Value of Information and Reward Specification in Active Inference and POMDPs (2408.06542v1)

Abstract: Expected free energy (EFE) is a central quantity in active inference which has recently gained popularity due to its intuitive decomposition of the expected value of control into a pragmatic and an epistemic component. While numerous conjectures have been made to justify EFE as a decision making objective function, the most widely accepted is still its intuitiveness and resemblance to variational free energy in approximate Bayesian inference. In this work, we take a bottom up approach and ask: taking EFE as given, what's the resulting agent's optimality gap compared with a reward-driven reinforcement learning (RL) agent, which is well understood? By casting EFE under a particular class of belief MDP and using analysis tools from RL theory, we show that EFE approximates the Bayes optimal RL policy via information value. We discuss the implications for objective specification of active inference agents.

• The paper demonstrates that expected free energy combines pragmatic and epistemic values to approximate Bayes optimal policies in POMDPs.
• It reframes active inference within a belief MDP using the performance difference lemma and regret bounds to align with RL methods.
• The findings advance objective specification by balancing information gain with task rewards, improving deployment in uncertain environments.

Value of Information and Reward Specification in Active Inference and POMDPs

The paper by Ran Wei investigates the nuanced relationship between active inference and reinforcement learning (RL) within the framework of partially observable Markov decision processes (POMDPs). The primary focus is on understanding how the expected free energy (EFE), as an objective in active inference, approximates a reward-driven RL policy by examining the value of information and its implications for agent behavior.

Introduction to Active Inference and POMDPs

Active inference, derived from the free energy principle, models agent behavior as minimizing free energy, encapsulating a fit between the environment and the agent's internal model. This framework has found applications across cognitive and neural science, machine learning, and robotics, often within the context of decision-making in POMDPs. Unlike RL, where agents maximize expected reward, active inference agents minimize EFE, which decomposes into pragmatic and epistemic values. The pragmatic value corresponds to expected future outcomes aligning with preferred observations, while the epistemic value involves reducing uncertainty by prioritizing actions that lead to significant belief updates.

Unifying Active Inference and RL Under Belief MDPs

The paper translates the EFE objective into a belief MDP framework, akin to the Bellman equation, facilitating a comparison between EFE-optimized policies and RL policies. It demonstrates that EFE-optimal action sequences can be framed as belief-action policies within a class of belief MDPs. This equivalence underscores the potential for using policy optimization techniques from RL within active inference.

Analysis Tools and Performance Gaps

The analysis leverages the performance difference lemma and simulation lemma from RL theory to quantify the gap between EFE-based policies and Bayes optimal policies. The key insight is that the epistemic value in EFE compensates for the exploration-exploitation trade-off inherent in Bayes optimal policies. The regret bound derived in the paper shows that the performance gap related to the Bayes optimal policy is significantly reduced by incorporating epistemic value into the reward function.

Value of Information in POMDPs

The value of information (VOI) theory articulated by Howard is extended to POMDPs, defining the expected value of perfect observation (EVPO) as the differential gain from utilizing information over naive policies that disregard future belief updates. The analysis demonstrates that EVPO is inherently non-negative since optimal policies leverage future observations to mitigate uncertainty effectively.

Main Results: EFE Approximates Bayes Optimal RL Policy

The paper's central thesis is substantiated by showing that EFE, by incorporating epistemic value, approximates the Bayes optimal policy. This is particularly salient in environments where state information is inherently partial and where the trade-off between information gain and pragmatic rewards is crucial. The regret bound reveals that epistemic value significantly closes the gap between the naive open-loop policy and Bayes optimal policy, with a linear adjustment due to the value of information term.

Objective Specification in Active Inference

The implications for specifying objectives in active inference are profound. The balance between pragmatic outcomes and epistemic value, often adjusted via a temperature parameter, ensures that the agent does not overly prioritize information gain at the expense of task-related rewards. This balance is essential for deploying active inference agents in real-world environments where goal achievement and information-seeking must be finely tuned.

Conclusion

This paper enriches the theoretical foundation of active inference by elucidating its approximation to Bayes optimal policies in POMDPs, facilitated by the inclusion of epistemic value. The findings encourage a nuanced approach to setting objectives in active inference, promoting a balanced integration of reward maximization and uncertainty reduction. These insights pave the way for more robust and effective applications of active inference in complex and partially observable environments.

Future Directions

Theoretical developments highlighted in this work open the door to further empirical validations and enhancements to active inference frameworks. Future research could focus on optimizing the temperature parameter dynamically and exploring more complex belief MDPs that continuously balance information gain and task-specific rewards. Additionally, integrating these insights into practical applications in robotics and adaptive systems could yield significant advancements in autonomous agent performance.