Generalization Properties of Learning with Random Features (1602.04474v5)

Abstract: We study the generalization properties of ridge regression with random features in the statistical learning framework. We show for the first time that $O(1/\sqrt{n})$ learning bounds can be achieved with only $O(\sqrt{n}\log n)$ random features rather than $O({n})$ as suggested by previous results. Further, we prove faster learning rates and show that they might require more random features, unless they are sampled according to a possibly problem dependent distribution. Our results shed light on the statistical computational trade-offs in large scale kernelized learning, showing the potential effectiveness of random features in reducing the computational complexity while keeping optimal generalization properties.

• The paper demonstrates that learning with random features can achieve optimal statistical rates ($O(1 / √n)$) using significantly fewer features ($O( √n log n)$) than previously thought.
• It presents a detailed analytical framework for kernel ridge regression with random features, decomposing excess risk and employing rigorous analytical tools.
• The results suggest random features are a principled approach for scaling kernel methods, opening avenues for future research like extending analysis to other loss functions and exploring data-dependent features.

Generalization Properties of Learning with Random Features

The paper "Generalization Properties of Learning with Random Features" by Alessandro Rudi and Lorenzo Rosasco addresses the computational and statistical challenges associated with large-scale machine learning, particularly in the context of kernel methods. The paper is grounded in the use of random features to approximate the computation-heavy exact kernel computations in machine learning models, specifically employing ridge regression within the statistical learning framework.

Core Contributions

The primary contribution of the paper is demonstrating that random features can achieve optimal learning rates comparable to traditional kernel ridge regression, but with significantly reduced computational costs. The authors establish that an $O(1/\sqrt{n})$ learning bound can be attained using $O(\sqrt{n}\log n)$ random features, rather than the $O(n)$ random features suggested by previous works. This insight directly reflects the trade-offs between statistical accuracy and computational efficiency in machine learning using kernel methods.

The paper further explores conditions under which faster convergence rates can be achieved. It suggests that by appropriately sampling random features from a potentially problem-dependent distribution, one might achieve faster learning rates. The researchers provide analytical and probabilistic results that offer a sharp analysis of kernel ridge regression in the presence of random feature-based approximation, showcasing their potential in large-scale applications.

Detailed Analytical Framework

The paper presents a detailed framework for the analysis, beginning with the reformulation of kernel ridge regression from an empirical risk minimization perspective using random features. By introducing random features, the authors address the computational bottlenecks inherent in kernel methods, particularly for large datasets. The approximation $K(x, x') \approx \phi_M(x)^\top \phi_M(x')$, where $\phi_M(x)$ is a feature map sampled from a specific distribution, permits the use of linear methods for non-linear problems, thus reducing complexity.

The paper provides a rigorous decomposition of the excess risk of the learning algorithm into multiple components, evaluating the variance due to noisy outputs, the complexity due to the random sampling, and the approximation error inherent in substituting random features for the true kernel. The authors deploy a suite of analytical tools alongside concentration and operator inequalities to derive their results, ensuring the comprehensiveness of their theoretical contributions.

Theoretical Implications and Future Directions

From a theoretical standpoint, the authors’ results suggest that the use of random features is not merely a heuristic but a principled approach which maintains statistical efficiency while providing computational savings. The improvement over prior results, particularly the reduction in the number of needed random features, underscores the practical applicability of the methods presented.

The paper’s findings provoke several avenues for future research. One potential direction includes extending the analysis to other loss functions beyond the quadratic one used here, which could broaden the applicability of random features in different machine learning settings. The exploration of data-dependent random features, or leverage scores, presents another promising area for achieving even greater computational efficiency.

In summary, the paper effectively marries theoretical precision with practical applicability, offering robust foundations for the use of random features in scaling kernel methods to handle large datasets without sacrificing statistical properties. This work promises to impact both the computational efficiency and versatility of machine learning tools in real-world applications.