Temporal plannability by variance of the episode length

TL;DR

This paper develops formulae for the average and variability of episode durations in stochastic decision processes, aiding planning and scheduling in highly variable environments.

Contribution

It introduces Bellman-type equations for episode duration statistics, extending Sobel's work, applicable to dynamic programming and reinforcement learning.

Findings

Derived formulas for mean and standard deviation of episode lengths.

Extended Sobel's Bellman equation to include duration variability.

Demonstrated principles through a toy problem simulation.

Abstract

Optimization of decision problems in stochastic environments is usually concerned with maximizing the probability of achieving the goal and minimizing the expected episode length. For interacting agents in time-critical applications, learning of the possibility of scheduling of subtasks (events) or the full task is an additional relevant issue. Besides, there exist highly stochastic problems where the actual trajectories show great variety from episode to episode, but completing the task takes almost the same amount of time. The identification of sub-problems of this nature may promote e.g., planning, scheduling and segmenting Markov decision processes. In this work, formulae for the average duration as well as the standard deviation of the duration of events are derived. The emerging Bellman-type equation is a simple extension of Sobel's work (1982). Methods of dynamic programming as well as methods of reinforcement learning can be applied for our extension. Computer demonstration on a toy problem serve to highlight the principle.