An Idiosyncrasy of Time-discretization in Reinforcement Learning

TL;DR

This work considers the relationship between the definitions of the continuous-time and discrete-time returns and acknowledges an idiosyncrasy with naively applying a discrete-time algorithm to a discretized continuous-time environment, and notes how a simple modification can better align the return definitions.

Abstract

Many reinforcement learning algorithms are built on an assumption that an agent interacts with an environment over fixed-duration, discrete time steps. However, physical systems are continuous in time, requiring a choice of time-discretization granularity when digitally controlling them. Furthermore, such systems do not wait for decisions to be made before advancing the environment state, necessitating the study of how the choice of discretization may affect a reinforcement learning algorithm. In this work, we consider the relationship between the definitions of the continuous-time and discrete-time returns. Specifically, we acknowledge an idiosyncrasy with naively applying a discrete-time algorithm to a discretized continuous-time environment, and note how a simple modification can better align the return definitions. This observation is of practical consideration when dealing with environments where time-discretization granularity is a choice, or situations where such granularity is inherently stochastic.

Figures (6)

• Figure 1: The resulting sum when applying a discrete-time algorithm to a discretized continuous-time domain. Note how rectangle heights may fall out of the function's range within an interval.
• Figure 2: A visualization of the left-point and right-point Riemann sum approximation errors for an exponential decay. Due to curvature, a right-point Riemann sum will always have lower error.
• Figure 3: Numerical integration approximation error on discounted random signals. Results are averaged over $10^6$ signals and shaded regions represent one standard error.
• Figure 4: Numerical integration approximation error on discounted random signals, with stochastic discretization intervals. Results are averaged over $10^6$ signals and shaded regions represent one standard error.
• Figure 5: Numerical integration approximation error on undiscounted products of random signals. Results are averaged over $10^6$ signals and shaded regions represent one standard error.