Two betting strategies that predict all compressible sequences

Size: px

Start display at page:

Download "Two betting strategies that predict all compressible sequences"

Madison Carr
8 years ago
Views:

1 Two betting strategies that predict all compressible sequences T. Petrović Seventh International Conference on Computability, Complexity and Randomness (CCR 2012) T. Petrović Predicting compressible sequences CCR / 17

com Seventh International Conference on Computability, Complexity

2 Outline 1 Betting Games Definition of Sequence-set Betting Comparison with Martingale Processes Comparison with Non-monotonic Betting 2 Algorithm that Constructs Two Betting Strategies T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

Betting 2 Algorithm that Constructs Two Betting Strategies T.

3 Betting Games Sequence-set Betting Definition of Sequence-set Betting (using mass-distribution) Start with(s, m) Betting decision (S 0,m 0 ), (S 1,m 1 ) m 0 + m 1 = m λ(s 0 ) = λ(s 1 ) = 1 2 λ(s) Sequence in S 0, next is(s 0,m 0 ) otherwise start with (S 1,m 1 ) Success: capital c = m/λ(s) rises unboundedly Decision tree descrbes betting strategy T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

0, next is(s 0,m 0 ) otherwise start with (S 1,m 1 ) Success: capital c = m/λ(s) rises unboundedly Decision

4 Betting Games Sequence-set Betting Definition of Sequence-set Betting (using mass-distribution) Start with(s, m) Betting decision (S 0,m 0 ), (S 1,m 1 ) m 0 + m 1 = m λ(s 0 ) = λ(s 1 ) = 1 2 λ(s) Sequence in S 0, next is(s 0,m 0 ) otherwise start with (S 1,m 1 ) Success: capital c = m/λ(s) rises unboundedly Decision tree descrbes betting strategy T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

5 Betting Games Comparison with Martingale Processes Martingale Processes Function d from words to reals Equivalence relation on words d v d w if for v v, w w, l(v ) = l(w ) we have d(v ) = d(w ) Fairness condition: 2 d(v) {v:v d w} = [d(v0) + d(v1)] {v:v d w} T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

l(w ) we have d(v ) = d(w ) Fairness condition: 2 d(v) {v:v d w} = [d(v0) + d(v1)]

6 Betting Games Comparison with Martingale Processes Modified Sequence-set Betting Replace λ(s 0 ) = λ(s 1 ) = 1 2 λ(s) with λ(s 0 ) + λ(s 1 ) = λ(s) Betting game equivalent to martingale processes T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

) + λ(s 1 ) = λ(s) Betting game equivalent to martingale processes T.

7 Betting Games Comparison with Martingale Processes Betting Strategy from Martingale Process (S,m) (V,d), V = {v : v d w} S α V α and m = d(w)λ(v ) Find v s.t. v v, d(v ) d(v) d divides V into V 1,...,V n Make according betting decisions No such v, make no further bets T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

8 Betting Games Comparison with Martingale Processes Martingale Process from Betting Strategy (S,m) (V,d), V = {v : v d w} S α V α and m = d(w)λ(v ) Betting decision (S 0,m 0 ), (S 1,m 1 ) Find n n > l(w), n max(l(w) : w S 0 S 1 } Set d(v ) = m 0 /λ(s 0 ) for S 0 v and d(v ) = m 1 /λ(s 1 ) for S 1 v where l(v ) = n No betting decision for (S,m) Set d(v ) = d(w) for V v T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

1 } Set d(v ) = m 0 /λ(s 0 ) for S 0 v and d(v ) = m 1 /λ(s 1 ) for S 1 v where l(v ) = n No betting decision

9 Betting Games Comparison with Non-monotonic Betting Comparison with Non-monotonic Betting Add to req. λ(s 0 ) = λ(s 1 ) = 1 2 λ(s) req. that for some n w S 0 w(n) = 0 w S 1 w(n) = 1 Betting decision (S 0,m 0 ), (S 1,m 1 ) places a bet on bit value at position n Next iteration choses different position since previously picked positions have all 0 s or all 1 s T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

that for some n w S 0 w(n) = 0 w S 1 w(n) = 1 Betting decision (S 0,m 0 ), (S 1,m 1 ) places a bet on bit

10 Betting Games Comparison with Non-monotonic Betting Unpredictable Compressible Sequence Step1:For node (S,m) wait for a bet No betting decision, strategy fails Choose node with less mass If node small enough, compress all go to Step1 T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

fails Choose node with less mass If node small enough, compress all go to

11 Algorithm that Constructs Two Betting Strategies Algorithm Outline Algorithm constructs betting decision trees for strategies A and B The inputs are some word s and masses ma, mb assigned to that prefix by A and B Calculates k from (s,ma,mb) Runs UTM to enumerate first prefixes p that have inputs shorter by k For each p adds betting decisions, nodes that contain p, p p contain only one word, same in A and B, for these start a new instance of algorithm If the set of prefixes is small, betting decisions such that either A or B double capital on that prefixes T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

adds betting decisions, nodes that contain p, p p contain only one word, same in A and B, for these start a new instance of algorithm If the set of prefixes is

12 Algorithm that Constructs Two Betting Strategies Notation Only the leaf nodes of b.d.t. that don t contain sequences on which another instance of algorithm was started are considered For each node (a i,s,m), (b j,s,m) two additional values are used Mass reserved to ensure that no node has mass 0, me The portion of size of the so far found prefixes belonging to the node L Mass ms is used for doubling the capital on compressible prefixes, mass assigned to node in b.d.t. m = ms + me Denote indexes of considered leaf nodes of A a 1,...,a x, of B b 1,...,b y (a i,s,ms,me,l) S a i,ms a i,me a i,l a i (b j,s,ms,me,l) S bj,ms bj,me bj,l bj T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

each node (a i,s,m), (b j,s,m) two additional values are used Mass reserved to ensure that no node has mass 0, me The portion of size of the so far found prefixes belonging to the

13 Algorithm that Constructs Two Betting Strategies Initialization The input for the algorithm instance is (s,ma,mb) Set mass used for doubling the capital m = min(ma, mb)/2 Set for A S a 1 = {s}, ms a 1 = m, me a 1 = ma m, L a 1 = 0 for B S b1 = {s}, ms b1 = m, me b1 = mb m, L b1 = 0 The capital for compressible prefixes will be c = 4(ma + mb)/λ(s) set k such that 2 k < m 2 (1 cs)/2λ(s)c 2 (1 + cs) 2 The cs is some constant 0 < cs < 1, in all iterations S a i S bj = {} or λ(s)λ(s a i S bj ) is between (1 ± cs)(λ(s a i ) + L a i )(λ(s bj ) + L bj ) The construction of b.d.t. starts by running the algorithm for (ε,1,1) T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

set k such that 2 k < m 2 (1 cs)/2λ(s)c 2 (1 + cs) 2 The cs is some constant 0 < cs < 1, in all iterations S a i S bj = {} or λ(s)λ(s a i S bj ) is between (1 ± cs)(λ(s a i ) +

14 Algorithm that Constructs Two Betting Strategies Preparation Step Find next p, S a i S bj p {} If for all a i,b j S a i S bj p = {} or S a i S bj \ p = {} skip rest For nodes a i,b j make n betting decisions, 2 n new leaf nodes a ig,b jh Evenly distribute mass and L Extend words not in intersection by n, distribute them ammong leaf nodes Extend words in intersection and distribute them to have S a ig S bjh p = {} or S a ig S bjh \ p = {} If λ(s a i S bj ) = r(λ(s a i ) + L a i )(λ(s bj ) + L bj ) then λ(s a ig S bjh ) = r(λ(s a ig ) + L a ig )(λ(s bjh ) + L bjh ) T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

$ammong leaf nodes Extend words in intersection and distribute them to have S a ig S bjh p = {} or S a ig S bjh \ p = {} If λ(s a i S bj ) = r(λ(s a i ) + L a i$

15 Algorithm that Constructs Two Betting Strategies Preparation Step Find next p, S a i S bj p {} If for all a i,b j S a i S bj p = {} or S a i S bj \ p = {} skip rest For nodes a i,b j make n betting decisions, 2 n new leaf nodes a ig,b jh Evenly distribute mass and L Extend words not in intersection by n, distribute them ammong leaf nodes Extend words in intersection and distribute them to have S a ig S bjh p = {} or S a ig S bjh \ p = {} If λ(s a i S bj ) = r(λ(s a i ) + L a i )(λ(s bj ) + L bj ) then λ(s a ig S bjh ) = r(λ(s a ig ) + L a ig )(λ(s bjh ) + L bjh ) T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

$ammong leaf nodes Extend words in intersection and distribute them to have S a ig S bjh p = {} or S a ig S bjh \ p = {} If λ(s a i S bj ) = r(λ(s a i ) + L a i$

16 Algorithm that Constructs Two Betting Strategies Mass Assignment Step Pick S ai Sbj, S ai Sbj \ p = {}, assign mass d ai, dbj from ms ai, msbj d ai + dbj = cλ (S ai Sbj ) If (ms ai cλ (S ai Sbj ))/(λ (S ai ) + Lai ) msbj /(λ (Sbj ) + Lbj ) then dbj = 0 If (msbj cλ (S ai Sbj ))/(λ (Sbj ) + Lbj ) ms ai /(λ (S ai ) + Lai ) then d ai = 0 otherwise find (ms ai d ai )/(λ (S ai ) + Lai ) = (msbj dbj )/(λ (Sbj ) + Lbj ) T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

dbj = 0 If (msbj cλ (S ai Sbj ))/(λ (Sbj ) + Lbj ) ms ai /(λ (S ai ) + Lai ) then d ai = 0 otherwise find (ms ai d ai )/(λ

17 Algorithm that Constructs Two Betting Strategies Betting Decisions Step Add the size of all words of a node that are extensions of p to L For nodes that contain words p 0 s.t. p p 0 add betting decisions: Leaf nodes with extensions of p contain only one word, assign mass from previous step, start another instance of the algorithm Distribute words unrelated to p ammong remaining leaf nodes. Distribute L and the remainder of mass evenly ammong the remaining leaf nodes. For the remaining nodes we have S ai Sbj = {} or λ (s)λ (S ai Sbj ) between (1 ± cs)(λ (S ai ) + Lai )(λ (Sbj ) + Lbj ) T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

p p 0 add betting decisions: Leaf nodes with extensions of p contain only one word, assign mass from previous step, start another instance of the algorithm Distribute

18 Algorithm that Constructs Two Betting Strategies Proof Sketch ms a i = ms a i ms a i f = λ(s) m msa i /(λ(s a i ) + L a i ) f = λ(s) m ms a i /(λ(s a i ) + L a i ) ms a i = m λ(s) (f f )(λ(s a i ) + L a i ) ms a i cλ(s a i S bj ) λ(s a i S bj ) is close to 1 λ(s) (λ(s a i ) + L a i )(λ(s bj ) + L bj ) g = λ(s) m ms b j /(λ(s bj ) + L bj ) cl bj (1 g) m λ(s) (λ(s b j ) + L bj ) Assume that g f L bj const.m 2 (1 f )(f f ) z 2 l(p i ) const.m 2 /2 2 k i=1 T. Petrović (tomeepx@gmail.com) Predicting compressible sequences CCR / 17

a i ) + L a i )(λ(s bj ) + L bj ) g = λ(s) m ms b j /(λ(s bj ) + L bj ) cl bj (1 g) m λ(s) (λ(s b j ) + L bj ) Assume that g f L bj

19 Appendix References M. Li, P. Vitanyi An Introduction to Kolmogorov Complexity and Its Applications, second edition Springer, J.M.Hitchcock, J.H. Lutz Why Computational Complexity Requires Stricter Martingales Theory of Computing Systems, W. Merkle, N. Mihailović, T.A. Slaman Some results on effective randomness Theory of Computing Systems, T. Petrović Predicting compressible sequences CCR / 17

Hitchcock, J.H. Lutz Why Computational Complexity Requires Stricter Martingales Theory of Computing Systems, 2006.

The Goldberg Rao Algorithm for the Maximum Flow Problem

The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }