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Title: Hyperbolic Discounting and Learning Over Multiple Horizons

Paper content:
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.
Here we extend earlier work of Kurth-Nelson and Redish and propose an efficient deep reinforcement learning agent that acts via hyperbolic discounting and other non-exponential discount mechanisms.
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 state-of-the-art methods.
The standard treatment of the reinforcement learning (RL) problem is the Markov Decision Process (MDP) which includes a discount factor 0 ≤ γ ≤ 1 that exponentially reduces the present value of future rewards (Bellman, 1957; Sutton & Barto, 1998) .
A reward r t received in t-time steps is devalued to γ t r t , a discounted utility model introduced by Samuelson (1937) .
This establishes a timepreference for rewards realized sooner rather than later.
The decision to exponentially discount future rewards by γ leads to value functions that satisfy theoretical convergence properties (Bertsekas, 1995) .
The magnitude of γ also plays a role in stabilizing learning dynamics of RL algorithms (Prokhorov & Wunsch, 1997; Bertsekas & Tsitsiklis, 1996) and has recently been treated as a hyperparameter of the optimization (OpenAI, 2018; Xu et al., 2018) .
However, both the magnitude and the functional form of this discounting function establish priors over the solutions learned.
The magnitude of γ chosen establishes an effective horizon for the agent of 1/(1 − γ), far beyond which rewards are neglected (Kearns & Singh, 2002) .
This effectively imposes a time-scale of the environment, which may not be accurate.
Further, the exponential discounting of future rewards is consistent with a prior belief that there is a known constant per-time-step hazard rate (Sozou, 1998) or probability of dying of 1 − γ (Lattimore & Hutter, 2011).
Additionally, discounting future values exponentially and according to a single discount factor γ does not harmonize with the measured value preferences in humans 1 and animals (Mazur, 1985; Ainslie, 1992; Green & Myerson, 2004; Maia, 2009) .
A wealth of empirical evidence has been amassed that humans, monkeys, rats and pigeons instead discount future returns hyperbolically, where d k (t) = 1 1+kt , for some positive k > 0 (Ainslie, 1975; 1992; Mazur, 1985; Frederick et al., 2002; Green et al., 1981; Green & Myerson, 2004) .
This discrepancy between the time-preferences of animals from the exponential discounted measure of value might be presumed irrational.
But Sozou (1998) showed that hyperbolic time-preferences is mathematically consistent with the agent maintaining some uncertainty over the prior belief of the hazard rate in the environment.
Hazard rate h(t) measures the per-time-step risk the agent incurs as it acts in the environment due to a potential early death.
Precisely, if s(t) is the probability that the agent is alive at time t then the hazard rate is h(t) = − d dt lns(t).
We consider the case where there is a fixed, but potentially unknown hazard rate h(t) = λ ≥ 0.
The prior belief of the hazard rate p(λ) implies a specific discount function Sozou (1998) .
Under this formalism, the canonical case in RL of discounting future rewards according to d(t) = γ t is consistent with the belief that there exists a single hazard rate λ = e −γ known with certainty.
Further details are available in Appendix A. Figure 1: Hyperbolic versus exponential discounting.
Humans and animals often exhibit hyperbolic discounts (blue curve) which have shallower discount declines for large horizons.
In contrast, RL agents often optimize exponential discounts (orange curve) which drop at a constant rate regardless of how distant the return.
Common RL environments are also characterized by risk, but often in a narrower sense.
In deterministic environments like the original Arcade Learning Environment (ALE) (Bellemare et al., 2013) stochasticity is often introduced through techniques like no-ops (Mnih et al., 2015) and sticky actions (Machado et al., 2018) where the action execution is noisy.
Physics simulators may have noise and the randomness of the policy itself induces risk.
But even with these stochastic injections the risk to reward emerges in a more restricted sense.
In Section 2 we show that a prior distribution reflecting the uncertainty over the hazard rate, has an associated discount function in the sense that an MDP with either this hazard distribution or the discount function, has the same value function for all policies.
This equivalence implies that learning policies with a discount function can be interpreted as making them robust to the associated hazard distribution.
Thus, discounting serves as a tool to ensure that policies deployed in the real world perform well even under risks they were not trained under.
We propose an algorithm that approximates hyperbolic discounting while building on successful Qlearning (Watkins & Dayan, 1992) tools and their associated theoretical guarantees.
We show learning many Q-values, each discounting exponentially with a different discount factor γ, can be aggregated to approximate hyperbolic (and other non-exponential) discount factors.
We demonstrate the efficacy of our approximation scheme in our proposed Pathworld environment which is characterized both by an uncertain per-time-step risk to the agent.
Conceptually, Pathworld emulates a foraging environment where an agent must balance easily realizable, small meals versus more distant, fruitful meals.
We then consider higher-dimensional deep RL agents in the ALE, where we measure the benefits of hyperbolic discounting.
This approximation mirrors the work of Kurth-Nelson & Redish (2009); Redish & Kurth-Nelson (2010) which empirically demonstrates that modeling a finite set of µAgents simultaneously can approximate hyperbolic discounting function.
Our method then generalizes to other non-hyperbolic discount functions and uses deep neural networks to model the different Q-values from a shared representation.
Surprisingly and in addition to enabling new non-exponential discounting schemes, we observe that learning a set of Q-values is beneficial as an auxiliary task (Jaderberg et al., 2016) .
Adding this multi-horizon auxiliary task often improves over a state-of-the-art baseline, Rainbow (Hessel et al., 2018) in the ALE (Bellemare et al., 2013) .
This work questions the RL paradigm of learning policies through a single discount function which exponentially discounts future rewards through the following contributions:
1. Hazardous MDPs.
We formulate MDPs with hazard present and demonstrate an equivalence between undiscounted values learned under hazards and (potentially nonexponentially) discounted values without hazard.
2. Hyperbolic (and other non-exponential)-agent.
A practical approach for training an agent which discounts future rewards by a hyperbolic (or other non-exponential) discount function and acts according to this.
3. Multi-horizon auxiliary task.
A demonstration of multi-horizon learning over many γ simultaneously as an effective auxiliary task.

assistant: text: A deep RL agent that learns hyperbolic (and other non-exponential) Q-values and a new multi-horizon auxiliary task.