Optimistic Regret Bounds for Online Learning in Adversarial Markov Decision Processes

Abstract: The Adversarial Markov Decision Process (AMDP) is a learning framework that deals with unknown and varying tasks in decision-making applications like robotics and recommendation systems. A major limitation of the AMDP formalism, however, is pessimistic regret analysis results in the sense that although the cost function can change from one episode to the next, the evolution in many settings is not adversarial. To address this, we introduce and study a new variant of AMDP, which aims to minimize regret while utilizing a set of cost predictors. For this setting, we develop a new policy search method that achieves a sublinear optimistic regret with high probability, that is a regret bound which gracefully degrades with the estimation power of the cost predictors. Establishing such optimistic regret bounds is nontrivial given that (i) as we demonstrate, the existing importance-weighted cost estimators cannot establish optimistic bounds, and (ii) the feedback model of AMDP is different (and more realistic) than the existing optimistic online learning works. Our result, in particular, hinges upon developing a novel optimistically biased cost estimator that leverages cost predictors and enables a high-probability regret analysis without imposing restrictive assumptions. We further discuss practical extensions of the proposed scheme and demonstrate its efficacy numerically.

• The paper introduces an innovative cost estimator that leverages future cost predictions to achieve sub-linear regret.
• It rigorously derives regret bounds for both full-information and bandit settings, scaling with prediction error metrics.
• The method offers practical insights for dynamic environments such as autonomous systems and financial markets.

Understanding Optimistic Learning in Adversarial MDPs

Introduction to AMDPs

In the world of reinforcement learning (RL), decision-making scenarios often involve interacting with environments that are not only complex but can also change unpredictably over time. This is where Adversarial Markov Decision Processes (AMDPs) come into play, providing a framework to handle environments where the rules can change variably across episodes. AMDPs extend the traditional Markov Decision Process (MDP) models by incorporating the flexibility to manage different and evolving cost functions across episodes, making them ideal for applications ranging from drone navigation to dynamic pricing systems.

The Problem with Traditional AMDP Approaches

The key challenge in AMDPs is designing learning strategies that can effectively adapt to changing environments without suffering excessive regret - essentially, the difference between the cost incurred and the best possible cost in hindsight. Traditional approaches to AMDPs have often leaned towards conservative strategies that, while safe, do not always capitalize on patterns or predictive insights that could minimize regret. Moreover, they tend to analyze regret pessimistically – assuming the worst-case scenario.

A New Approach: Optimistic Policy Learning

This paper presents an innovative twist to the traditional AMDP by incorporating optimistic learning. Here's the core idea: by using predictive estimators of future costs, the learning algorithm can foresee and better adapt to potential changes in the environment. This approach not only offers a sub-linear optimistic regret but does so with a high probability, ensuring that the learning performance improves robustly over time.

The Power of Prediction:

The model introduces a novel cost estimator that utilizes predictions about future costs. If these predictions are accurate, the algorithm can achieve remarkably low regret – close to zero in the best-case scenarios. On the flip side, even if the predictions aren't perfect, the model still guarantees that the regret remains sub-linear.

Diving Deeper: Key Contributions and Results

1. Innovative Cost Estimator: At the heart of the proposed method is a new cost estimator that leverages information about the cost predicted for future states and actions. This estimator stands out because it intelligently combines observed data with predictions to improve estimation accuracy.
2. Promising Regret Bounds: The paper presents detailed analysis under both full-information and bandit settings (where limited information is available):
   • In a full-information scenario, where the true costs are fully observable, the proposed method provides regret bounds that scale with the square root of the cumulative prediction error.
   • In the more challenging bandit setting, the regret bounds become slightly looser, scaling with the cube root of the error, but they still remain sub-linear, demonstrating the effectiveness of the method even with minimal information.
3. Practical Extensions: The methodology isn't just theoretical. The paper discusses practical extensions and shows how the approach can be adapted for continuous learning and situations where the environment dynamics are unknown.

Implications and Future Directions

The optimistic approach to learning in AMDPs not only challenges the conservative nature of traditional AMDP methods but also opens up new possibilities for more effective learning in non-stationary environments. This could significantly impact various real-world applications, from autonomous vehicles adapting to varying traffic conditions to financial models adjusting to market dynamics.

Looking ahead, the next steps could involve exploring more complex environments where the cost dynamics are influenced by additional external factors, or integrating more advanced prediction models to further enhance the learning efficiency and accuracy.

In Conclusion

The shift from pessimistic to optimistic learning in AMDPs outlined in this paper highlights a significant advancement in reinforcement learning. By smartly incorporating predictions about future costs, the proposed method offers a promising avenue for developing more robust and adaptive learning algorithms that are not just theoretically sound but also practically viable in diverse and dynamic environments.