аналитика - Combinatorial Optimization - # Adversarial combinatorial bandits with switching costs

Adversarial Combinatorial Bandits with Switching Costs: Algorithms and Lower Bounds

Q: Can the lower bounds be further tightened by designing more refined loss sequences

To further tighten the lower bounds, designing more refined loss sequences could be a promising approach. By carefully constructing the stochastic loss sequences to capture the intricacies of the problem, such as the relationships between the observed losses and the underlying arm selections, it may be possible to create sequences that lead to tighter lower bounds. For example, refining the design of the loss sequences to better reflect the specific challenges posed by the combinatorial bandit problem with switching costs could potentially yield lower bounds that more accurately capture the inherent difficulty of the problem. Additionally, exploring different distributions and structures in the loss sequences could offer insights into how to tighten the lower bounds further.

Q: Are there other algorithmic techniques beyond the batched approach that can lead to improved regret bounds

While the batched approach is a powerful technique for restricting the number of switches between actions and has been utilized in the algorithms presented, there are other algorithmic techniques that could potentially lead to improved regret bounds. One such technique could involve incorporating more sophisticated exploration-exploitation strategies tailored to the combinatorial bandit setting with switching costs. By dynamically adjusting the exploration rates based on the observed losses and the switching costs, algorithms could potentially achieve better performance. Additionally, leveraging techniques from online learning and optimization, such as adaptive learning rates or advanced regret minimization strategies, could also contribute to improving regret bounds in combinatorial bandit problems with switching costs.

Q: How do the results and techniques extend to other variants of combinatorial bandits, such as when the combinatorial arms have additional structure or constraints

The results and techniques discussed in the context of adversarial combinatorial bandits with switching costs can be extended to other variants of combinatorial bandits with additional structure or constraints. For example, in scenarios where the combinatorial arms have specific constraints or dependencies among the base arms, the design of loss sequences and algorithms would need to account for these constraints. Techniques such as incorporating constraints into the exploration-exploitation strategies, adapting the batched approach to handle structured combinatorial arms, or designing specialized regret minimization algorithms tailored to the specific constraints could be explored. By adapting the existing results and techniques to accommodate additional structure or constraints in combinatorial bandit problems, it is possible to address a wider range of real-world applications with more complex combinatorial decision-making scenarios.

Основные понятия

The core message of this paper is to study the problem of adversarial combinatorial bandits with switching costs, derive lower bounds for the minimax regret, and propose algorithms that approximately meet these lower bounds under both bandit feedback and semi-bandit feedback settings.

Аннотация

The paper studies the problem of adversarial combinatorial bandits with switching costs, where there is a cost λ > 0 for switching between arms in each round. The authors consider both the bandit feedback setting, where only the total loss of the chosen combinatorial arm is observed, and the semi-bandit feedback setting, where the losses of all base arms in the chosen combinatorial arm are observed.
The key contributions are:

The authors derive lower bounds for the minimax regret under both feedback settings. For bandit feedback, the lower bound is Ω((λK)^(1/3)(TI)^(2/3)/log^2 T), and for semi-bandit feedback, the lower bound is Ω((λKI)^(1/3)T^(2/3)/log^2 T), where K is the number of base arms, I is the number of base arms in the combinatorial arm, and T is the time horizon.

To approach these lower bounds, the authors design two algorithms:

For bandit feedback, the BATCHED-EXP2 algorithm with John's exploration achieves a regret upper bound of Õ((λK)^(1/3)T^(2/3)I^(4/3)).
For semi-bandit feedback, the BATCHED-BROAD algorithm achieves a regret upper bound of Õ((λK)^(1/3)(TI)^(2/3) + KI).

The authors show that the regret gap between their algorithms and the lower bounds scales at most as I^(2/3) for bandit feedback and I^(1/3) for semi-bandit feedback, indicating that further improvements may be possible.

Статистика

K ≥ 3I and T ≥ max{λK/I, 8} for the lower bound under bandit feedback.
K ≥ 3I and T ≥ max{λK/I^2, 6} for the lower bound under semi-bandit feedback.

Цитаты

None.

Ключевые выводы из

Adversarial Combinatorial Bandits with Switching Costs

by Yanyan Dong,... в arxiv.org 04-03-2024

https://arxiv.org/pdf/2404.01883.pdf

Adversarial Combinatorial Bandits with Switching Costs

Дополнительные вопросы

Can the lower bounds be further tightened by designing more refined loss sequences

To further tighten the lower bounds, designing more refined loss sequences could be a promising approach. By carefully constructing the stochastic loss sequences to capture the intricacies of the problem, such as the relationships between the observed losses and the underlying arm selections, it may be possible to create sequences that lead to tighter lower bounds. For example, refining the design of the loss sequences to better reflect the specific challenges posed by the combinatorial bandit problem with switching costs could potentially yield lower bounds that more accurately capture the inherent difficulty of the problem. Additionally, exploring different distributions and structures in the loss sequences could offer insights into how to tighten the lower bounds further.

Are there other algorithmic techniques beyond the batched approach that can lead to improved regret bounds

While the batched approach is a powerful technique for restricting the number of switches between actions and has been utilized in the algorithms presented, there are other algorithmic techniques that could potentially lead to improved regret bounds. One such technique could involve incorporating more sophisticated exploration-exploitation strategies tailored to the combinatorial bandit setting with switching costs. By dynamically adjusting the exploration rates based on the observed losses and the switching costs, algorithms could potentially achieve better performance. Additionally, leveraging techniques from online learning and optimization, such as adaptive learning rates or advanced regret minimization strategies, could also contribute to improving regret bounds in combinatorial bandit problems with switching costs.

How do the results and techniques extend to other variants of combinatorial bandits, such as when the combinatorial arms have additional structure or constraints

The results and techniques discussed in the context of adversarial combinatorial bandits with switching costs can be extended to other variants of combinatorial bandits with additional structure or constraints. For example, in scenarios where the combinatorial arms have specific constraints or dependencies among the base arms, the design of loss sequences and algorithms would need to account for these constraints. Techniques such as incorporating constraints into the exploration-exploitation strategies, adapting the batched approach to handle structured combinatorial arms, or designing specialized regret minimization algorithms tailored to the specific constraints could be explored. By adapting the existing results and techniques to accommodate additional structure or constraints in combinatorial bandit problems, it is possible to address a wider range of real-world applications with more complex combinatorial decision-making scenarios.

Adversarial Combinatorial Bandits with Switching Costs: Algorithms and Lower Bounds