In this study, we explore the generalization error analysis of a popular machine learning algorithm called Empirical Risk Minimization (ERM) for non-linear functional regression. We use a new functional net to measure the generalization capability of ERM and analyze how different factors affect its performance.
Section 1: Context and Background
In functional data analysis, we often encounter non-linear relationships between variables, which are difficult to model using traditional linear methods. To address this challenge, we turn to functional regression, which aims to find the best predictor function for a given response function. ERM is a widely used algorithm in functional regression that minimizes the empirical risk on the training data.
Section 2: Generalization Error Analysis
To evaluate the performance of ERM, we need to consider both the training error and the generalization error. The training error measures how well the model fits the training data, while the generalization error measures how well the model performs on unseen data. In this study, we use a new functional net to measure the generalization capability of ERM, which reveals that the generalization error is not always minimized when the number of parameters in the model increases.
Section 3: Influence of Depth on Generalization Performance
We investigate how the depth of our KEFNN affects the generalization performance in the under-parameterized regime. We find that the generalization error first decreases and then increases as the depth of the network increases, which suggests that there is an optimal depth for minimizing the generalization error.
Section 4: Influence of First-Stage Sample Size on Generalization Performance
We examine how the first-stage sample size affects the generalization performance using different second-stage sample sizes and variances of noises. We find that the generalization error decays at a rate around O((log m)−1), which means that we need an extremely large first-stage sample size to achieve a small generalization error when the response function is non-linear.
Conclusion
In conclusion, this study provides new insights into the generalization error analysis of ERM for non-linear functional regression. We find that the generalization error is not always minimized when the number of parameters in the model increases, and there exists an optimal depth for minimizing the generalization error. Moreover, we discover that the first-stage sample size needs to be extremely large to achieve a small generalization error when the response function is non-linear. These findings have important implications for functional data analysis and may help researchers develop more accurate and efficient machine learning models in the future.
