Nonparametric General Reinforcement Learning

Reinforcement learning problems are often phrased in terms of Markov decision processes (MDPs). In this thesis we go beyond MDPs and consider reinforcement learning in environments that are non-Markovian, non-ergodic and only partially observable. Our focus is not on practical algorithms, but rather on the fundamental underlying problems: How do we balance exploration and exploitation? How do we explore optimally? When is an agent optimal? We follow the nonparametric realizable paradigm: we assume the data is drawn from an unknown source that belongs to a known countable class of candidates. First, we consider the passive (sequence prediction) setting, learning from data that is not independent and identically distributed. We collect results from artificial intelligence, algorithmic information theory, and game theory and put them in a reinforcement learning context: they demonstrate how an agent can learn the value of its own policy. Next, we establish negative results on Bayesian reinforcement learning agents, in particular AIXI. We show that unlucky or adversarial choices of the prior cause the agent to misbehave drastically. Therefore Legg-Hutter intelligence and balanced Pareto optimality, which depend crucially on the choice of the prior, are entirely subjective. Moreover, in the class of all computable environments every policy is Pareto optimal. This undermines all existing optimality properties for AIXI. However, there are Bayesian approaches to general reinforcement learning that satisfy objective optimality guarantees: We prove that Thompson sampling is asymptotically optimal in stochastic environments in the sense that its value converges to the value of the optimal policy. We connect asymptotic optimality to regret given a recoverability assumption on the environment that allows the agent to recover from mistakes. Hence Thompson sampling achieves sublinear regret in these environments. AIXI is known to be incomputable. We quantify this using the arithmetical hierarchy, and establish upper and corresponding lower bounds for incomputability. Further, we show that AIXI is not limit computable, thus cannot be approximated using finite computation. However there are limit computable ε-optimal approximations to AIXI. We also derive computability bounds for knowledge-seeking agents, and give a limit computable weakly asymptotically optimal reinforcement learning agent. Finally, our results culminate in a formal solution to the grain of truth problem: A Bayesian agent acting in a multi-agent environment learns to predict the other agents' policies if its prior assigns positive probability to them (the prior contains a grain of truth). We construct a large but limit computable class containing a grain of truth and show that agents based on Thompson sampling over this class converge to play ε-Nash equilibria in arbitrary unknown computable multi-agent environments

Nonparametric General Reinforcement Learning

Reinforcement learning problems are often phrased in terms of Markov decision processes (MDPs). In this thesis we go beyond MDPs and consider reinforcement learning in environments that are non-Markovian, non-ergodic and only partially observable. Our focus is not on practical algorithms, but rather on the fundamental underlying problems: How do we balance exploration and exploitation? How do we explore optimally? When is an agent optimal? We follow the nonparametric realizable paradigm: we assume the data is drawn from an unknown source that belongs to a known countable class of candidates. First, we consider the passive (sequence prediction) setting, learning from data that is not independent and identically distributed. We collect results from artificial intelligence, algorithmic information theory, and game theory and put them in a reinforcement learning context: they demonstrate how an agent can learn the value of its own policy. Next, we establish negative results on Bayesian reinforcement learning agents, in particular AIXI. We show that unlucky or adversarial choices of the prior cause the agent to misbehave drastically. Therefore Legg-Hutter intelligence and balanced Pareto optimality, which depend crucially on the choice of the prior, are entirely subjective. Moreover, in the class of all computable environments every policy is Pareto optimal. This undermines all existing optimality properties for AIXI. However, there are Bayesian approaches to general reinforcement learning that satisfy objective optimality guarantees: We prove that Thompson sampling is asymptotically optimal in stochastic environments in the sense that its value converges to the value of the optimal policy. We connect asymptotic optimality to regret given a recoverability assumption on the environment that allows the agent to recover from mistakes. Hence Thompson sampling achieves sublinear regret in these environments. AIXI is known to be incomputable. We quantify this using the arithmetical hierarchy, and establish upper and corresponding lower bounds for incomputability. Further, we show that AIXI is not limit computable, thus cannot be approximated using finite computation. However there are limit computable ε-optimal approximations to AIXI. We also derive computability bounds for knowledge-seeking agents, and give a limit computable weakly asymptotically optimal reinforcement learning agent. Finally, our results culminate in a formal solution to the grain of truth problem: A Bayesian agent acting in a multi-agent environment learns to predict the other agents' policies if its prior assigns positive probability to them (the prior contains a grain of truth). We construct a large but limit computable class containing a grain of truth and show that agents based on Thompson sampling over this class converge to play ε-Nash equilibria in arbitrary unknown computable multi-agent environments