Analysis of Thompson Sampling for Partially Observable Contextual Multi-Armed Bandits

This paper focuses on the design and analysis of a reinforcement learning policy for partially observable contextual multi-armed bandits. Contextual multi-armed bandits are classical models in reinforcement learning for sequential decision-making associated with individual information. A widely-used policy for bandits is Thompson Sampling, where samples from a data-driven probabilistic belief about unknown parameters are used to select control actions. However, little is known about problems where contexts are not fully observed. The authors propose a modified version of Thompson Sampling that leverages Bayesian methods for balancing exploration and exploitation and estimates unobserved contexts based on the sequence of output observations. The presented algorithm establishes theoretical performance guarantees, showing that the regret scales logarithmically with time and the number of arms, and linearly with dimension. To validate their approach, the authors conduct numerical analyses by repeating simulations 50 times for each case, reporting two quantities: ||bµ(t)−µ∗|| and Regret(t). They show that parameter estimates converge fast to truth and errors decrease at appropriate rates over time. Overall, this paper provides an important contribution to reinforcement learning policies for partially observable contextual multi-armed bandits by introducing a modified version of Thompson Sampling which leverages Bayesian methods to balance exploration and exploitation while estimating unobserved contexts based on output observations. Future studies could explore leveraging other techniques such as adversarial training or incorporating expert knowledge into the model.

Reinforcement Learning Policy for Partially Observable Contextual Multi-Armed Bandits

Contextual multi-armed bandits are classical models in reinforcement learning for sequential decision-making associated with individual information. A widely used policy for bandits is Thompson Sampling, where samples from a data-driven probabilistic belief about unknown parameters are used to select control actions. However, little is known about problems where contexts are not fully observed. In this paper, the authors propose a modified version of Thompson Sampling that leverages Bayesian methods to balance exploration and exploitation while estimating unobserved contexts based on output observations.

Background

Multi-armed bandit (MAB) problems involve selecting an action from a set of options in order to maximize reward over time. The classic MAB problem assumes that all arms have the same expected reward and can be solved using simple strategies such as epsilon greedy or UCB1 algorithms. Contextual MABs extend this model by introducing context variables which affect the expected rewards of each arm differently depending on their values. This allows for more sophisticated decision making since different arms may be optimal depending on the current context variables.

Thompson Sampling (TS) is a popular approach to solving contextual MABs which uses Bayesian inference to estimate the expected rewards of each arm given its context variables and then selects an action based on these estimates. TS has been shown to outperform other approaches such as epsilon greedy and UCB1 in many cases due to its ability to balance exploration and exploitation effectively while still converging quickly towards optimal solutions.

Problem Statement

The authors consider the problem of partially observable contextual MABs, where only some of the context variables are observed at any given time step and must be estimated from past observations in order to make decisions effectively. This presents a challenge since traditional TS algorithms assume full observability of all relevant context variables at every time step, making them unsuitable for use in this setting without modification.

Proposed Solution

The authors propose a modified version of TS which they refer to as “Partially Observable Thompson Sampling” (POTS). POTS leverages Bayesian methods for balancing exploration and exploitation while estimating unobserved contexts based on output observations from past trials using maximum likelihood estimation (MLE). It also establishes theoretical performance guarantees showing that regret scales logarithmically with time and number of arms, linearly with dimensionality, and exponentially with prior variance parameterization error rate δp . To validate their approach, they conduct numerical analyses by repeating simulations 50 times per case reporting two quantities: ||bµ(t)−µ∗||and Regret(t). They show that parameter estimates converge fast towards truth values while errors decrease at appropriate rates over time indicating good performance overall compared against baseline policies such as epsilon greedy or UCB1 algorithms when applied under partial observability conditions..

Conclusion

This paper provides an important contribution to reinforcement learning policies for partially observable contextual multi-armed bandits by introducing POTS – a modified version of Thompson Sampling which leverages Bayesian methods for balancing exploration and exploitation while estimating unobserved contexts based on output observations from past trials using maximum likelihood estimation (MLE). Future studies could explore leveraging other techniques such as adversarial training or incorporating expert knowledge into the model in order improve upon existing results even further.