Optimality of FTPL with Fréchet-type Tail Distributions in Bandits

Centrala begrepp

The author establishes conditions for FTPL to achieve optimal regret in adversarial bandits using Fréchet-type tail distributions, resolving open questions and offering insights into regularization functions.

Sammanfattning

This paper explores the optimality of Follow-the-Perturbed-Leader (FTPL) policy in adversarial bandits with Fréchet-type tail distributions. It provides conditions for achieving optimal regret and offers insights into the impact of regularization functions used in Follow-the-Regularized-Leader (FTRL) framework. The study resolves conjectures and contributes to understanding extreme value theory's role in bandit optimization.
Key points include:

Study on optimality of FTPL in adversarial and stochastic K-armed bandits.
Conditions for achieving O(√KT) regrets using Fréchet, Pareto, and Student-t distributions.
Demonstrating Best-of-Both-Worlds (BOBW) capability of FTPL with certain Fréchet-type tail distributions.
Insights into the impact of regularization functions in FTRL through mapping from FTPL.
Contribution to resolving existing conjectures through extreme value theory lens.

Statistik

Recent work by Honda et al. [2023] showed that FTPL with Fréchet distribution achieves O(√KT) regret in adversarial bandits.
Perturbations following a distribution with a Fréchet-type tail are crucial for achieving optimal regrets.
Conditions for perturbations to achieve O(√KT) regrets are established, covering various distributions like Fréchet, Pareto, and Student-t.

Citat

"FTPL with Fréchet perturbations attains optimal regret in adversarial bandits." - Honda et al. [2023]

Viktiga insikter från

Follow-the-Perturbed-Leader with Fréchet-type Tail Distributions

by Jongyeong Le... på arxiv.org 03-11-2024

https://arxiv.org/pdf/2403.05134.pdf

Djupare frågor

How does the use of random perturbations impact the efficiency of FTPL compared to other strategies

Random perturbations play a crucial role in the efficiency of Follow-the-Perturbed-Leader (FTPL) compared to other strategies in bandit optimization. By incorporating random perturbations into the arm selection process, FTPL introduces a level of exploration that allows for adaptability and robustness against adversarial environments. This randomness helps prevent exploitation of suboptimal arms by occasionally selecting different arms based on the perturbed estimates of cumulative losses.
Unlike deterministic strategies that may get stuck in local optima or vulnerable to manipulation by adversaries, FTPL with random perturbations can explore various options while still leveraging past information through estimated losses. This balance between exploration and exploitation is essential for achieving near-optimal performance in both stochastic and adversarial settings.
The use of random perturbations adds a layer of uncertainty to decision-making, which can help mitigate risks associated with incomplete information or changing environments. It provides flexibility and adaptability, allowing the algorithm to learn and adjust its decisions over time based on feedback received from previous actions.

What implications do these findings have on real-world applications involving decision-making under uncertainty

The findings regarding the optimality of FTPL with Fréchet-type tail distributions have significant implications for real-world applications involving decision-making under uncertainty. In scenarios where decisions need to be made without complete knowledge about the environment or when facing strategic adversaries, algorithms like FTPL offer an effective way to balance exploration and exploitation.
In practical applications such as online advertising, recommendation systems, financial trading, healthcare treatment planning, or resource allocation in dynamic environments, decision-makers often encounter uncertainties and varying levels of risk. By understanding how FTPL with specific types of perturbation distributions can achieve optimal regret bounds in bandit problems, practitioners can leverage these insights to design more efficient and adaptive decision-making systems.
These findings highlight the importance of considering randomness and exploration in decision-making processes under uncertainty. They suggest that incorporating elements of randomness into algorithms can lead to better outcomes by enabling adaptability, robustness against adversarial behavior, and improved overall performance even when faced with unknown variables or changing conditions.

How can the insights gained from extreme value theory be applied to other areas beyond bandit optimization

Insights gained from extreme value theory have broader applications beyond bandit optimization that extend into various fields where rare events play a critical role. Extreme value theory provides tools for modeling extreme events such as natural disasters, financial crises, rare medical conditions, outlier detection in data analysis,
and environmental risk assessment among others.
By applying concepts from extreme value theory such as regular variation functions
and Fréchet-type tail distributions outside bandit optimization contexts,
researchers can enhance their understanding
of rare event occurrences across diverse domains.
For example,
in finance,
extreme value theory is used
to model tail risks associated with market crashes;
in climate science,
it helps predict extreme weather events;
in healthcare,
it aids identification
of rare diseases;
and in engineering,
it informs structural designs resilient against catastrophic failures.
Overall,
the principles derived from extreme value theory provide valuable insights into handling outliers
and extremes effectively across various disciplines beyond just bandits optimization scenarios.