Free Energy and the Generalized Optimality Equations for Sequential Decision Making

The free energy functional has recently been proposed as a variational principle for bounded rational decision-making, since it instantiates a natural trade-off between utility gains and information processing costs that can be axiomatically derived. Here we apply the free energy principle to general decision trees that include both adversarial and stochastic environments. We derive generalized sequential optimality equations that not only include the Bellman optimality equations as a limit case, but also lead to well-known decision-rules such as Expectimax, Minimax and Expectiminimax. We show how these decision-rules can be derived from a single free energy principle that assigns a resource parameter to each node in the decision tree. These resource parameters express a concrete computational cost that can be measured as the amount of samples that are needed from the distribution that belongs to each node. The free energy principle therefore provides the normative basis for generalized optimality equations that account for both adversarial and stochastic environments.Comment: 10 pages, 2 figure

A Minimum Relative Entropy Principle for Learning and Acting

This paper proposes a method to construct an adaptive agent that is universal with respect to a given class of experts, where each expert is an agent that has been designed specifically for a particular environment. This adaptive control problem is formalized as the problem of minimizing the relative entropy of the adaptive agent from the expert that is most suitable for the unknown environment. If the agent is a passive observer, then the optimal solution is the well-known Bayesian predictor. However, if the agent is active, then its past actions need to be treated as causal interventions on the I/O stream rather than normal probability conditions. Here it is shown that the solution to this new variational problem is given by a stochastic controller called the Bayesian control rule, which implements adaptive behavior as a mixture of experts. Furthermore, it is shown that under mild assumptions, the Bayesian control rule converges to the control law of the most suitable expert.Comment: 36 pages, 11 figure

Abstraction in decision-makers with limited information processing capabilities

A distinctive property of human and animal intelligence is the ability to form abstractions by neglecting irrelevant information which allows to separate structure from noise. From an information theoretic point of view abstractions are desirable because they allow for very efficient information processing. In artificial systems abstractions are often implemented through computationally costly formations of groups or clusters. In this work we establish the relation between the free-energy framework for decision making and rate-distortion theory and demonstrate how the application of rate-distortion for decision-making leads to the emergence of abstractions. We argue that abstractions are induced due to a limit in information processing capacity.Comment: Presented at the NIPS 2013 Workshop on Planning with Information Constraint

A conversion between utility and information

Rewards typically express desirabilities or preferences over a set of alternatives. Here we propose that rewards can be defined for any probability distribution based on three desiderata, namely that rewards should be real-valued, additive and order-preserving, where the latter implies that more probable events should also be more desirable. Our main result states that rewards are then uniquely determined by the negative information content. To analyze stochastic processes, we define the utility of a realization as its reward rate. Under this interpretation, we show that the expected utility of a stochastic process is its negative entropy rate. Furthermore, we apply our results to analyze agent-environment interactions. We show that the expected utility that will actually be achieved by the agent is given by the negative cross-entropy from the input-output (I/O) distribution of the coupled interaction system and the agent's I/O distribution. Thus, our results allow for an information-theoretic interpretation of the notion of utility and the characterization of agent-environment interactions in terms of entropy dynamics.Comment: AGI-2010. 6 pages, 1 figur