Is there a gap between weak learning and PAC-learning? For concreteness lets use the definitions of PAC and weak-learning as in the notes of Avrim Blum (http://www.cs.cmu.edu/~avrim/ML12/lect0208.txt) and also his notes on SQ-Learning (http://www.cs.cmu.edu/~avrim/ML12/lect0321.txt)

It seems to me that the following are true,


That what is difficult to PAC-learn is also difficult to weak learn
If a binary valued/classification concept class has a weak learner then AdaBoost can produce a PAC-learner for it. 


My questions are two fold,


But its not clear if one can boost a weak learner for an arbitrary concept class into a PAC learner for the same class. Or what is the best known statement in this direction? 
Why is SQDim presented as a measure of hardness of weak learning? Is that a limitation of theory that we cannot get hardness of PAC-learning from SQDim? (..except in the case of binary classification when AdaBoost will lift from weak to PAC..)
I think you're confusing two notions of hardness: computational and statistical. 

In Q1, you point out, essentially, that boosting produces an improper learner. I think what you're asking then, is: Is there a concept class which is (computationally) hard to learn properly but can be efficiently learned via boosting? I believe there should be such examples, but can't think of one off-hand.

In Q2, please note that SQDim refers to hardness in a statistical sense -- i.e., the minimal number of statistical queries required. (We don't have any unconditional computational hardness results but plenty of statistical ones.)