Bandits with Switching Costs: T^{2/3} Regret
(1310.2997v2)
Published 11 Oct 2013 in cs.LG and math.PR
Abstract: We study the adversarial multi-armed bandit problem in a setting where the player incurs a unit cost each time he switches actions. We prove that the player's $T$-round minimax regret in this setting is $\widetilde{\Theta}(T{2/3})$, thereby closing a fundamental gap in our understanding of learning with bandit feedback. In the corresponding full-information version of the problem, the minimax regret is known to grow at a much slower rate of $\Theta(\sqrt{T})$. The difference between these two rates provides the \emph{first} indication that learning with bandit feedback can be significantly harder than learning with full-information feedback (previous results only showed a different dependence on the number of actions, but not on $T$.) In addition to characterizing the inherent difficulty of the multi-armed bandit problem with switching costs, our results also resolve several other open problems in online learning. One direct implication is that learning with bandit feedback against bounded-memory adaptive adversaries has a minimax regret of $\widetilde{\Theta}(T{2/3})$. Another implication is that the minimax regret of online learning in adversarial Markov decision processes (MDPs) is $\widetilde{\Theta}(T{2/3})$. The key to all of our results is a new randomized construction of a multi-scale random walk, which is of independent interest and likely to prove useful in additional settings.
Insightful Overview of "Bandits with Switching Costs: T2/3 Regret"
The paper "Bandits with Switching Costs: T2/3 Regret" by Dekel, Ding, Koren, and Peres addresses the complex problem of multi-armed bandits (MAB) in an environment where switching actions induces a cost. The authors decisively close a gap in the literature by showing that the minimax regret grows as Θ(T2/3) instead of the Θ(T) observed in full-information feedback settings. This finding elucidates the increased difficulty in learning scenarios with bandit feedback compared to those with full informational access.
Core Contributions
Regret Bound Analysis: The primary contribution lies in the establishment of a regret bound Θ(T2/3) for multi-armed bandits with switching costs. This result is significant because it quantitatively demonstrates the added complexity of learning when switching costs are factored in, distinguishing it clearly from full-feedback scenarios which have a Θ(T) bound.
Lower-Bound Proposition: The paper proposes a non-trivial lower bound of Ω(T2/3), confirming the tightness of their upper bounds. The authors apply Yao's Minimax Principle and construct a stochastic sequence of loss functions that effectively demonstrate this bound under adversarial conditions.
Algorithmic Implications: The results extend to settings described as adversarial Markov decision processes (MDPs) in online learning frameworks, emphasizing a broader relevance by implying a regret bound of Θ(T2/3).
Novel Use of Stochastic Processes: A significant technical novelty is their use of a random multi-scale random walk (MRW) for constructing the adversary's loss sequence. This process permits tight control over dependencies within the adversarial loss sequence, crucial for ensuring the proposed bound's validity.
Theoretical and Practical Implications
This work not only strengthens conceptual understanding by demonstrating a tangible distinction in difficulty between bandit and full-feedback models but also propels algorithmic developments within adversarial bandit settings. The use of switching costs models real-world scenarios such as financial decision-making and industrial process optimization where transaction costs exist.
Future Directions
Exploration of Alternative Cost Structures: A natural progression is to explore varying cost structures beyond the fixed costs used in this paper, such as state-dependent or time-dependent costs.
Broader Applications of MRW: The multi-scale random walk's potential utility in other online learning paradigms suggests valuable future research paths, particularly in developing alternative adversaries or loss functions in machine learning frameworks.
Complex Action Spaces: The extension of this framework to more complex and dynamic action spaces, such as those observed in deep reinforcement learning environments, offers notable promise.
In summary, this paper significantly advances the theory of online learning in constrained scenarios. By establishing a robust theoretical foundation for the performance of bandit algorithms with switching costs, the authors have set the stage for sophisticated models that could be leveraged in various domains experiencing similar constraints.