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

Contents 1.Resource-bounded dimension 2.Learning models 3.A few results on the size of learnable classes 4.Consequences Work in progress

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

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

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

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

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

Resource-bounded dimensions Restricting to effectively computable gales we have: –computable in polynomial time dim p –computable in quasi-polynomial time dim p 2 –computable in polynomial space dim pspace Each of this effective dimensions is “the right one” for a set of sequences (complexity class)

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

What for? We use dim p to estimate size of subclasses of E (and call it dimension in E) Important: Every set has a dimension Notice that dim p (X)<1 implies X  E Same for dim p 2 inside of EXP (dimension in EXP), etc I will also mention a dimension to be used inside PSPACE

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

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

Contents 1.Resource-bounded dimension 2.Learning models 3.A few results on the size of learnable classes 4.Consequences

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

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

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

Littlestone model (Online mistake-bound model) Let the concept be T  {0,1} n The learner receives a series of cases x 1, x 2,... from {0,1} n For each of them the learner guesses whether it belongs to T After guessing on case x i the learner receives the correct answer

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

PAC-learning A PAC-learner is a polynomial-time probabilistic algorithm A that given n, , and  produces a list of random membership 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?

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 A L (n,r, ,  ) with Pr( ||A L (n, r, ,  )  L =n || 1-  * r is the size of the representation of L =n

What can be PAC-learned AC 0 Everything can be PAC NP -learned Note: We are specially interested in learning parts of P/poly= languages that have a polynomial representation

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)

Contents 1.Resource-bounded dimension 2.Learning models 3.A few results on the size of learnable classes 4.Consequences

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

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

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

Littlestone Theorem For each a  1/2 every class that is Littlestone learnable with at most a2 n mistakes has dimension  H(a) H(a)= -a log a –(1-a) log(1-a) E =DTIME(2 O(n) )

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

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

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

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

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