What is the sample complexity of a learning algorithm?

1 Sample complexity

Sample complexity is the minimum number of data points that a learning algorithm needs to achieve a desired level of accuracy and confidence. Accuracy is the fraction of correct predictions that the algorithm makes on new data, and confidence is the probability that the accuracy is above a certain threshold. The sample complexity depends on the algorithm, the data distribution, and the learning task.

3 Factors influencing sample complexity

The sample complexity of a learning algorithm is determined by multiple factors, such as the complexity of the algorithm, the complexity of the data distribution, and the complexity of the learning task. The complexity of the algorithm can be measured by parameters, hypothesis space, or VC dimension. The complexity of the data distribution can be measured by entropy, dimensionality, or noise level. The complexity of the learning task can be measured by number of classes, difficulty in separating them, or loss function. All these complexities must be taken into account to avoid overfitting and to capture the variability and structure of data while achieving a good performance in a learning task.

4 Examples of sample complexity bounds

Deriving sample complexity bounds for different types of learning algorithms and problems can be done in a few ways. For example, when it comes to binary classification with a finite hypothesis space, the sample complexity is bounded by O(log(|H|)/epsilon), where H is the hypothesis space and epsilon is the maximum acceptable generalization error with high confidence. This bound is based on the Hoeffding inequality and the union bound. On the other hand, when it comes to binary classification with an infinite hypothesis space, the sample complexity is bounded by O(VC(H)/epsilon), where VC(H) is the VC dimension of the hypothesis space and epsilon is still the maximum acceptable generalization error with high confidence. This bound is based on the VC theory and the Sauer-Shelah lemma. Lastly, for regression with a linear model, the sample complexity is bounded by O(d/epsilon), where d is the dimensionality of the input space and epsilon is still the maximum acceptable mean squared error with high confidence. This bound is based on the Bernstein inequality and the Rademacher complexity.

5 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?