Optimistic Regret Bounds for Online Learning in Adversarial Markov Decision Processes

Core Concepts

This paper proposes a novel policy search scheme that utilizes optimistic cost predictors to achieve sub-linear regret bounds in Adversarial Markov Decision Processes (AMDPs), where the cost functions can change adversarially across episodes.

Abstract

The paper introduces a new variant of the AMDP framework, which aims to minimize regret while utilizing a set of cost predictors. The authors develop a new policy search method, OREPS-OPIX, that achieves optimistic regret bounds, where the regret scales with the estimation power of the cost predictors.
Key highlights:

OREPS-OPIX utilizes a novel cost estimator that leverages the cost predictors and enables high-probability regret analysis without restrictive assumptions.
In the full information setting, OREPS-OPIX achieves a sub-linear optimistic regret bound of ˜O(√(∑_t ∥σ_t∥_∞^2)), where σ_t = c_t - M_t is the prediction error.
In the bandit feedback setting, OREPS-OPIX achieves an expected regret bound of ˜O(L^(1/3)(∑_t ∥σ_t∥_2^2 + ∥σ_t∥_1)^(2/3)) and a high probability regret bound of ˜O(L^(1/4)(∑_t ∥σ_t∥_∞^2 + ∥σ_t∥_1)^(3/4) + √(∑_t ∥σ_t∥_1^2)).
The authors also introduce anytime extensions and handle the case of unknown transition dynamics.
Numerical experiments demonstrate the benefits of the proposed method in terms of regret reduction and variance reduction compared to existing approaches.

Stats

The cost functions are bounded, i.e., c_t ∈ [0, 1]^(|X| x |A|) for t = 1, 2, ..., T.
The state space X is partitioned into L non-overlapping layers X_0, X_1, ..., X_L.
The state transition function Pr(x'|x, a) is stationary.

Quotes

"Motivated by this shortcoming, we propose to study a new formulation for RL with time-varying cost functions where the aim is to learn a policy that minimizes its regret while resorting to a given set of time-varying predictive estimators of the cost functions, denoted by {c_t}^T_t=1 and {M_t}^T_t=1, respectively."
"Crucial to the establishment of these results is the development of a new cost estimator. This new estimator leverages the bandit information about the cost as well as the set of predictive estimators to update the policy."

Key Insights Distilled From

Optimistic Regret Bounds for Online Learning in Adversarial Markov Decision Processes

by Sang Bin Moo... at arxiv.org 05-06-2024

https://arxiv.org/pdf/2405.02188.pdf

Deeper Inquiries

How can the proposed method be extended to handle non-stationary transition dynamics, where the transition probabilities also change over time

To extend the proposed method to handle non-stationary transition dynamics where the transition probabilities change over time, we can incorporate adaptive techniques for estimating the transition probabilities. One approach could involve using online learning algorithms to update the transition probabilities based on the observed data. By continuously updating the transition probabilities as new data becomes available, the algorithm can adapt to the changing dynamics of the environment. Additionally, techniques such as confidence intervals or Bayesian updating can be employed to account for uncertainty in the transition probabilities. This adaptive approach would allow the algorithm to effectively learn and optimize policies in environments with non-stationary transition dynamics.

Can the optimistic regret bounds be further improved by incorporating more sophisticated cost predictors or by relaxing the assumptions on the cost predictors

The optimistic regret bounds of the OREPS-OPIX algorithm can potentially be improved by incorporating more sophisticated cost predictors or by relaxing the assumptions on the cost predictors. One way to enhance the performance of the algorithm is to use ensemble methods for cost prediction, where multiple predictors are combined to provide more accurate estimates. Additionally, incorporating techniques such as deep learning or reinforcement learning for cost prediction could improve the accuracy of the predictors. Relaxing the assumptions on the cost predictors, such as allowing for more flexibility in the prediction models or considering different types of cost functions, could also lead to better regret bounds. By exploring these avenues, the algorithm could achieve even more efficient and effective decision-making in adversarial environments.

What are the potential applications of the proposed OREPS-OPIX algorithm beyond the AMDP setting, and how can it be adapted to those domains

The proposed OREPS-OPIX algorithm has potential applications beyond the Adversarial Markov Decision Process (AMDP) setting. One possible application is in online reinforcement learning tasks where decision-making involves unknown and changing environments. For example, the algorithm could be applied to dynamic pricing strategies in e-commerce, adaptive resource allocation in cloud computing, or personalized recommendation systems. By adapting the algorithm to these domains, it could optimize policies in real-time based on changing conditions and user preferences. Additionally, the algorithm could be extended to multi-agent systems, hierarchical decision-making problems, or other complex environments where adaptive learning is crucial. By leveraging the principles of optimistic regret bounds and cost prediction, the algorithm could offer significant advantages in a wide range of applications requiring online decision-making.