find a balance between them and achieve near-optimal performance in high-dimensional contextual bandits.
Near-Optimal Performance
Our main result shows that by using a combination of optimistic and pessimistic advice, we can achieve near-optimal performance in high-dimensional contextual bandits. This means that our algorithm performs close to the best possible performance, while also being simple to implement and understand.
Implications and Future Work
Our study has implications for a wide range of applications, including personalized recommendations, sequential treatment allocation, and online advertising. In these settings, it is often important to balance optimism and pessimism in order to make the best decisions possible. Our work provides a framework for achieving this balance and improving performance.
For future work, we consider exploring more advanced models of contextual bandits, such as multi-armed bandits with multiple contexts or contextual bandits with non-linear losses. We also plan to study the applicability of our results to real-world scenarios, where the contexts may be complex and high-dimensional.
Conclusion
In conclusion, this study provides a comprehensive analysis of the linear contextual bandit problem using both optimistic and pessimistic perspectives. By combining these two approaches, we are able to achieve near-optimal performance in high-dimensional contextual bandits. Our results have implications for a wide range of applications and provide a valuable contribution to the field of decision-making under uncertainty.
