Convergence of Expected Utilities with Algorithmic Probability Distributions

CONVERGENCE OF EXPECTED UTILITIES WITH ALGORITHMIC PR OBABILITY DISTRIBUTIONS PETER DE BLANC DEP AR TMENT OF MA THEMA TICS TEMPLE UNIVERSITY Date : October 22, 2018. Thanks to Nick Ha y at Cornell University for help with the Recursion Theorem. 0 CONVERG ENCE OF EXP ECTED UTILITIES WITH ALGORITHMIC PROBAB ILITY DISTRIBUTIONS 1 1. Abstract W e consider an ag ent interacting with an unknown en vironment. The environ- men t is a function whic h maps natural num b ers to natural num b ers; the agent’s set of h yp otheses ab out the environment con tains all such functions which are com- putable and co mpatible with a finite set of known input-output pairs, and the ag e n t assigns a po sitive probability to each such hypothesis (P robability distributions over all computable functions are us e d in theoretical AI systems such a s AIXI (Hutter, 2007)). W e do not r equire tha t this probability distribution be computable, but it m ust b e bo unded b elow b y a positive computable function. The agent ha s a utilit y function on outputs from the en vironment. W e sho w that if this utilit y function is b ounded below in absolute v alue b y an unbo unded computable function, then the expec ted utilit y of any input is undefined. This implies that a co mputable ut ility function will have co nv ergent exp ected utilities iff that function is b o unded. 2. Not a tion Here we set up our notation. Let R b e the set of partial µ -recursive functions, a nd let S = R ∩ N N ; that is , S is the s e t of to ta l µ -recursive functions o n a sing le argument. W e will use an index R = { φ n } ∞ n =0 , wher e our index φ is a G¨ odel num b ering of the computable functions. Let h ∈ N N be the true function that describ es the environmen t. The ag e nt has some finite set of tested inputs I , s o the agent knows the v alue of h ( i ) for all i ∈ I . Let S I = { f ∈ S : ( ∀ i ∈ I ) , f ( i ) = h ( i ) } . Tha t is, S I is the s e t of hypotheses whic h agree with the ag ent’s knowledge of tested inputs. Let H be our s et o f hypotheses abo ut the en viro nmen t, with S I ⊆ H ⊆ N N . Let p : H → R b e our probability distr ibution o n the hypothesis set. W e r equire that there exist s o me co mputable function ¯ p : N → Q such tha t ( ∀ φ n ∈ S I ) , 0 < ¯ p ( n ) ≤ p ( φ n ). Let U : N → R be the a gent’s utilit y function. W e supp ose that U is unbounded, and that there exists an un b ounded computable function ¯ U : N → Q : ( ∀ n ∈ N ) , | ¯ U ( n ) | ≤ | U ( n ) | . 3. Proof of Diver gence of Expected Utilities Fix k ∈ ( N − I ), a n input whose expec ted utilit y we will co nsider. Then define B : N → N , with (1) ( ∀ x ∈ N ) , B ( x ) = max { φ n ( k ) : n ∈ N , n ≤ x } The s equence { B ( j ) } ∞ j =0 can be thought of as an ana log of the Busy Beav er nu mbers (Rad´ o, 1 9 62). This sequence will b e used in proving our main result. Note that B is not a computable function, as we are abo ut to show. Lemma 1. Le t f ∈ S . Then B ( x ) > f ( x ) infinitely often. Pr o of. Supp ose not. Then B ( x ) > f ( x ) only finitely man y times, s o let F ( x ) = 1 + f ( x ) + max { B ( x ) − f ( x ) : x ∈ N } . Then F ∈ S , and ( ∀ x ∈ N ) , F ( x ) > B ( x ). Let Q : N 2 → N , Q ( i, x ) = F ( i ). By a co rollar y of the r ecursion theorem (Kleene, 1938), there exists p ∈ N suc h that ( ∀ x ∈ N ) , φ p ( x ) = Q ( p, x ) = F ( p ). 2 PETER DE BLANC DEP AR TMENT OF M A THEMA TICS TEM PLE UNI VERSITY By equation 1, B ( p ) ≥ φ p ( k ) = F ( p ). This contradicts our statement that ( ∀ x ∈ N ) , F ( x ) > B ( x ).  The e x pe c ted utility of inputting k to the e nvironment is: (2) E U ( h ( k )) = X f ∈ H p ( f ) U ( f ( k )) Theorem 1 . The series in e quation 2 do es not c onver ge. Pr o of. T o es ta blish that this series does not conv erge, we will sho w that infinitely many of its terms hav e abs olute v alue ≥ 1. W e will do this by constructing a sequence of hypothes e s in S I whose utilit y grows very quickly - as quickly as the function B - faster than their proba bilities can shrink. By equation 1, for each j ∈ N there exists u j ∈ N , u j ≤ j such that φ u j ( k ) = B ( j ). Now we define a map o n function indices G : N → N s uch that: φ G ( n ) ( x ) =  h ( x ) if x ∈ I min { y ∈ N : | ¯ U ( y ) | ≥ | φ n ( k ) |} otherwise Because | ¯ U | is an unbounded function, φ G ( n ) is a total function a s long as φ n ( k ) is defined. By definition, ( ∀ n ∈ N ) , φ G ( n ) ∈ S I ⊆ H . So our sequence of h yp otheses will b e  φ G ( u j )  ∞ j =1 . Becaus e k / ∈ I , w e hav e: (3) ( ∀ j ∈ N ) , | U ( φ G ( u j ) ( k )) | ≥ | ¯ U ( φ G ( u j ) ( k )) | ≥ φ u j ( k ) = B ( j ) W e’re almost done. W e no w let q : N → N , with: (4) q ( x ) = ⌈ sup { 1 ¯ p ( G ( y )) : y ∈ N , y ≤ x }⌉ Then q is a nondecreasing function in S with the pro pe rty that ( ∀ x ∈ N ) , q ( x ) ≥ ¯ p ( G ( x )) − 1 . B y L emma 1, B ( x ) ≥ q ( x ) infinitely often. It follows fro m equations 3 and 4 that: (5) | U ( φ G ( u j ) ( k )) | ≥ B ( j ) ≥ q ( j ) ≥ q ( u j ) ≥ ¯ p ( G ( u j )) − 1 ≥ p ( φ G ( u j )) − 1 for infinitely many j ∈ N . Because p ( G ( u j )) is alw ays pos itive, a nd by the a bove equa tion, it follows that: (6) | p ( φ G ( u j )) U ( φ G ( u j ) ( k )) | ≥ 1 infinitely often. Because these ar e all terms o f the series in equa tion 2, the serie s can not con verge.  4. Remarks W e ha ve shown that utilit y functions which ar e b ounded below in absolute v alue by a n unbounded computable function do not hav e conv ergent exp ected utilities. Because a ll c o mputable functions are bo unded b elow in absolute v alue by them- selves, it follows that all unbounded computable utilit y functions hav e div ergent exp ected utilities. Since b ounded utility functions a lwa ys hav e conv ergent exp ected utilities, we k now that a c omputable utility function has c onver gent exp e ct e d utilities iff it is b ounde d. CONVERG ENCE OF EXP ECTED UTILITIES WITH ALGORITHMIC PROBAB ILITY DISTRIBUTIONS 3 References [1] Kleene, S., On Notation for Or dinal Numb ers , The Journal of Symbolic Logic, 3 (1938), 150-155. [2] Hutter, M ., Universal Algorithmic Intel ligenc e: A mathematic al top-down appr o ach , Artifi- cial General In telligence (2007 ), Springer [3] Rad´ o, T., On non-c omputable functions , Bell Systems T ech. J. 41, 3 (May 1962).