Distribution-Dependent Sample Complexity of Large Margin Learning
This provides a foundational theoretical framework for comparing learning rules and improving methods like active learning, though it is incremental in refining existing sample complexity analysis.
The paper tackles the problem of characterizing the sample complexity of large-margin classification with L2 regularization by introducing the margin-adapted dimension, showing that it tightly governs distribution-specific upper and lower bounds, with the lower bounds proven for sub-Gaussian distributions with independent features.
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.