Bad Universal Priors and Notions of Optimality

A big open question of algorithmic information theory is the choice of the universal Turing machine (UTM). For Kolmogorov complexity and Solomonoff induction we have invariance theorems: the choice of the UTM changes bounds only by a constant. For the universally intelligent agent AIXI (Hutter, 2005) no invariance theorem is known. Our results are entirely negative: we discuss cases in which unlucky or adversarial choices of the UTM cause AIXI to misbehave drastically. We show that Legg-Hutter intelligence and thus balanced Pareto optimality is entirely subjective, and that every policy is Pareto optimal in the class of all computable environments. This undermines all existing optimality properties for AIXI. While it may still serve as a gold standard for AI, our results imply that AIXI is a relative theory, dependent on the choice of the UTM.Comment: COLT 201

Universal Reinforcement Learning Algorithms: Survey and Experiments

Many state-of-the-art reinforcement learning (RL) algorithms typically assume that the environment is an ergodic Markov Decision Process (MDP). In contrast, the field of universal reinforcement learning (URL) is concerned with algorithms that make as few assumptions as possible about the environment. The universal Bayesian agent AIXI and a family of related URL algorithms have been developed in this setting. While numerous theoretical optimality results have been proven for these agents, there has been no empirical investigation of their behavior to date. We present a short and accessible survey of these URL algorithms under a unified notation and framework, along with results of some experiments that qualitatively illustrate some properties of the resulting policies, and their relative performance on partially-observable gridworld environments. We also present an open-source reference implementation of the algorithms which we hope will facilitate further understanding of, and experimentation with, these ideas.Comment: 8 pages, 6 figures, Twenty-sixth International Joint Conference on Artificial Intelligence (IJCAI-17

On the Computability of Solomonoff Induction and Knowledge-Seeking

Solomonoff induction is held as a gold standard for learning, but it is known to be incomputable. We quantify its incomputability by placing various flavors of Solomonoff's prior M in the arithmetical hierarchy. We also derive computability bounds for knowledge-seeking agents, and give a limit-computable weakly asymptotically optimal reinforcement learning agent.Comment: ALT 201

Fast DD-classification of functional data

A fast nonparametric procedure for classifying functional data is introduced. It consists of a two-step transformation of the original data plus a classifier operating on a low-dimensional hypercube. The functional data are first mapped into a finite-dimensional location-slope space and then transformed by a multivariate depth function into the $DD$-plot, which is a subset of the unit hypercube. This transformation yields a new notion of depth for functional data. Three alternative depth functions are employed for this, as well as two rules for the final classification on $[0,1]^q$. The resulting classifier has to be cross-validated over a small range of parameters only, which is restricted by a Vapnik-Cervonenkis bound. The entire methodology does not involve smoothing techniques, is completely nonparametric and allows to achieve Bayes optimality under standard distributional settings. It is robust, efficiently computable, and has been implemented in an R environment. Applicability of the new approach is demonstrated by simulations as well as a benchmark study

Extremal Mechanisms for Local Differential Privacy

Local differential privacy has recently surfaced as a strong measure of privacy in contexts where personal information remains private even from data analysts. Working in a setting where both the data providers and data analysts want to maximize the utility of statistical analyses performed on the released data, we study the fundamental trade-off between local differential privacy and utility. This trade-off is formulated as a constrained optimization problem: maximize utility subject to local differential privacy constraints. We introduce a combinatorial family of extremal privatization mechanisms, which we call staircase mechanisms, and show that it contains the optimal privatization mechanisms for a broad class of information theoretic utilities such as mutual information and $f$-divergences. We further prove that for any utility function and any privacy level, solving the privacy-utility maximization problem is equivalent to solving a finite-dimensional linear program, the outcome of which is the optimal staircase mechanism. However, solving this linear program can be computationally expensive since it has a number of variables that is exponential in the size of the alphabet the data lives in. To account for this, we show that two simple privatization mechanisms, the binary and randomized response mechanisms, are universally optimal in the low and high privacy regimes, and well approximate the intermediate regime.Comment: 52 pages, 10 figures in JMLR 201

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