A Theory of Universal AI

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1 A Theory of Universa AI Literature Marcus Hutter Kircherr, Li, and Vitanyi Presenter Prashant J. Doshi CS594: Optima Decision Maing A Theory of Universa Artificia Inteigence p.1/18

2 Roadmap Caim Bacground Key Concepts The AI Mode in Functiona Form The AI Mode in Recursive Form The Universa AI Mode Exampe Constants and Limits Appication Seuentia Decision Theory Concusions A Theory of Universa Artificia Inteigence p.2/18

3 Caim Deveopment of a universay optima AI mode Universa Optima faster parameteress, unbiased, mode-free No other program can earn or sove the tas A Theory of Universa Artificia Inteigence p.3/18

4 Bacground Decision Theory Soves the probem of rationa agent behaviour in uncertain ords given an environment Knon prior probabiity distribution over the environment Soomonoff s Universa Induction Soves the probem of seuence prediction for an unnon prior distribution Predict the continuation of a given binary seuence A Theory of Universa Artificia Inteigence p.4/18

5 . # < (. ; 98 ( Bacground Soomonoff s resut: is finite and expected Eucidean distance beteen (-, ' )+* ( $&% "! Convergence 43 )/1 2 is the Universa Probabiity Distribution ;, 9: is the Komogorov Compexity of $>= A Theory of Universa Artificia Inteigence p.5/18

6 O B D C B A?@ I N P O K L B A C B D Q Q@ S Key Concepts The Cybernetic or Agent Mode KML? J I EGFH is partia function / chronoogica Turing machine J I FH I ERFH is partia function / chronoogica Turing machine A Theory of Universa Artificia Inteigence p.6/18

7 e c b d c T X b e c b d c e d c c b X b b [ d c e d c e b b Key Concepts History : [ X a` [ T a` X]_^ ^ ^ T] X\ T\ X[ T[ YZ X U TVU Probabiity of input given the history X [f TG[f e Z X U T U X U T U ^ e [ ^^ T ` [ a` X U [ a` T U X ` T X U T U e Z X [f Tg[f T[ X[ T\ \ X U \ T U X\ ^ ^ ^ A Theory of Universa Artificia Inteigence p.7/18

8 i h u v o z n z z y # The AI Mode in Functiona Form Tas: Derive hich maximizes the tota credit over a predefined ifetime(t) For a non deterministic environment s ts r _npo m Optima poicy } n o 6 n { _npo xu v n For a prior distribution over environments be the set of a environments that ~4 Let produces the history ~ ~ ~ ~ ~ # ~ ~ ~ ~ ~ npo m ˆ r ~ ~ _n ƒ m A Theory of Universa Artificia Inteigence p.8/18

9 Ž v n ; yy v v n n y i h The AI Mode in Functiona Form AI maximizes expected future reard over the next (horizon) cyces Œg Š Z Optima poicy ~ ~ _n ƒ m xu ˆ 9 v n ~ v n ~ ~ ~ ~ ~ n ƒ xu ~ Best output are finite Š and, is computabe if A Theory of Universa Artificia Inteigence p.9/18

10 Œ Š Ž Š u v v u v u u ž The AI Mode in Recursive Form to Tas: Derive expected reard sum in cyces using expected reard sum in cyces to Œ ~ ~ ~ ~ v ƒ $ a ˆ ~ ~ ƒ m Optima expected reard ~ ~ ƒ m ~ ~ ƒ m ~ ~ ~ ~ ƒ $ ˆ ~ ~ ƒ output œ Vš ~ ~ ƒ m xu ~ Expectimax seuence xu ~ ž ~ ~ A Theory of Universa Artificia Inteigence p.1/18

11 Ÿ Ÿ c f The AI Mode in Recursive Form Functiona AI Recursive AI e cm e X [f TR[f G ªp«G A Theory of Universa Artificia Inteigence p.11/18

12 ; yy ; ; yy ; The Universa AI Mode Tas: Repace the true but unnon prior probabiity ith mode In the Functiona AI _npo m xu ~ ˆ ˆ 9 9 _npo m ;9 8 ( xu ~ ˆ ˆ 9 9 ˆ A Theory of Universa Artificia Inteigence p.12/18

13 u u ž u u ž The Universa AI Mode Tas: Repace the true but unnon prior probabiity ith mode In the Recursive AI ˆ xu ~ ž ~ ~ ˆ xu ~ ž ~ ~ A Theory of Universa Artificia Inteigence p.13/18

14 s. # Ž Z The Universa AI Mode Tas: Sho the convergence of AI to AI Utiize the Soomonoff s resut generaized from arbitrary aphabet are pure spectators = 2 to an (, ' )+* ( $t%! 3 )/ 2 Outputs of the AI of the AI mode converge to the outputs mode ateast for the bounded horizon T T Œ Š A Theory of Universa Artificia Inteigence p.14/18

15 d d Œ µ ³ X T ² ± ¾» º] º ] \ ¼ \ º[ ¹ ¹ ¹ ¹ Exampe Constants and Limits 1 1 ½ (a) The agents interface is ide (b) The interface can be sufficienty expored (c) The death is far aay (d) Most input/output combinations do not occur These imits are never used in proofs but eare ony interested in theorems hich do not degenerate under the above imits A Theory of Universa Artificia Inteigence p.15/18

16 vã Â vã vã Â n o À À Á Á Í Ù Û Ú Ê Ç Î Ð Ì Ë Í Ù Û É å â Ç Í Ù Û É å â Ù Û Ç Ü Ç Ç É Í Í Ò É Ò å â Ç ÒÇ ç Ç Ç å Appication Seuentia Decision Theory (MDP) Beman euation for optima poicy Ä À sá ˆ / Å / Ä À sá xu / v Appy the AI mode Ò É+Ù Ø Ç ÔÖÕ Ó ÍÏÎ ÉÊ ÆÈÇ Ò>Û Ò Î Ç Ñ Ò>Û á Íæ +ß É Ù Þ á +ß äã Í áéü à ÍÏÎ +ß É Ù Þ ÍÉÜ Ý Ò>Û ÒÛ Î á â è ã Íæ É Ù Þ Ù á Í áéç à ÍÏÎ É+Ù Þ ÍÉç Ý ÙG Î Ù A Theory of Universa Artificia Inteigence p.16/18

17 Appication Observations e use the compete history as the environment state The AI mode does not assume Marovian property stationary environment accessibe environment Other appications Game Paying Function Minimization Supervised Learning Bod Caim: AI is the most genera mode A Theory of Universa Artificia Inteigence p.17/18

18 Concusion A parameteress mode of AI based on Decision Theory and Agorithmic Probabiity is presented Maes minima assumptions about the environment Is the AI Future or mode computabe? Derive vaue and reard bounds for AI Appy AI mode to more probem casses mode A Theory of Universa Artificia Inteigence p.18/18

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