Assessing Policy Performance in Approximate MDPs

Markov Decision Processes (MDPS) are mathematical models used to make decisions in situations where outcomes are partly random and partly under the control of a decision maker. These models are widely used in various fields, such as robotics, economics, and artificial intelligence, to determine the best course of action when dealing with uncertain environments.

A key challenge in working with MDPs is that often the exact model of the system is either unknown or too complex to use directly. In such cases, we rely on an approximate model. However, it is crucial to know how well the decisions derived from this approximate model will perform when applied to the original, true model.

#The Problem

This article explores the problem of designing a control policy in infinite-horizon discounted cost MDPs using only an approximate model. We are interested in understanding the performance of an optimal policy from the approximate model when it is implemented in the true model. In simpler terms, we want to know: if we find a good solution in a simpler version of the problem, how well does that solution work in the actual, more complicated problem?

#Existing Work

In the past, different methods have been proposed to address this question. Some researchers focused on simplifying the models into finite state approximations, while others developed techniques such as state aggregation and state discretization. Although these approaches have made important contributions, they have primarily dealt with MDPs that have bounded per-step costs.

Another line of research examined how changes in the model affect the optimal policy. If the models converge in some way, do the Optimal Policies derived from them also converge towards the optimal policy for the true model? This issue has received considerable attention, leading to a deeper understanding of the continuity of policies and Value Functions across varying model parameters.

In reinforcement learning, where the model is often unknown and needs to be learned from data, similar concepts emerge. Researchers study various approximations and metrics that help in making decisions when the exact model is not available.

#Key Concepts

1. Markov Decision Process (MDP): A framework for modeling decision-making situations where outcomes are determined by both random factors and the actions taken by the decision maker.

2. Optimal Policy: A strategy that specifies the best action to take in each state to minimize costs or maximize rewards over time within the MDP framework.

3. Approximate Model: A simplified version of the true model that is easier to work with, but may not capture all the nuances of the true system.

4. Value Function: A function that estimates the expected cost or reward of being in a given state and following a certain policy thereafter.

5. Weighted Norm: A method to measure the difference between functions, which is particularly useful when the cost is unbounded.

#Approach

Our approach involves deriving bounds that quantify how well the optimal policy from the approximate model performs in the original model. We start by considering two MDPs, one representing the true model and the other representing the approximate model.

Next, we derive bounds on the performance loss incurred by applying the optimal policy from the approximate model to the true model. By utilizing Weighted Norms, we can capture differences more effectively, especially in situations where costs can be unbounded.

#New Insights and Methodology

1. Bellman Operators: These are tools used to express the relationships between value functions in MDPs. We introduce new functionals, which we call Bellman mismatch functionals, to study the difference between the value functions of the original and Approximate Models.

2. Policy Stability: Stability conditions are crucial in ensuring that the policies derived from the approximate model can perform well in the true model. We relax common assumptions about stability to allow for a broader range of applicable situations.

3. Affine Transformations: By examining transformations of the cost structure, we can create tighter bounds on the performance of the policies. This flexibility allows us to better align the approximate model with the characteristics of the true model.

4. Examples and Applications: We provide practical examples that illustrate our findings. These include scenarios such as inventory management and linear quadratic regulation (LQR), showcasing situations where our bounds yield valuable insights.

#Inventory Management Example

Consider an inventory management system where we want to minimize costs related to holding stocks and fulfilling demand. We can define two models: one that represents the true cost structure and another that serves as an approximation.

Using our framework, we analyze the performance of the optimal policy derived from the approximate model when implemented within the true model. We demonstrate that our weighted-norm bounds provide tighter estimates of performance loss compared to classical methods.

#Linear Quadratic Regulation Example

In the context of control systems, consider an LQR problem where we aim to minimize costs related to system states and control actions. We construct both a true model and a simplified approximate model for analysis.

Through our methodology, we show how the derived bounds facilitate understanding of how the control solutions derived from the approximate model relate to optimal solutions in the true model. Even in the case of unbounded costs, our approach allows us to establish meaningful guarantees on performance.

#Conclusion

We have explored the challenges of designing policies in MDPs when only approximate models are available. By deriving bounds based on the relationships between the approximate and true models, we offer a deeper understanding of the performance of derived policies.

Through the introduction of new functional forms and stability conditions, we facilitate a more flexible and powerful framework for analyzing model approximations. The applicability of our approach spans various domains, from robotics to economics, offering valuable insights for decision-makers dealing with uncertainty and approximations.

As we move forward, further research can expand upon these findings, exploring more complex models and diverse applications. By continuing to refine our understanding of model approximation in MDPs, we pave the way for better decision-making strategies in uncertain environments.