comp4620/8620: Advanced Topics in AI Foundations of Artificial Intelligence
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1 comp4620/8620: Advanced Topics in AI Foundations of Artificial Intelligence Marcus Hutter Australian National University Canberra, ACT, 0200, Australia ANU
2 Foundations of Artificial Intelligence Marcus Hutter Table of Contents 1. A SHORT TOUR THROUGH THE COURSE 2. INFORMATION THEORY & KOLMOGOROV COMPLEXITY 3. BAYESIAN PROBABILITY THEORY 4. ALGORITHMIC PROBABILITY & UNIVERSAL INDUCTION 5. MINIMUM DESCRIPTION LENGTH 6. THE UNIVERSAL SIMILARITY METRIC 7. BAYESIAN SEQUENCE PREDICTION 8. UNIVERSAL RATIONAL AGENTS 9. THEORY OF RATIONAL AGENTS 10. APPROXIMATIONS & APPLICATIONS 11. DISCUSSION
3 A Short Tour Through the Course Marcus Hutter 1 A SHORT TOUR THROUGH THE COURSE
4 A Short Tour Through the Course Marcus Hutter Informal Definition of (Artificial) Intelligence Intelligence measures an agent s ability to achieve goals in a wide range of environments. [S. Legg and M. Hutter] Emergent: Features such as the ability to learn and adapt, or to understand, are implicit in the above definition as these capacities enable an agent to succeed in a wide range of environments. The science of Artificial Intelligence is concerned with the construction of intelligent systems/artifacts/agents and their analysis. What next? Substantiate all terms above: agent, ability, utility, goal, success, learn, adapt, environment,... Never trust a theory === if it is not supported by an experiment ===== experiment theory
![Hutter] Emergent: Features such as the ability to learn and adapt, or to understand, are implicit in the above definition as these capacities enable an agent to succeed in a wide range](/docs-images/45/7511782/images/page_4.jpg "of environments. The science of Artificial Intelligence is concerned with the construction of intelligent systems/artifacts/agents and their analysis. What next?")
5 A Short Tour Through the Course Marcus Hutter Induction Prediction Decision Action Having or acquiring or learning or inducing a model of the environment an agent interacts with allows the agent to make predictions and utilize them in its decision process of finding a good next action. Induction infers general models from specific observations/facts/data, usually exhibiting regularities or properties or relations in the latter. Example Induction: Find a model of the world economy. Prediction: Use the model for predicting the future stock market. Decision: Decide whether to invest assets in stocks or bonds. Action: Trading large quantities of stocks influences the market.
6 A Short Tour Through the Course Marcus Hutter Science Induction Occam s Razor Grue Emerald Paradox: Hypothesis 1: All emeralds are green. Hypothesis 2: All emeralds found till y2020 are green, thereafter all emeralds are blue. Which hypothesis is more plausible? H1! Justification? Occam s razor: take simplest hypothesis consistent with data. is the most important principle in machine learning and science. Problem: How to quantify simplicity? Beauty? Elegance? Description Length! [The Grue problem goes much deeper. This is only half of the story]
7 A Short Tour Through the Course Marcus Hutter Information Theory & Kolmogorov Complexity Quantification/interpretation of Occam s razor: Shortest description of object is best explanation. Shortest program for a string on a Turing machine T leads to best extrapolation=prediction. K T (x) = min p {l(p) : T (p) = x} Prediction is best for a universal Turing machine U. Kolmogorov-complexity(x) = K(x) = K U (x) K T (x) + c T
8 A Short Tour Through the Course Marcus Hutter Bayesian Probability Theory Given (1): Models P (D H i ) for probability of observing data D, when H i is true. Given (2): Prior probability over hypotheses P (H i ). Goal: Posterior probability P (H i D) of H i, after having seen data D. Solution: Bayes rule: P (H i D) = P (D H i) P (H i ) i P (D H i) P (H i ) (1) Models P (D H i ) usually easy to describe (objective probabilities) (2) But Bayesian prob. theory does not tell us how to choose the prior P (H i ) (subjective probabilities)
9 A Short Tour Through the Course Marcus Hutter Algorithmic Probability Theory Epicurus: If more than one theory is consistent with the observations, keep all theories. uniform prior over all H i? Refinement with Occam s razor quantified in terms of Kolmogorov complexity: P (H i ) := 2 K T /U (H i ) Fixing T we have a complete theory for prediction. Problem: How to choose T. Choosing U we have a universal theory for prediction. Observation: Particular choice of U does not matter much. Problem: Incomputable.
10 A Short Tour Through the Course Marcus Hutter Inductive Inference & Universal Forecasting Solomonoff combined Occam, Epicurus, Bayes, and Turing into one formal theory of sequential prediction. M(x) = probability that a universal Turing machine outputs x when provided with fair coin flips on the input tape. A posteriori probability of y given x is M(y x) = M(xy)/M(x). Given ẋ 1,.., ẋ t 1, the probability of x t is M(x t ẋ 1...ẋ t 1 ). Immediate applications : - Weather forecasting: x t {sun,rain}. - Stock-market prediction: x t {bear,bull}. - Continuing number sequences in an IQ test: x t N. Optimal universal inductive reasoning system!
11 A Short Tour Through the Course Marcus Hutter The Minimum Description Length Principle Approximation of Solomonoff, since M is incomputable: M(x) 2 K U (x) (quite good) K U (x) K T (x) (very crude) Predict y of highest M(y x) is approximately same as MDL: Predict y of smallest K T (xy).
12 A Short Tour Through the Course Marcus Hutter Application: Universal Clustering Question: When is object x similar to object y? Universal solution: x similar to y x can be easily (re)constructed from y K(x y) := min{l(p) : U(p, y) = x} is small. Universal Similarity: Symmetrize&normalize K(x y). Normalized compression distance: Approximate K K U by K T. Practice: For T choose (de)compressor like lzw or gzip or bzip(2). Multiple objects similarity matrix similarity tree. Applications: Completely automatic reconstruction (a) of the evolutionary tree of 24 mammals based on complete mtdna, and (b) of the classification tree of 52 languages based on the declaration of human rights and (c) many others. [Cilibrasi&Vitanyi 05]
13 A Short Tour Through the Course Marcus Hutter Sequential Decision Theory Setup: For t = 1, 2, 3, 4,... Given sequence x 1, x 2,..., x t 1 (1) predict/make decision y t, (2) observe x t, (3) suffer loss Loss(x t, y t ), (4) t t + 1, goto (1) Goal: Minimize expected Loss. Greedy minimization of expected loss is optimal if: Important: Decision y t does not influence env. (future observations). Loss function is known. Problem: Expectation w.r.t. what? Solution: W.r.t. universal distribution M if true distr. is unknown.
14 A Short Tour Through the Course Marcus Hutter Example: Weather Forecasting Observation x t X = {sunny, rainy} Decision y t Y = {umbrella, sunglasses} Loss sunny rainy umbrella sunglasses Taking umbrella/sunglasses does not influence future weather (ignoring butterfly effect)
15 A Short Tour Through the Course Marcus Hutter Agent Model with Reward if actions/decisions a influence the environment q r 1 o 1 r 2 o 2 r 3 o 3 r 4 o 4 r 5 o 5 r 6 o 6... work Agent p tape... work Environment q tape... y 1 y 2 y 3 y 4 y 5 y 6...
16 A Short Tour Through the Course Marcus Hutter Rational Agents in Known Environment Setup: Known deterministic or probabilistic environment Fields: AI planning & sequential decision theory & control theory Greedy maximization of reward r (= Loss) no longer optimal. Example: Chess Exploration versus exploitation problem. Agent has to be farsighted. Optimal solution: Maximize future (expected) reward sum, called value. Problem: Things drastically change if environment is unknown
17 A Short Tour Through the Course Marcus Hutter Rational Agents in Unknown Environment Additional problem: (probabilistic) environment unknown. Fields: reinforcement learning and adaptive control theory Bayesian approach: Mixture distribution ξ. 1. What performance does Bayes-optimal policy imply? It does not necessarily imply self-optimization (Heaven&Hell example). 2. Computationally very hard problem. 3. Choice of horizon? Immortal agents are lazy. Universal Solomonoff mixture universal agent AIXI. Represents a formal (math., non-comp.) solution to the AI problem? Most (all AI?) problems are easily phrased within AIXI.
18 A Short Tour Through the Course Marcus Hutter Computational Issues: Universal Search Levin search: Fastest algorithm for inversion and optimization problems. Theoretical application: Assume somebody found a non-constructive proof of P=NP, then Levin-search is a polynomial time algorithm for every NP (complete) problem. Practical (OOPS) applications (J. Schmidhuber) Mazes, towers of hanoi, robotics,... FastPrg: The asymptotically fastest and shortest algorithm for all well-defined problems. Computable Approximations of AIXI: AIXItl and AIξ and MC-AIXI-CTW and ΦMDP. Human Knowledge Compression Prize: ( C=)
19 A Short Tour Through the Course Marcus Hutter Monte-Carlo AIXI Applications without providing any domain knowledge, the same agent is able to self-adapt to a diverse range of interactive environments. Normalised Average Reward per Cycle [VNH + 11] Experience Optimal Cheese Maze Tiger 4x4 Grid TicTacToe Biased RPS Kuhn Poker Pacman
![Normalised Average Reward per Cycle 1 0.8 0.6 0.4 0.2 0 [VNH + 11] www.youtube.com/watch?](/docs-images/45/7511782/images/page_19.jpg "v=yfsmhtmgdke Experience Optimal Cheese Maze Tiger 4x4 Grid TicTacToe Biased RPS Kuhn Poker Pacman 100")
20 A Short Tour Through the Course Marcus Hutter Discussion at End of Course What has been achieved? Made assumptions. General and personal remarks. Open problems. Philosophical issues.
21 A Short Tour Through the Course Marcus Hutter Exercises 1. [C10] What is the probability p that the sun will rise tomorrow, 2. [C15] Justify Laplace rule (p = n+1 n+2, where n= #days sun rose in past) 3. [C05] Predict sequences: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,? 3,1,4,1,5,9,2,6,5,3,?, 1,2,3,4,? 4. [C10] Argue in (1) and (3) for different continuations.
22 A Short Tour Through the Course Marcus Hutter Introductory Literature [HMU06] J. E. Hopcroft, R. Motwani, and J. D. Ullman. Introduction to Automata Theory, Language, and Computation. Addison-Wesley, 3rd edition, [RN10] [LV08] [SB98] S. J. Russell and P. Norvig. Artificial Intelligence. A Modern Approach. Prentice-Hall, Englewood Cliffs, NJ, 3rd edition, M. Li and P. M. B. Vitányi. An Introduction to Kolmogorov Complexity and its Applications. Springer, Berlin, 3rd edition, R. S. Sutton and A. G. Barto. Reinforcement Learning: An Introduction. MIT Press, Cambridge, MA, [Leg08] S. Legg. Machine Super Intelligence. PhD Thesis, Lugano, [Hut05] M. Hutter. Universal Artificial Intelligence: Sequential Decisions based on Algorithmic Probability. Springer, Berlin, See for more.
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