Explore no more: Improved high-probability regret bounds for non-stochastic bandits (1506.03271v3)

Abstract: This work addresses the problem of regret minimization in non-stochastic multi-armed bandit problems, focusing on performance guarantees that hold with high probability. Such results are rather scarce in the literature since proving them requires a large deal of technical effort and significant modifications to the standard, more intuitive algorithms that come only with guarantees that hold on expectation. One of these modifications is forcing the learner to sample arms from the uniform distribution at least $\Omega(\sqrt{T})$ times over $T$ rounds, which can adversely affect performance if many of the arms are suboptimal. While it is widely conjectured that this property is essential for proving high-probability regret bounds, we show in this paper that it is possible to achieve such strong results without this undesirable exploration component. Our result relies on a simple and intuitive loss-estimation strategy called Implicit eXploration (IX) that allows a remarkably clean analysis. To demonstrate the flexibility of our technique, we derive several improved high-probability bounds for various extensions of the standard multi-armed bandit framework. Finally, we conduct a simple experiment that illustrates the robustness of our implicit exploration technique.

Exploration Strategies in Non-Stochastic Multi-Armed Bandits: An Analysis of Implicit Exploration

The paper "Explore no more: Improved high-probability regret bounds for non-stochastic bandits" by Gergely Neu offers a significant contribution to the paper of non-stochastic multi-armed bandit (MAB) problems, specifically concerning regret minimization and performance guarantees that hold with high probability. Traditional approaches have heavily relied on explicit exploration mechanisms by insisting on a uniform sampling of arms. This paper challenges that notion by demonstrating that high-probability regret bounds can be achieved without such explicit exploration via a novel approach termed Implicit eXploration (IX).

Key Contributions

1. Implicit eXploration (IX) Strategy: The core innovation introduced in this research is the IX strategy, which proposes a biasing of loss estimates without the need for uniform arm sampling. IX recalibrates loss estimation by introducing a subtle bias that maintains variance control, thus allowing for cleaner analyses and the derivation of improved high-probability bounds.
2. Improved High-Probability Regret Bounds: Utilizing the IX approach, the paper establishes remarkable bounds that improve upon traditional results. For instance, it claims bounds achieving sublinear regret growth, with leading constants notably smaller than those previously established (e.g., a factor of $2\sqrt{2}$ compared to the best-known factor of 5.15).
3. Versatile Framework: The adaptability of the IX method is demonstrated across various extensions of the MAB framework, such as bandits with expert advice and bandits with side-observations. This versatility underscores the broad applicability and robustness of the IX strategy against a variety of adversarial and structured feedback scenarios.
4. Practical Implications and Experiments: An empirical evaluation underlines the effectiveness of the IX-based algorithms compared to traditional methods like Exp3.P and demonstrates robustness without explicit exploration, maintaining strong performance even in stochastic settings.

Implications for Theory and Practice

The introduction of IX is likely to shift the paradigm of exploration-exploitation strategies in non-stochastic bandit problems. Theoretically, this paper prompts a reevaluation of exploration needs and suggests that the prevalent assumption of mandatory explicit exploration could be relaxed under certain conditions, without compromising high-confidence performance guarantees.

Practically, the insights gained from this research could drive enhancements in applications where exploration cost is high or impractical. For example, scenarios in financial portfolio design or adaptive system management, where exploratory actions bear direct costs, may benefit from adopting IX strategies to optimize decision-making without the overhead of unwarranted exploration.

Future Directions

Future research could delve into several intriguing paths opened by this paper:

• Adaptive Learning Rates: The current results hinge on deterministic learning rates. An exploration into adaptive rates with high-probability guarantees would complement the existing theoretical framework.
• Extension to Linear Bandits: While IX shows promise in advancing non-stochastic MAB understanding, its potential extension to the more complex linear bandit models presents an opportunity for further investigation. This could offer advancements in optimizing decision-making across higher-dimensional action spaces.
• Comparative Analysis with Other Implicit Strategies: The robustness and comparative efficacy of IX relative to other implicit strategies, such as those employing log-based estimations, warrant additional exploration. This could fine-tune the understanding of conditions and performance trade-offs in various problem settings.

In summary, the paper's significant theoretical developments and practical insights could reshape exploration strategies in MAB problems and stimulate further advancements in online learning algorithms. The IX strategy offers not just an incremental performance improvement but a substantial rethinking of how exploration can be achieved in adversarial setups.