Ensemble Sampling Analysis for Linear Bandits

Core Concepts

Ensemble sampling for linear bandits with small ensembles is effective and efficient, providing a rigorous analysis that does not require ensemble size to scale linearly with the horizon T.

Abstract

Ensemble sampling is a valuable method for balancing exploration and exploitation in sequential decision-making tasks. It has practical applications in deep reinforcement learning, online recommendation systems, behavioral sciences, and marketing. The analysis of ensemble sampling has been challenging but this study provides a breakthrough by showing that small ensembles are sufficient for optimal performance. The algorithm selects actions based on randomly chosen parameters from an ensemble of perturbed models, achieving regret bounds that are competitive with other methods. The theoretical results presented in this study lay the foundation for justifying the practical effectiveness of ensemble sampling in various structured settings.

Stats

Ensemble sampling with an ensemble size logarithmic in T and linear in the number of features d incurs regret no worse than order (d log T)5/2√T. For a d-dimensional linear bandit with an action set X of cardinality K, the Bayesian regret incurred scales as BR(T) ≤ C√dT log K + CT√KrK log(mT)(d ∧ log K). The regret bound holds with probability 1 − δ, showcasing the efficiency of small ensembles in ensemble sampling.

Quotes

"A lot of work has attempted to analyze ensemble sampling, but none of them has been successful." - Qin et al. (2022) "Our contribution is a guarantee that ensemble sampling with an ensemble size logarithmic in T and linear in the number of features d incurs regret no worse than order (d log T)5/2√T." - Authors

Key Insights Distilled From

Ensemble sampling for linear bandits

by Davi... at arxiv.org 03-07-2024

https://arxiv.org/pdf/2311.08376.pdf

Deeper Inquiries

How can the findings on ensemble sampling be extended to other structured settings beyond linear bandits?

The findings on ensemble sampling for linear bandits can be extended to other structured settings by adapting the algorithm and analysis to suit the specific characteristics of those settings. For example, in generalized linear bandits, where the relationship between actions and rewards is more complex than in linear bandits, one could modify the perturbation scheme or ensemble selection process to accommodate this complexity. Similarly, in kernelized bandits or deep learning applications, adjustments may need to be made to account for non-linear relationships or high-dimensional feature spaces.

What are potential drawbacks or limitations of using small ensembles in ensemble sampling?

Using small ensembles in ensemble sampling may lead to limited diversity among models, which can hinder exploration and exploitation balance. With fewer models in the ensemble, there is a risk of missing out on important patterns or variations in the data that could impact decision-making. Additionally, small ensembles may not capture enough uncertainty about the underlying structure of the problem domain, leading to suboptimal performance compared to larger ensembles. Lastly, smaller ensembles might struggle with representing complex functions accurately due to their reduced capacity.

How can the concept of optimism be further explored and applied in optimizing random algorithms beyond Thompson sampling?

The concept of optimism can be further explored and applied in optimizing random algorithms by incorporating it into various aspects of algorithm design. One approach is through optimistic initialization strategies that bias initial estimates towards optimistic values based on prior knowledge or assumptions about model parameters. Another way is by integrating optimism into exploration-exploitation trade-offs during action selection processes - favoring actions with higher potential rewards even if they have not been extensively explored yet. Optimism can also guide adaptive learning rates or step sizes within optimization algorithms based on confidence levels about parameter estimates.