Resource bounded dimension and learning

1. Resource bounded dimension and learning Joint work with Ricard Gavaldà, María López-Valdés, and Vinodchandran N. Variyam Elvira Mayordomo, U. Zaragoza CIRM, 2009

2. Contents • Resource-bounded dimension • Learning models • A few results on the size of learnable classes • Consequences Work in progress

3. Effective dimension • Effective dimension is based in a characterization of Hausdorff dimension on  given by Lutz (2000) • The characterization is a very clever way to deal with a single covering using gambling

4. Hausdorff dimension in (Lutz characterization) Let s(0,1). An s-gale is such that It is the capital corresponding to a fixed strategy and a the house taking a fraction of d(w) is an s-gale iff ||(1-s)|w|d(w) is a martingale

5. Hausdorff dimension (Lutz characterization) • An s-gale d succeeds on x   if • limsupi d (x[0..i-1])=  • d succeeds on A   if d succeeds • on each x A • dimH(A) = inf {s | there is an s-gale that succeeds on A} The smaller the s the harder to succeed

6. Effectivizing Hausdorff dimension • We restrict to constructive or effective gales and get the corresponding “dimensions” that are meaningful in subsets of  we are interested in

7. Constructive dimension • If we restrict to constructive gales we get constructive dimension (dim) • The characterization you are used to: For each x  dim(x) = liminfn For each A dim(A)= supxA dim(x) K (x[1..n]) n log||

8. Resource-bounded dimensions • Restricting to effectively computable gales we have: • computable in polynomial time dimp • computable in quasi-polynomial time dimp2 • computable in polynomial space dimpspace • Each of this effective dimensions is “the right one” for a set of sequences (complexity class)

9. In Computational Complexity • A complexity class is a set of languages (a set of infinite sequences) P, NP, PSPACE E= DTIME (2n) EXP = DTIME (2p(n)) • dimp(E)= 1 • dimp2(EXP)= 1

10. What for? • We use dimp to estimate size of subclasses of E (and call it dimension in E) Important: Every set has a dimension Notice that dimp(X)<1 implies XE • Same for dimp2 inside of EXP (dimension in EXP), etc • I will also mention a dimension to be used inside PSPACE

11. My goal today • I will use resource-bounded dimension to estimate the size of interesting subclasses of E, EXP and PSPACE • If I show that X a subclass of E has dimension 0 (or dimension <1) in E this means: • X is quite smaller than E (most elements of E are outside of X) • It is easy to construct an element out of X (I can even combine this with other dim 0 properties) • Today I will be looking at learnable subclasses

12. My goal today • We want to use dimension to compare the power of different learning models • We also want to estimate the amount of languages that can be learned

13. Contents • Resource-bounded dimension • Learning models • A few results on the size of learnable classes • Consequences

14. Learning algorithms • The teacher has a finite set T with T{0,1}n in mind, the concept • The learner goal is to identify exactly T, by asking queries to the teacher or making guesses about T • The teacher is faithful but adversarial • The learner goal is to identify exactly T • Learner=algorithm, limited resources

15. Learning … • Learning algorithms are extensively used in practical applications • It is quite interesting as an alternative formalism for information content

16. Two learning models • Online mistake-bound model (Littlestone) • PAC- learning (Valiant)

17. Littlestone model (Online mistake-bound model) • Let the concept be T{0,1}n • The learner receives a series of cases x1, x2, ... from {0,1}n • For each of them the learner guesses whether it belongs to T • After guessing on case xi the learner receives the correct answer

18. Littlestone model • “Online mistake-bound model” • The following are restricted • The maximum number of mistakes • The time to guess case xi in terms of n and i

19. PAC-learning • A PAC-learner is a polynomial-time probabilistic algorithm A that given n, , and  produces a list of randommembership queries q1, …, qt to the concept T{0,1}n and from the answers it computes a hypothesis A(n, , ) that is “- close to the concept with probability 1- ” Membership query q: is q in the concept?

20. PAC-learning • An algorithm A PAC-learns a class C if • A is a probabilistic algorithm running in polynomial time • for every L in C and for every n, (T= L=n) • for every >0 and every >0 • A outputs a concept AL(n,r,,) with Pr( ||AL(n, r, , )  L=n||<  2n ) > 1-  * r is the size of the representation of L=n

21. What can be PAC-learned • AC0 • Everything can be PACNP-learned • Note: We are specially interested in learning parts of P/poly= languages that have a polynomial representation

22. Related work • Lindner, Schuler, and Watanabe (2000) study the size of PAC-learnable classes using resource-bounded measure • Hitchcock (2000) looked at the online mistake-bound model for a particular case (sublinear number of mistakes)

23. Contents • Resource-bounded dimension • Learning models • A few results on the size of learnable classes • Consequences

24. Our result Theorem If EXP≠MA then every PAC-learnable subclass of P/poly has dimension 0 in EXP In other words: If weak pseudorandom generators exist then every PAC-learnable class (with polynomial representations) has dimension 0 in EXP

25. Immediate consequences • From [Regan et al] If strong pseudorandom generators exist then P/poly has dimension 1 in EXP So under this hypothesis most of P/poly cannot be PAC-learned

26. Further results • Every class that can be PAC-learned with polylog space has dimension 0 in PSPACE

27. Littlestone Theorem For each a1/2 every class that is Littlestone learnable with at most a2n mistakes has dimension  H(a) H(a)= -a log a –(1-a) log(1-a) E =DTIME(2O(n))

28. Can we Littlestone-learn P/poly? • We mentioned From [Regan et al] If strong pseudorandom generators exist then P/poly has dimension 1 in EXP

29. Can we Littlestone-learn P/poly? If strong pseudorandom generators exist then (for every ) P/poly is not learnable with less than (1-)2n-1 mistakes in the Littlestone model

30. Both results • For every <1/2, a class that can be Littlestone-learned with at most 2n mistakes has dimension <1 in E • If weak pseudorandom generators exist then every PAC-learnable class (with polynomial representations) has dimension 0 in EXP

31. Comparison • It is not clear how to go from PAC to Littlestone (or vice versa) • We can go • from Equivalence queries to PAC • from Equivalence queries to Littlestone

32. Directions • Look at other models for exact learning (membership, equivalence). • Find quantitative results that separate them.