Bandits with heavy tail (1209.1727v1)

Abstract: The stochastic multi-armed bandit problem is well understood when the reward distributions are sub-Gaussian. In this paper we examine the bandit problem under the weaker assumption that the distributions have moments of order 1+\epsilon, for some $\epsilon \in (0,1]$. Surprisingly, moments of order 2 (i.e., finite variance) are sufficient to obtain regret bounds of the same order as under sub-Gaussian reward distributions. In order to achieve such regret, we define sampling strategies based on refined estimators of the mean such as the truncated empirical mean, Catoni's M-estimator, and the median-of-means estimator. We also derive matching lower bounds that also show that the best achievable regret deteriorates when \epsilon <1.

• The paper introduces a novel framework for multi-armed bandits under heavy-tailed rewards, challenging the traditional sub-Gaussian assumption.
• It derives regret bounds using robust estimators such as truncated empirical mean, Catoni’s M-estimator, and the median-of-means to mitigate heavy-tail effects.
• Optimal lower bound analysis reveals that logarithmic regret is achievable even with minimal moment conditions, guiding robust decision-making under uncertainty.

Regret Bounds for Multi-Armed Bandits with Heavy-Tailed Reward Distributions

The paper explores the classical multi-armed bandit problem under the scenario where reward distributions are characterized by heavy tails. Specifically, it challenges the traditional assumption of sub-Gaussian distributions that are prevalent in bandit problem analysis. The authors propose analyzing the problem when the distributions only possess finite moments of order 1+, thereby allowing for potential infinite variance.

Key Contributions

1. Theoretical Framework: The paper begins by establishing a theoretical framework for the multi-armed bandit problem under heavy-tailed reward distributions. Traditional bandit approaches typically assume sub-Gaussian rewards for tractability, ensuring that each arm’s reward distribution has a well-defined moment generating function. This paper departs from that approach, proposing new strategies suitable for distributions where moments beyond the first are undefined or infinite.
2. Regret Bound Formulation: Central to the paper is the derivation of regret bounds in this general setting. The authors introduce sampling strategies based on robust estimators such as:
   • Truncated empirical mean
   • Catoni's M-estimator
   • Median-of-means estimator

Each estimator targets mitigating the effects of heavy tails in the reward distributions of the arms.

1. Regret Analysis: The paper presents upper bounds for expected regret, demonstrating that even under heavy-tailed conditions, it is possible to achieve regrets similar to those under sub-Gaussian assumptions, provided certain conditions are met. Notably, they prove that logarithmic regret can still be achieved if the distributions merely have finite variance. Furthermore, the analysis extends to cases with tail distributions exhibiting only finite first-order moments, achieving logarithmic regret with a dependency on $\Delta_i^{1/\alpha}$ where $<1$.
2. Lower Bound Analysis: Matching lower bounds are derived, demonstrating the optimality of the regret bounds under the specified moment conditions. The work elucidates that the deterioration of regret bounds is tied directly to the tail heaviness, specifically when $\alpha<1$.

Implications and Future Work

The implications of these findings are considerable within theoretical and applied contexts of decision-making under uncertainty, particularly in environments characterized by significant outlier events or extensive variability. In practice, these results suggest adopting more robust mean estimators can significantly improve performance in real-world applications ranging from finance to algorithmic trading and bioinformatics, where heavy-tailed phenomena are commonplace.

Future developments in this line of research could further investigate:

• Improved computational efficiency for the proposed estimators
• Extension to contextual bandits or reinforcement learning scenarios with heavy-tailed reward signals
• Exploration of adaptive or online approaches for automatically estimating the tail index to guide estimator selection in runtime

Theoretical advancements in understanding the trade-offs between computational complexity and estimator robustness will continue to enhance the applicability and implementation of bandit solutions in these complex environments.