On Explore-Then-Commit Strategies (1605.08988v2)

Abstract: We study the problem of minimising regret in two-armed bandit problems with Gaussian rewards. Our objective is to use this simple setting to illustrate that strategies based on an exploration phase (up to a stopping time) followed by exploitation are necessarily suboptimal. The results hold regardless of whether or not the difference in means between the two arms is known. Besides the main message, we also refine existing deviation inequalities, which allow us to design fully sequential strategies with finite-time regret guarantees that are (a) asymptotically optimal as the horizon grows and (b) order-optimal in the minimax sense. Furthermore we provide empirical evidence that the theory also holds in practice and discuss extensions to non-gaussian and multiple-armed case.

• The paper shows that explore-then-commit strategies incur asymptotically double the regret of fully sequential methods in two-armed Gaussian bandit problems.
• It leverages deviation inequalities and empirical experiments to compare fixed-design, modified Best Arm Identification, and SPRT-inspired methods.
• The findings imply that real-time feedback systems outperform phased exploration, prompting further research into adaptive sequential decision-making.

Insights into Explore-Then-Commit Strategies

The paper "On Explore-Then-Commit Strategies" by Aurélien Garivier, Emilie Kaufmann, and Tor Lattimore examines the efficiency of Explore-Then-Commit (ETC) strategies in solving two-armed bandit problems with Gaussian rewards. The primary objective is to understand how strategies that incorporate a phased approach of exploration followed by exploitation perform relative to fully sequential strategies.

The central argument presented is that ETC strategies are notably suboptimal when minimizing regret, a conclusion reached by leveraging deviation inequalities and empirical experiments. The paper methodically compares ETC strategies against fully sequential strategies, providing comprehensive analyses under conditions where the difference in means between arms is either known or unknown. The theoretical implications are significant: the regret associated with ETC strategies can be, asymptotically, twice that of optimal sequential strategies.

Key Numerical Results

The authors provide substantial mathematical analysis, resulting in several bounds for regret. When the difference in means, denoted as $\Delta$, is known, the optimal regret bound using ETC strategies is shown to be $R^\pi_\mu(T) \sim \log(T)/\Delta$. Through a novel strategy inspired by the Sequential Probability Ratio Test (SPRT), the paper achieves an improvement upon fixed-design strategies, yet still reveals that fully sequential strategies can further halve the regret to $R^\pi_\mu(T) \sim \log(T)/(2\Delta)$.

Conversely, when $\Delta$ is unknown, fixed-design strategies struggle without proper tuning, yet ETC strategies utilizing a modified Best Arm Identification algorithm can achieve asymptotic optimality within their class, with a regret bound of $R^\pi_\mu(T) \sim 4 \log(T)/\Delta$. Despite this, fully sequential strategies, such as UCB, remain more efficient.

Implications and Future Directions

The implications of this work extend beyond theoretical constructs to practical applications in decision-making systems, such as dynamic content adjustment on websites based on user interaction. The evidence against strict phased strategies indicates that systems benefiting from real-time feedback loops should embrace fully sequential strategies for optimal performance.

The paper also explores broader applications beyond the Gaussian two-armed setting, proposing that ETC strategies might be similarly suboptimal in more complex or non-Gaussian scenarios. However, the work suggests that there remains room for improvement in sequential strategies when multiple arms are involved, hinting at future research directions in extending the framework to higher dimensions and model complexities.

In conclusion, this work presents a rigorous examination of ETC strategies, highlighting their inefficiencies in regret minimization, and setting the stage for further exploration into alternative sequential strategies that can adaptively balance exploration and exploitation for optimal decision-making outcomes.