Online Bandit Learning against an Adaptive Adversary: from Regret to Policy Regret (1206.6400v1)

Abstract: Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to adapt to the online algorithm's actions. We define the alternative notion of policy regret, which attempts to provide a more meaningful way to measure an online algorithm's performance against adaptive adversaries. Focusing on the online bandit setting, we show that no bandit algorithm can guarantee a sublinear policy regret against an adaptive adversary with unbounded memory. On the other hand, if the adversary's memory is bounded, we present a general technique that converts any bandit algorithm with a sublinear regret bound into an algorithm with a sublinear policy regret bound. We extend this result to other variants of regret, such as switching regret, internal regret, and swap regret.

• The paper introduces policy regret, a novel metric that more accurately evaluates performance against adaptive adversaries than traditional regret measures.
• It demonstrates that mini-batching techniques enable sublinear policy regret bounds for online bandit algorithms facing adversaries with bounded memory.
• The study extends its insights to switching, internal, and swap regrets, offering significant theoretical and practical implications for robust algorithm design.

Online Bandit Learning against an Adaptive Adversary: From Regret to Policy Regret

The paper "Online Bandit Learning against an Adaptive Adversary: from Regret to Policy Regret" by Raman Arora, Ofer Dekel, and Ambuj Tewari, investigates the efficacy of online learning algorithms in adversarial environments. The authors propose the novel concept of policy regret as opposed to traditional notions of regret, particularly in settings where adversaries adapt to the player's strategies, thereby challenging the assumptions of standard regret metrics in online bandit learning.

Key Contributions

1. Reconceptualizing Regret in Adversarial Settings: The authors critique the standard regret measure, which fails to hold intuitive relevance when an adversary adapts to the learner's strategy. They introduce policy regret as a more meaningful metric that compares player's loss to that of the best possible action sequence, taking into account the adaptive nature of the adversary.
2. Sublinear Policy Regret with Bounded Memory Adversaries: The paper demonstrates that no online bandit algorithm consistently achieves sublinear policy regret against adversaries with unbounded memory capabilities. However, they derive a general method for achieving sublinear policy regret against adversaries with bounded memory using a mini-batching technique. This method leverages existing bandit algorithms with proven sublinear regret bounds.
3. Specific Applications and Bounds: The authors extend their findings to several bandit problem variants. For instance:
   • In the k-armed bandit problem, the mini-batching technique yields a policy regret bound of $O(T^{2/3})$.
   • Bandit convex optimization achieves a policy regret bound of $O(T^{4/5})$.
   • Bandit linear optimization and submodular optimization result in a policy regret bound of $O(T^{3/4})$, with potential improvement to $O(T^{2/3})$ if the adversary's memory bound is known.
4. Implications for Other Regret Forms: Beyond external regret, the paper develops insights for switching, internal, and swap regrets, demonstrating the adaptability of their technique across various learning frameworks.

Theoretical and Practical Implications

The implications of this paper are multifaceted. Theoretically, it challenges the current paradigms governing online learning evaluation metrics, potentially reshaping the benchmarks against which learning algorithms are assessed in adversarial contexts. Practically, the demonstrated methodology offers robust strategies to design algorithms that perform optimally in environments populated by strategic adversaries with bounded adaptability, relevant for fields such as finance, cybersecurity, and automated decision-making.

Future Directions

The paper leaves open several lines of inquiry. There is a notable absence of lower bounds for policy regret, which raises questions about the optimality of the presented upper bounds. Furthermore, the necessity of the mini-batching technique warrants further exploration; it remains undetermined whether the original bandit algorithms, unmodified, could also exhibit non-trivial policy regret bounds. Continued research in this area could unveil more sophisticated algorithms optimized specifically for minimizing policy regret.

In summary, this work provides a pivotal perspective on redefining success metrics in online learning under adversarial conditions. The introduction of policy regret represents a significant step forward in understanding how adaptive challenges can be systematically addressed through strategic algorithmic innovations.