Second Order Bounds for Contextual Bandits with Function Approximation (2409.16197v3)

Abstract: Many works have developed no-regret algorithms for contextual bandits with function approximation, where the mean reward function over context-action pairs belongs to a function class. Although there are many approaches to this problem, one that has gained in importance is the use of algorithms based on the optimism principle such as optimistic least squares. It can be shown the regret of this algorithm scales as square root of the product of the eluder dimension (a statistical measure of the complexity of the function class), the logarithm of the function class size and the time horizon. Unfortunately, even if the variance of the measurement noise of the rewards at each time is changing and is very small, the regret of the optimistic least squares algorithm scales with square root of the time horizon. In this work we are the first to develop algorithms that satisfy regret bounds of scaling not with the square root of the time horizon, but the square root of the sum of the measurement variances in the setting of contextual bandits with function approximation when the variances are unknown. These bounds generalize existing techniques for deriving second order bounds in contextual linear problems.

• The paper develops variance-aware algorithms that integrate second-order bounds using filtered least squares to achieve sublinear regret.
• It introduces adaptive methods for both known and unknown variance settings, ensuring sharper regret bounds in low-noise scenarios.
• The analysis leverages eluder dimension theory to enhance theoretical insights and practical applications in decision-making and reinforcement learning.

Second Order Bounds for Contextual Bandits with Function Approximation

Overview

The paper "Second Order Bounds for Contextual Bandits with Function Approximation" by Aldo Pacchiano addresses significant advancements in creating algorithms for contextual bandit problems with function approximation under a mean reward realizability assumption. The work aims to develop efficient algorithms with sublinear regret bounds that also incorporate variance-awareness.

Problem Context

Contextual bandits are a crucial model in decision-making processes, where a learning agent makes sequential decisions based on the context received in each round. Traditional bandit algorithms tend to overlook the richness of contextual information, which is often critical in fields such as robotics, medical trials, and personalized recommendation systems.

Contributions

The primary contributions of the paper are twofold:

1. Variance-Aware Algorithms with Known Variance:
   • Algorithm: The introduction of a second-order optimistic least squares algorithm that accounts for the variance in the reward. This algorithm computes filtered least squares estimates and constructs confidence sets based on those estimates.
   • Regret Bound: The algorithm achieves a regret bound of the order $$\mathcal{O}\left(\sigma \sqrt{(F, B/T)T \log(T|F|/\delta)} + B (F, B/T) \log(T)\log(T|F|/\delta)\right)$$.
   • Significance: The bounds are tighter compared to existing ones, providing sharp regret guarantees that scale with the variance of the reward noise.
2. Variance-Aware Algorithms with Unknown Variance:
   • Algorithm: Extending the previous approach, the author introduces variance estimation procedures to handle unknown variances. The cumulative variance is estimated adaptively, and the confidence sets are adjusted accordingly.
   • Regret Bound: This enhanced approach obtains a regret bound of the order $$\mathcal{O}\left( \sqrt{\left(\sum_{t=1}^T \sigma_t^2\right) \log(T)\log\left(T|F|/\delta\right)} + B (F, B/T) \log(T)\log(T|F|/\delta)\right)$$.
   • Implications: Successfully tackles the challenge of unknown variances, a realistic assumption for many practical settings.

Theoretical Implications

Eluder Dimension

A critical component of the theoretical analysis is the eluder dimension, a measure of the complexity of the function class $F$. The paper rigorously extends eluder dimension-based analysis to derive second-order bounds, which improves the understandability and practical utility of these bounds in varied contexts.

Variance Dependent Bounds

The variance-aware approach ensures that the regret bounds are not overestimated when the noise in observations is low, resulting in practical performance improvements. This addresses a key shortcoming in traditional algorithms where regret scales with $\sqrt{T}$, irrespective of the actual variance.

Speculations and Future Directions

Impact on Reinforcement Learning

While this paper focuses on bandits with function approximation, the techniques developed could extend to reinforcement learning (RL). In RL, where the state-action space can be vast and rewards noisy, variance-aware methods might yield even more substantial improvements.

Refined Analysis for Sharper Bounds

Future work might refine the existing analysis to improve on the $\mathcal{O}(\cdot)$ factors in the bounds, especially reducing the dependence on the log factors. It is suggested that a more in-depth examination of the algorithms could yield sharper bounds, essential for theoretical advancements and practical efficiency.

Practical Implementations

Despite the theoretical nature of this work, the algorithms can inspire practical implementations in industries like healthcare and personalized recommendation systems, where context and function approximation are vital. Real-world testing and evaluation of these algorithms could provide insights into their actual performance and further refine their design.

Conclusion

This paper represents a significant step in advancing the field of contextual bandits with function approximation. By introducing variance-aware second-order bounds and addressing both known and unknown variance scenarios, it opens new avenues for research and practical application in adaptive learning systems. The blend of rich theoretical insights and potential for practical implementation makes this work a valuable contribution to the field.