Value Iteration for Long-run Average Reward in Markov Decision Processes

Markov decision processes (MDPs) are standard models for probabilistic systems with non-deterministic behaviours. Long-run average rewards provide a mathematically elegant formalism for expressing long term performance. Value iteration (VI) is one of the simplest and most efficient algorithmic approaches to MDPs with other properties, such as reachability objectives. Unfortunately, a naive extension of VI does not work for MDPs with long-run average rewards, as there is no known stopping criterion. In this work our contributions are threefold. (1) We refute a conjecture related to stopping criteria for MDPs with long-run average rewards. (2) We present two practical algorithms for MDPs with long-run average rewards based on VI. First, we show that a combination of applying VI locally for each maximal end-component (MEC) and VI for reachability objectives can provide approximation guarantees. Second, extending the above approach with a simulation-guided on-demand variant of VI, we present an anytime algorithm that is able to deal with very large models. (3) Finally, we present experimental results showing that our methods significantly outperform the standard approaches on several benchmarks

Examining average and discounted reward optimality criteria in reinforcement learning

In reinforcement learning (RL), the goal is to obtain an optimal policy, for which the optimality criterion is fundamentally important. Two major optimality criteria are average and discounted rewards, where the later is typically considered as an approximation to the former. While the discounted reward is more popular, it is problematic to apply in environments that have no natural notion of discounting. This motivates us to revisit a) the progression of optimality criteria in dynamic programming, b) justification for and complication of an artificial discount factor, and c) benefits of directly maximizing the average reward. Our contributions include a thorough examination of the relationship between average and discounted rewards, as well as a discussion of their pros and cons in RL. We emphasize that average-reward RL methods possess the ingredient and mechanism for developing the general discounting-free optimality criterion (Veinott, 1969) in RL.Comment: 14 pages, 3 figures, 10-page main conten

A Reinforcement Learning Model of Precommitment in Decision Making

Addiction and many other disorders are linked to impulsivity, where a suboptimal choice is preferred when it is immediately available. One solution to impulsivity is precommitment: constraining one's future to avoid being offered a suboptimal choice. A form of impulsivity can be measured experimentally by offering a choice between a smaller reward delivered sooner and a larger reward delivered later. Impulsive subjects are more likely to select the smaller-sooner choice; however, when offered an option to precommit, even impulsive subjects can precommit to the larger-later choice. To precommit or not is a decision between two conditions: (A) the original choice (smaller-sooner vs. larger-later), and (B) a new condition with only larger-later available. It has been observed that precommitment appears as a consequence of the preference reversal inherent in non-exponential delay-discounting. Here we show that most models of hyperbolic discounting cannot precommit, but a distributed model of hyperbolic discounting does precommit. Using this model, we find (1) faster discounters may be more or less likely than slow discounters to precommit, depending on the precommitment delay, (2) for a constant smaller-sooner vs. larger-later preference, a higher ratio of larger reward to smaller reward increases the probability of precommitment, and (3) precommitment is highly sensitive to the shape of the discount curve. These predictions imply that manipulations that alter the discount curve, such as diet or context, may qualitatively affect precommitment

Stochastic models of evidence accumulation in changing environments

Organisms and ecological groups accumulate evidence to make decisions. Classic experiments and theoretical studies have explored this process when the correct choice is fixed during each trial. However, we live in a constantly changing world. What effect does such impermanence have on classical results about decision making? To address this question we use sequential analysis to derive a tractable model of evidence accumulation when the correct option changes in time. Our analysis shows that ideal observers discount prior evidence at a rate determined by the volatility of the environment, and the dynamics of evidence accumulation is governed by the information gained over an average environmental epoch. A plausible neural implementation of an optimal observer in a changing environment shows that, in contrast to previous models, neural populations representing alternate choices are coupled through excitation. Our work builds a bridge between statistical decision making in volatile environments and stochastic nonlinear dynamics.Comment: 26 pages, 7 figure

Temporal-Difference Reinforcement Learning with Distributed Representations

Temporal-difference (TD) algorithms have been proposed as models of reinforcement learning (RL). We examine two issues of distributed representation in these TD algorithms: distributed representations of belief and distributed discounting factors. Distributed representation of belief allows the believed state of the world to distribute across sets of equivalent states. Distributed exponential discounting factors produce hyperbolic discounting in the behavior of the agent itself. We examine these issues in the context of a TD RL model in which state-belief is distributed over a set of exponentially-discounting “micro-Agents”, each of which has a separate discounting factor (γ). Each µAgent maintains an independent hypothesis about the state of the world, and a separate value-estimate of taking actions within that hypothesized state. The overall agent thus instantiates a flexible representation of an evolving world-state. As with other TD models, the value-error (δ) signal within the model matches dopamine signals recorded from animals in standard conditioning reward-paradigms. The distributed representation of belief provides an explanation for the decrease in dopamine at the conditioned stimulus seen in overtrained animals, for the differences between trace and delay conditioning, and for transient bursts of dopamine seen at movement initiation. Because each µAgent also includes its own exponential discounting factor, the overall agent shows hyperbolic discounting, consistent with behavioral experiments