  We obtain a tight distribution-specific characterization of the sample
complexity of large-margin classification with L2 regularization: We introduce
the margin-adapted dimension, which is a simple function of the second order
statistics of the data distribution, and show distribution-specific upper and
lower bounds on the sample complexity, both governed by the margin-adapted
dimension of the data distribution. The upper bounds are universal, and the
lower bounds hold for the rich family of sub-Gaussian distributions with
independent features. We conclude that this new quantity tightly characterizes
the true sample complexity of large-margin classification. To prove the lower
bound, we develop several new tools of independent interest. These include new
connections between shattering and hardness of learning, new properties of
shattering with linear classifiers, and a new lower bound on the smallest
eigenvalue of a random Gram matrix generated by sub-Gaussian variables. Our
results can be used to quantitatively compare large margin learning to other
learning rules, and to improve the effectiveness of methods that use sample
complexity bounds, such as active learning.
