Commonly in reinforcement learning (RL), rewards are discounted over time using an exponential function to model time preference, thereby bounding the expected long-term reward. In contrast, in economics and psychology, it has been shown that humans often adopt a hyperbolic discounting scheme, which is optimal when a specific task termination time distribution is assumed. In this work, we propose a theory for continuous-time model-based reinforcement learning generalized to arbitrary discount functions. This formulation covers the case in which there is a non-exponential random termination time. We derive a Hamilton-Jacobi-Bellman (HJB) equation characterizing the optimal policy and describe how it can be solved using a collocation method, which uses deep learning for function approximation. Further, we show how the inverse RL problem can be approached, in which one tries to recover properties of the discount function given decision data. We validate the applicability of our proposed approach on two simulated problems. Our approach opens the way for the analysis of human discounting in sequential decision-making tasks.

Commonly in reinforcement learning (RL), rewards are discounted over time using an exponential function to model time preference, thereby bounding the expected long-term reward. In contrast, in economics and psychology, it has been shown that humans often adopt a hyperbolic discounting scheme, which is optimal when a specific task termination time distribution is assumed. In this work, we propose a theory for continuous-time model-based reinforcement learning generalized to arbitrary discount functions. This formulation covers the case in which there is a non-exponential random termination time. We derive a Hamilton–Jacobi–Bellman (HJB) equation characterizing the optimal policy and describe how it can be solved using a collocation method, which uses deep learning for function approximation.

Commonly in reinforcement learning (RL), rewards are discounted over time using an exponential function to model time preference, thereby bounding the expected long-term reward. In contrast, in economics and psychology, it has been shown that humans often adopt a hyperbolic discounting scheme, which is optimal when a specific task termination time distribution is assumed. In this work, we propose a theory for continuous-time model-based reinforcement learning generalized to arbitrary discount functions. This formulation covers the case in which there is a non-exponential random termination time. We derive a Hamilton–Jacobi–Bellman (HJB) equation characterizing the optimal policy and describe how it can be solved using a collocation method, which uses deep learning for function approximation.

The discounting mechanism in Reinforcement Learning determines the relative importance of future and present rewards. While exponential discounting is widely used in practice, non-exponential discounting methods that align with human behavior are often desirable for creating human-like agents. However, non-exponential discounting methods cannot be directly applied in modern on-policy actor-critic algorithms. To address this issue, we propose Universal Generalized Advantage Estimation (UGAE), which allows for the computation of GAE advantage values with arbitrary discounting. Additionally, we introduce Beta-weighted discounting, a continuous interpolation between exponential and hyperbolic discounting, to increase flexibility in choosing a discounting method. To showcase the utility of UGAE, we provide an analysis of the properties of various discounting methods. We also show experimentally that agents with non-exponential discounting trained via UGAE outperform variants trained with Monte Carlo advantage estimation. Through analysis of various discounting methods and experiments, we demonstrate the superior performance of UGAE with Beta-weighted discounting over the Monte Carlo baseline on standard RL benchmarks. UGAE is simple and easily integrated into any advantage-based algorithm as a replacement for the standard recursive GAE.