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.

The desirable gambles framework provides a foundational approach to imprecise probability theory but relies heavily on linear utility assumptions. This paper introduces {\em function-coherent gambles}, a generalization that accommodates non-linear utility while preserving essential rationality properties. We establish core axioms for function-coherence and prove a representation theorem that characterizes acceptable gambles through continuous linear functionals. The framework is then applied to analyze various forms of discounting in intertemporal choice, including hyperbolic, quasi-hyperbolic, scale-dependent, and state-dependent discounting. We demonstrate how these alternatives to constant-rate exponential discounting can be integrated within the function-coherent framework. This unified treatment provides theoretical foundations for modeling sophisticated patterns of time preference within the desirability paradigm, bridging a gap between normative theory and observed behavior in intertemporal decision-making under genuine uncertainty.

The aim of inverse reinforcement learning (IRL) is to infer an agent's preferences from observing their behaviour. Usually, preferences are modelled as a reward function, $R$, and behaviour is modelled as a policy, $\pi$. One of the central difficulties in IRL is that multiple preferences may lead to the same observed behaviour. That is, $R$ is typically underdetermined by $\pi$, which means that $R$ is only partially identifiable. Recent work has characterised the extent of this partial identifiability for different types of agents, including optimal and Boltzmann-rational agents. However, work so far has only considered agents that discount future reward exponentially: this is a serious limitation, especially given that extensive work in the behavioural sciences suggests that humans are better modelled as discounting hyperbolically. In this work, we newly characterise partial identifiability in IRL for agents with non-exponential discounting: our results are in particular relevant for hyperbolical discounting, but they also more generally apply to agents that use other types of (non-exponential) discounting. We significantly show that generally IRL is unable to infer enough information about $R$ to identify the correct optimal policy, which entails that IRL alone can be insufficient to adequately characterise the preferences of such agents.

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.

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.

Conditional value-at-risk (CVaR) precisely characterizes the influence that rare, catastrophic events can exert over decisions. Such characterizations are important for both normal decision-making and for psychiatric conditions such as anxiety disorders - especially for sequences of decisions that might ultimately lead to disaster. CVaR, like other well-founded risk measures, compounds in complex ways over such sequences - and we recently formalized three structurally different forms in which risk either averages out or multiplies. Unfortunately, existing cognitive tasks fail to discriminate these approaches well; here, we provide examples that highlight their unique characteristics, and make formal links to temporal discounting for the two of the approaches that are time consistent. These examples can ground future experiments with the broader aim of characterizing risk attitudes, especially for longer horizon problems and in psychopathological populations. Introduction Given the many uncertainties in the present and future, we had to evolve sophisticated ways of handling risk. Individual appetites or aversion for risk differ substantially, with various forms of psychopathology arising at the extremes of these preferences. Psychology and neuroscience have focused on single risky decisions (typically just one spin of the wheel of outrageous fortune). Historically, heuristics dominated [1]; however, recently, axiomatically justifiable forms of risk sensitivity from the finance industry are starting to permeate.

Assume an individual is represented by a multidimensional utility function that maps to the customer satisfaction domain. This function contains non-linear features and numerous feedback loops which may be negative, positive or either depending on market conditions. To illustrate, let's consider hyperbolic discounting, a well-established non-linear feature from behavior economics. As an exponential, small changes in the market interest rate can cause large changes in value perception. Each individual will have a different response ranging from almost none to dramatic changes in consumption and investment behavior. A change in interest rates could dramatically alter the cluster membership.

Reinforcement learning (RL) typically defines a discount factor as part of the Markov Decision Process. The discount factor values future rewards by an exponential scheme that leads to theoretical convergence guarantees of the Bellman equation. However, evidence from psychology, economics and neuroscience suggests that humans and animals instead have hyperbolic time-preferences. In this work we revisit the fundamentals of discounting in RL and bridge this disconnect by implementing an RL agent that acts via hyperbolic discounting. We demonstrate that a simple approach approximates hyperbolic discount functions while still using familiar temporal-difference learning techniques in RL. Additionally, and independent of hyperbolic discounting, we make a surprising discovery that simultaneously learning value functions over multiple time-horizons is an effective auxiliary task which often improves over a strong value-based RL agent, Rainbow.

An important use of machine learning is to learn what people value. What posts or photos should a user be shown? Which jobs or activities would a person find rewarding? In each case, observations of people's past choices can inform our inferences about their likes and preferences. If we assume that choices are approximately optimal according to some utility function, we can treat preference inference as Bayesian inverse planning. That is, given a prior on utility functions and some observed choices, we invert an optimal decision-making process to infer a posterior distribution on utility functions. However, people often deviate from approximate optimality. They have false beliefs, their planning is sub-optimal, and their choices may be temporally inconsistent due to hyperbolic discounting and other biases. We demonstrate how to incorporate these deviations into algorithms for preference inference by constructing generative models of planning for agents who are subject to false beliefs and time inconsistency. We explore the inferences these models make about preferences, beliefs, and biases. We present a behavioral experiment in which human subjects perform preference inference given the same observations of choices as our model. Results show that human subjects (like our model) explain choices in terms of systematic deviations from optimal behavior and suggest that they take such deviations into account when inferring preferences.