Bandit algorithms: Letting go of logarithmic regret for statistical robustness
This addresses a critical limitation in bandit algorithms for decision-making under uncertainty, revealing inherent statistical fragility in existing methods and offering a solution that balances performance and robustness.
The paper establishes a fundamental trade-off between regret and statistical robustness in stochastic multi-armed bandits, showing that algorithms with logarithmic regret are inconsistent, while consistent algorithms suffer super-logarithmic regret. It proposes distribution-oblivious algorithms that achieve asymptotic regret arbitrarily close to logarithmic while maintaining robustness and consistency.
We study regret minimization in a stochastic multi-armed bandit setting and establish a fundamental trade-off between the regret suffered under an algorithm, and its statistical robustness. Considering broad classes of underlying arms' distributions, we show that bandit learning algorithms with logarithmic regret are always inconsistent and that consistent learning algorithms always suffer a super-logarithmic regret. This result highlights the inevitable statistical fragility of all `logarithmic regret' bandit algorithms available in the literature---for instance, if a UCB algorithm designed for $σ$-subGaussian distributions is used in a subGaussian setting with a mismatched variance parameter, the learning performance could be inconsistent. Next, we show a positive result: statistically robust and consistent learning performance is attainable if we allow the regret to be slightly worse than logarithmic. Specifically, we propose three classes of distribution oblivious algorithms that achieve an asymptotic regret that is arbitrarily close to logarithmic.