Distribution-Dependent Sample Complexity of Large Margin Learning

TL;DR

This paper introduces the margin-adapted dimension, a data-dependent measure that precisely characterizes the sample complexity of large-margin classifiers with L2 regularization, providing both upper and lower bounds.

Contribution

It defines the margin-adapted dimension based on second order data statistics and establishes tight distribution-specific bounds on sample complexity for large-margin learning.

Findings

The margin-adapted dimension governs sample complexity bounds.

Upper bounds are universal across distributions.

Lower bounds hold for sub-Gaussian distributions with independent features.

Abstract

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.