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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.