We present a statistical method that PAC learns the class of stochastic perceptrons with arbitrary monotonic activation func(cid:173) tion and weights Wi E {-I, 0, I} when the probability distribution that generates the input examples is member of a family that we call k-blocking distributions. Such distributions represent an impor(cid:173) tant step beyond the case where each input variable is statistically independent since the 2k-blocking family contains all the Markov distributions of order k. By stochastic percept ron we mean a per(cid:173) ceptron which, upon presentation of input vector x, outputs 1 with probability fCLJi WiXi - B). Because the same algorithm works for any monotonic (nondecreasing or nonincreasing) activation func(cid:173) tion f on Boolean domain, it handles the well studied cases of sigmolds and the "usual" radial basis functions.

It is well established that humans decision making and instrumental control uses multiple systems, some which use habitual action selection and some which require deliberate planning. Deliberate planning systems use predictions of action-outcomes using an internal model of the agent's environment, while habitual action selection systems learn to automate by repeating previously rewarded actions. Habitual control is computationally efficient but may be inflexible in changing environments. Conversely, deliberate planning may be computationally expensive, but flexible in dynamic environments. This paper proposes a general architecture comprising both control paradigms by introducing an arbitrator that controls which subsystem is used at any time. This system is implemented for a target-reaching task with a simulated two-joint robotic arm that comprises a supervised internal model and deep reinforcement learning. Through permutation of target-reaching conditions, we demonstrate that the proposed is capable of rapidly learning kinematics of the system without a priori knowledge, and is robust to (A) changing environmental reward and kinematics, and (B) occluded vision. The arbitrator model is compared to exclusive deliberate planning with the internal model and exclusive habitual control instances of the model. The results show how such a model can harness the benefits of both systems, using fast decisions in reliable circumstances while optimizing performance in changing environments. In addition, the proposed model learns very fast. Keywords: Machine Learning, Reinforcement Learning, Supervised Learning, Habitual controller, Planning, Internal Models, Decision Making 1. Introduction Much of the current reinforcement learning (RL) literature is in the domain of model-free control. Such a learning agent learns a value function from interacting with the environment, usually updating a proposed value function from a temporal difference between the previous expectation and a new experience [1, 2]. The value function is like a big lookup-table that can quickly supply evaluations for possible actions and hence provide guidance for actions in a fast and somewhat automated way. Such a decision system can be characterized as habitual. While habitual action selection takes time to learn and requires that similar previous situations have been encountered sufficiently, the advantage is that decisions and correspondingly actions can be generated in a timely manner. In contrast, a system that has some internal models of the environment can be used to derive a value function on demand for a specific situation.