Restless Linear Bandits (2405.10817v1)

Abstract: A more general formulation of the linear bandit problem is considered to allow for dependencies over time. Specifically, it is assumed that there exists an unknown $\mathbb{R}^d$-valued stationary $\varphi$-mixing sequence of parameters $(\theta_t,~t \in \mathbb{N})$ which gives rise to pay-offs. This instance of the problem can be viewed as a generalization of both the classical linear bandits with iid noise, and the finite-armed restless bandits. In light of the well-known computational hardness of optimal policies for restless bandits, an approximation is proposed whose error is shown to be controlled by the $\varphi$-dependence between consecutive $\theta_t$. An optimistic algorithm, called LinMix-UCB, is proposed for the case where $\theta_t$ has an exponential mixing rate. The proposed algorithm is shown to incur a sub-linear regret of $\mathcal{O}\left(\sqrt{d n\mathrm{polylog}(n) }\right)$ with respect to an oracle that always plays a multiple of $\mathbb{E}\theta_t$. The main challenge in this setting is to ensure that the exploration-exploitation strategy is robust against long-range dependencies. The proposed method relies on Berbee's coupling lemma to carefully select near-independent samples and construct confidence ellipsoids around empirical estimates of $\mathbb{E}\theta_t$.

• The paper introduces LinMix-UCB, a novel algorithm that handles time-dependent parameters with exponential mixing to achieve sub-linear regret bounds.
• It leverages confidence ellipsoids constructed via regularized least-squares and Berbee’s coupling lemma to effectively balance exploration and exploitation.
• The approach is applicable in dynamic scenarios like online advertising and dynamic pricing, offering scalable, adaptive decision-making strategies.

Exploring Time-Dependent Linear Bandits: LinMix-UCB

In recent work, researchers explored a new way of approaching the linear bandit problem. This isn't your classic linear bandit; instead, the parameters influencing the pay-offs are allowed to have dependencies over time. This subtle change opens up new avenues for improving decision-making strategies in various applications such as online advertising, recommendation systems, and dynamic pricing.

Overview

Traditionally, linear bandit models assume that the noise impacting the pay-off is independent and identically distributed (iid). However, in practice, this assumption often falls flat. Dependencies over time are common in real data, making it critical to adapt our models to reflect this reality.

In this new approach, the parameters $(\theta_t,~ t \in N)$ are assumed to form an $R^d$-valued stationary sequence with $\varphi$-mixing properties. This means that while $\theta_t$ retains some of the familiar properties, it also incorporates dependencies over time.

Key Challenges and Contributions

Handling Dependencies

The main hurdle with these time-dependent parameters is to manage the exploration-exploitation trade-off effectively despite the long-range dependencies. The researchers tackled this using a novel algorithm called LinMix-UCB. Specifically designed for settings where $\theta_t$ has an exponential mixing rate, the algorithm promises a sub-linear regret of $O(\sqrt{d n \text{polylog}(n)})$ with respect to an oracle that always plays a multiple of $\theta_t$.

Theoretical Underpinnings

• Approximation Strategy: The paper outlines that exact computation of optimal policies is computationally hard for restless bandits. Hence, the authors propose an approximation strategy whose error depends on the $\varphi$-dependence between consecutive $\theta_t$.
• Confidence Ellipsoids: Leveraging Berbee's coupling lemma, the researchers carefully select near-independent samples to construct confidence ellipsoids around empirical estimates of $\theta_t$. This enables robust predictions and decision-making.
• Optimization in Play: The LinMix-UCB algorithm follows the principle of Optimism in the Face of Uncertainty (OFU). By updating confidence ellipsoids every few steps, it ensures that the exploration-exploitation balance remains optimally aligned despite long-range dependencies.

Detailed Breakdown

Algorithm Mechanics

The LinMix-UCB algorithm is designed to ensure robust performance in an environment with time-dependent linear dynamics. Here's how it works:

1. Initialization and Segmentation: The pay-offs are collected at specific intervals, allowing the system to gather enough data points for effective updating.
2. Confidence Ellipsoids: For each segment, the algorithm calculates empirical estimates of $\theta^*$ using a regularized least-squares method and constructs confidence ellipsoids to account for uncertainty.
3. Action Selection: At each time step, the action is chosen based on the current confidence ellipsoid, ensuring that the selected action maximizes the expected pay-off.

Numerical Results

Here are some strong numerical results highlighted in the paper:

• Regret Bounds: The LinMix-UCB algorithm achieves a sub-linear regret bound of $O(\sqrt{d n \text{polylog}(n)})$ for the finite horizon.
• Exponential Mixing: Leveraging an exponential mixing rate, the algorithm shows efficiency in handling temporal dependencies.

Practical and Theoretical Implications

The implications of this research stretch both into practice and theory:

• Practical Deployment: In real-world applications like dynamic pricing or online ad placements, where parameters change over time, LinMix-UCB offers a scalable and efficient way to adapt decision-making strategies.
• Theoretical Insights: The work adds depth to the paper of bandit problems in non-iid settings, providing a framework to understand and quantify the impact of temporal dependencies.

Future Directions

While the algorithm has shown promising results, there are several areas ripe for further exploration:

• Adaptive Parameter Estimation: One key challenge is the requirement to know the mixing rate parameters a priori. Future research could focus on methodologies to estimate these parameters on the fly.
• Complex Bandit Settings: Extending the LinMix-UCB framework to other types of bandit problems, such as contextual bandits with non-iid contexts, could open up new frontiers in the field.

Overall, this work provides a significant step forward in addressing the limitations of classical linear bandit models and opens up new possibilities for enhancing sequential decision-making strategies in complex, real-world environments.