arXiv subjects

Marcus Hutter

Publications and source records attributed to Marcus Hutter.

At least 109 records · Page 6Linked to original sources

arXiv2013

Sparse Adaptive Dirichlet-Multinomial-like Processes

Online estimation and modelling of i.i.d. data for short sequences over large or complex "alphabets" is a ubiquitous (sub)problem in machine learning, information theory, data compression, statistical language processing, and document analysis. The Dirichlet-Multinomial distribution (also called Polya urn scheme) and extensions thereof are widely applied for online i.i.d. estimation. Good a-priori choices for the parameters in this regime are difficult to obtain though. I derive an optimal adaptive choice for the main parameter via tight, data-dependent redundancy bounds for a related model. The 1-line recommendation is to set the 'total mass' = 'precision' = 'concentration' parameter to m/2ln[(n+1)/m], where n is the (past) sample size and m the number of different symbols observed (so far). The resulting estimator (i) is simple, (ii) online, (iii) fast, (iv) performs well for all m, small, middle and large, (v) is independent of the base alphabet size, (vi) non-occurring symbols induce no redundancy, (vii) the constant sequence has constant redundancy, (viii) symbols that appear only finitely often have bounded/constant contribution to the redundancy, (ix) is competitive with (slow) Bayesian mixing over all sub-alphabets.

cs.IT↗

arXiv2012

Optimistic Agents are Asymptotically Optimal

We use optimism to introduce generic asymptotically optimal reinforcement learning agents. They achieve, with an arbitrary finite or compact class of environments, asymptotically optimal behavior. Furthermore, in the finite deterministic case we provide finite error bounds.

cs.AI↗

arXiv2012

Probabilities on Sentences in an Expressive Logic

Automated reasoning about uncertain knowledge has many applications. One difficulty when developing such systems is the lack of a completely satisfactory integration of logic and probability. We address this problem directly. Expressive languages like higher-order logic are ideally suited for representing and reasoning about structured knowledge. Uncertain knowledge can be modeled by using graded probabilities rather than binary truth-values. The main technical problem studied in this paper is the following: Given a set of sentences, each having some probability of being true, what probability should be ascribed to other (query) sentences? A natural wish-list, among others, is that the probability distribution (i) is consistent with the knowledge base, (ii) allows for a consistent inference procedure and in particular (iii) reduces to deductive logic in the limit of probabilities being 0 and 1, (iv) allows (Bayesian) inductive reasoning and (v) learning in the limit and in particular (vi) allows confirmation of universally quantified hypotheses/sentences. We translate this wish-list into technical requirements for a prior probability and show that probabilities satisfying all our criteria exist. We also give explicit constructions and several general characterizations of probabilities that satisfy some or all of the criteria and various (counter) examples. We also derive necessary and sufficient conditions for extending beliefs about finitely many sentences to suitable probabilities over all sentences, and in particular least dogmatic or least biased ones. We conclude with a brief outlook on how the developed theory might be used and approximated in autonomous reasoning agents. Our theory is a step towards a globally consistent and empirically satisfactory unification of probability and logic.

cs.LO↗

arXiv2012

Sparse Sequential Dirichlet Coding

This short paper describes a simple coding technique, Sparse Sequential Dirichlet Coding, for multi-alphabet memoryless sources. It is appropriate in situations where only a small, unknown subset of the possible alphabet symbols can be expected to occur in any particular data sequence. We provide a competitive analysis which shows that the performance of Sparse Sequential Dirichlet Coding will be close to that of a Sequential Dirichlet Coder that knows in advance the exact subset of occurring alphabet symbols. Empirically we show that our technique can perform similarly to the more computationally demanding Sequential Sub-Alphabet Estimator, while using less computational resources.

cs.IT↗

arXiv2012

One Decade of Universal Artificial Intelligence

The first decade of this century has seen the nascency of the first mathematical theory of general artificial intelligence. This theory of Universal Artificial Intelligence (UAI) has made significant contributions to many theoretical, philosophical, and practical AI questions. In a series of papers culminating in book (Hutter, 2005), an exciting sound and complete mathematical model for a super intelligent agent (AIXI) has been developed and rigorously analyzed. While nowadays most AI researchers avoid discussing intelligence, the award-winning PhD thesis (Legg, 2008) provided the philosophical embedding and investigated the UAI-based universal measure of rational intelligence, which is formal, objective and non-anthropocentric. Recently, effective approximations of AIXI have been derived and experimentally investigated in JAIR paper (Veness et al. 2011). This practical breakthrough has resulted in some impressive applications, finally muting earlier critique that UAI is only a theory. For the first time, without providing any domain knowledge, the same agent is able to self-adapt to a diverse range of interactive environments. For instance, AIXI is able to learn from scratch to play TicTacToe, Pacman, Kuhn Poker, and other games by trial and error, without even providing the rules of the games. These achievements give new hope that the grand goal of Artificial General Intelligence is not elusive. This article provides an informal overview of UAI in context. It attempts to gently introduce a very theoretical, formal, and mathematical subject, and discusses philosophical and technical ingredients, traits of intelligence, some social questions, and the past and future of UAI.

cs.AI↗

arXiv2012

Can Intelligence Explode?

The technological singularity refers to a hypothetical scenario in which technological advances virtually explode. The most popular scenario is the creation of super-intelligent algorithms that recursively create ever higher intelligences. It took many decades for these ideas to spread from science fiction to popular science magazines and finally to attract the attention of serious philosophers. David Chalmers' (JCS 2010) article is the first comprehensive philosophical analysis of the singularity in a respected philosophy journal. The motivation of my article is to augment Chalmers' and to discuss some issues not addressed by him, in particular what it could mean for intelligence to explode. In this course, I will (have to) provide a more careful treatment of what intelligence actually is, separate speed from intelligence explosion, compare what super-intelligent participants and classical human observers might experience and do, discuss immediate implications for the diversity and value of life, consider possible bounds on intelligence, and contemplate intelligences right at the singularity.

cs.AI↗

arXiv2012

PAC Bounds for Discounted MDPs

We study upper and lower bounds on the sample-complexity of learning near-optimal behaviour in finite-state discounted Markov Decision Processes (MDPs). For the upper bound we make the assumption that each action leads to at most two possible next-states and prove a new bound for a UCRL-style algorithm on the number of time-steps when it is not Probably Approximately Correct (PAC). The new lower bound strengthens previous work by being both more general (it applies to all policies) and tighter. The upper and lower bounds match up to logarithmic factors.

cs.LG↗

arXiv2012

A Bayesian View of the Poisson-Dirichlet Process

The two parameter Poisson-Dirichlet Process (PDP), a generalisation of the Dirichlet Process, is increasingly being used for probabilistic modelling in discrete areas such as language technology, bioinformatics, and image analysis. There is a rich literature about the PDP and its derivative distributions such as the Chinese Restaurant Process (CRP). This article reviews some of the basic theory and then the major results needed for Bayesian modelling of discrete problems including details of priors, posteriors and computation. The PDP allows one to build distributions over countable partitions. The PDP has two other remarkable properties: first it is partially conjugate to itself, which allows one to build hierarchies of PDPs, and second using a marginalised relative the CRP, one gets fragmentation and clustering properties that lets one layer partitions to build trees. This article presents the basic theory for understanding the notion of partitions and distributions over them, the PDP and the CRP, and the important properties of conjugacy, fragmentation and clustering, as well as some key related properties such as consistency and convergence. This article also presents a Bayesian interpretation of the Poisson-Dirichlet process based on an improper and infinite dimensional Dirichlet distribution. This means we can understand the process as just another Dirichlet and thus all its sampling properties emerge naturally. The theory of PDPs is usually presented for continuous distributions (more generally referred to as non-atomic distributions), however, when applied to discrete distributions its remarkable conjugacy property emerges. This context and basic results are also presented, as well as techniques for computing the second order Stirling numbers that occur in the posteriors for discrete distributions.

math.ST↗

arXiv2012

3D Model Assisted Image Segmentation

The problem of segmenting a given image into coherent regions is important in Computer Vision and many industrial applications require segmenting a known object into its components. Examples include identifying individual parts of a component for process control work in a manufacturing plant and identifying parts of a car from a photo for automatic damage detection. Unfortunately most of an object's parts of interest in such applications share the same pixel characteristics, having similar colour and texture. This makes segmenting the object into its components a non-trivial task for conventional image segmentation algorithms. In this paper, we propose a "Model Assisted Segmentation" method to tackle this problem. A 3D model of the object is registered over the given image by optimising a novel gradient based loss function. This registration obtains the full 3D pose from an image of the object. The image can have an arbitrary view of the object and is not limited to a particular set of views. The segmentation is subsequently performed using a level-set based method, using the projected contours of the registered 3D model as initialisation curves. The method is fully automatic and requires no user interaction. Also, the system does not require any prior training. We present our results on photographs of a real car.

cs.CV↗

arXiv2012

Adaptive Context Tree Weighting

We describe an adaptive context tree weighting (ACTW) algorithm, as an extension to the standard context tree weighting (CTW) algorithm. Unlike the standard CTW algorithm, which weights all observations equally regardless of the depth, ACTW gives increasing weight to more recent observations, aiming to improve performance in cases where the input sequence is from a non-stationary distribution. Data compression results show ACTW variants improving over CTW on merged files from standard compression benchmark tests while never being significantly worse on any individual file.

cs.IT↗

arXiv2011

Principles of Solomonoff Induction and AIXI

We identify principles characterizing Solomonoff Induction by demands on an agent's external behaviour. Key concepts are rationality, computability, indifference and time consistency. Furthermore, we discuss extensions to the full AI case to derive AIXI.

cs.AI↗

arXiv2011

No Free Lunch versus Occam's Razor in Supervised Learning

The No Free Lunch theorems are often used to argue that domain specific knowledge is required to design successful algorithms. We use algorithmic information theory to argue the case for a universal bias allowing an algorithm to succeed in all interesting problem domains. Additionally, we give a new algorithm for off-line classification, inspired by Solomonoff induction, with good performance on all structured problems under reasonable assumptions. This includes a proof of the efficacy of the well-known heuristic of randomly selecting training data in the hope of reducing misclassification rates.

cs.LG↗

arXiv2011

(Non-)Equivalence of Universal Priors

Ray Solomonoff invented the notion of universal induction featuring an aptly termed "universal" prior probability function over all possible computable environments. The essential property of this prior was its ability to dominate all other such priors. Later, Levin introduced another construction --- a mixture of all possible priors or `universal mixture'. These priors are well known to be equivalent up to multiplicative constants. Here, we seek to clarify further the relationships between these three characterisations of a universal prior (Solomonoff's, universal mixtures, and universally dominant priors). We see that the the constructions of Solomonoff and Levin define an identical class of priors, while the class of universally dominant priors is strictly larger. We provide some characterisation of the discrepancy.

cs.IT↗

arXiv2011

Context Tree Switching

This paper describes the Context Tree Switching technique, a modification of Context Tree Weighting for the prediction of binary, stationary, n-Markov sources. By modifying Context Tree Weighting's recursive weighting scheme, it is possible to mix over a strictly larger class of models without increasing the asymptotic time or space complexity of the original algorithm. We prove that this generalization preserves the desirable theoretical properties of Context Tree Weighting on stationary n-Markov sources, and show empirically that this new technique leads to consistent improvements over Context Tree Weighting as measured on the Calgary Corpus.

cs.IT↗

arXiv2011

Feature Reinforcement Learning In Practice

Following a recent surge in using history-based methods for resolving perceptual aliasing in reinforcement learning, we introduce an algorithm based on the feature reinforcement learning framework called PhiMDP. To create a practical algorithm we devise a stochastic search procedure for a class of context trees based on parallel tempering and a specialized proposal distribution. We provide the first empirical evaluation for PhiMDP. Our proposed algorithm achieves superior performance to the classical U-tree algorithm and the recent active-LZ algorithm, and is competitive with MC-AIXI-CTW that maintains a bayesian mixture over all context trees up to a chosen depth.We are encouraged by our ability to compete with this sophisticated method using an algorithm that simply picks one single model, and uses Q-learning on the corresponding MDP. Our PhiMDP algorithm is much simpler, yet consumes less time and memory. These results show promise for our future work on attacking more complex and larger problems.

cs.AI↗

arXiv2011

Axioms for Rational Reinforcement Learning

We provide a formal, simple and intuitive theory of rational decision making including sequential decisions that affect the environment. The theory has a geometric flavor, which makes the arguments easy to visualize and understand. Our theory is for complete decision makers, which means that they have a complete set of preferences. Our main result shows that a complete rational decision maker implicitly has a probabilistic model of the environment. We have a countable version of this result that brings light on the issue of countable vs finite additivity by showing how it depends on the geometry of the space which we have preferences over. This is achieved through fruitfully connecting rationality with the Hahn-Banach Theorem. The theory presented here can be viewed as a formalization and extension of the betting odds approach to probability of Ramsey and De Finetti.

cs.LG↗

arXiv2011

Time Consistent Discounting

A possibly immortal agent tries to maximise its summed discounted rewards over time, where discounting is used to avoid infinite utilities and encourage the agent to value current rewards more than future ones. Some commonly used discount functions lead to time-inconsistent behavior where the agent changes its plan over time. These inconsistencies can lead to very poor behavior. We generalise the usual discounted utility model to one where the discount function changes with the age of the agent. We then give a simple characterisation of time-(in)consistent discount functions and show the existence of a rational policy for an agent that knows its discount function is time-inconsistent.

cs.AI↗

arXiv2011

Universal Prediction of Selected Bits

Many learning tasks can be viewed as sequence prediction problems. For example, online classification can be converted to sequence prediction with the sequence being pairs of input/target data and where the goal is to correctly predict the target data given input data and previous input/target pairs. Solomonoff induction is known to solve the general sequence prediction problem, but only if the entire sequence is sampled from a computable distribution. In the case of classification and discriminative learning though, only the targets need be structured (given the inputs). We show that the normalised version of Solomonoff induction can still be used in this case, and more generally that it can detect any recursive sub-pattern (regularity) within an otherwise completely unstructured sequence. It is also shown that the unnormalised version can fail to predict very simple recursive sub-patterns.

cs.LG↗