Resource bounded dimension and learning

Elvira Mayordomo, U. Zaragoza

CIRM, 2009

Joint work with Ricard Gavaldà, María López-Valdés, andVinodchandran 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 anda 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 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

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) = liminfn

For each A dim (A)= supxA dim (x)

K (x[1..n])

n log||

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)

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

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 XE

• Same for dimp2 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 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

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

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 AL(n,r,,) with

Pr( ||AL(n, r, , ) L=n||< 2n ) > 1-

* r is the size of the representation of L=n

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

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

TheoremIf 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

LittlestoneTheoremFor each a1/2 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))

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-)2n-1 mistakes in the Littlestone model

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

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.