How Does Independence Help Generalization? Sample Complexity of ERM on Product Distributions

While many classical notions of learnability (e.g., PAC learnability) are distribution-free, utilizing the specific structures of an input distribution may improve learning performance. For example, a product distribution on a multi-dimensional input space has a much simpler structure than a correlated distribution. A recent paper [GHTZ21] shows that the sample complexity of a general learning problem on product distributions is polynomial in the input dimension, which is exponentially smaller than that on correlated distributions. However, the learning algorithm they use is not the standard Empirical Risk Minimization (ERM) algorithm. In this note, we characterize the sample complexity of ERM in a general learning problem on product distributions. We show that, even though product distributions are simpler than correlated distributions, ERM still needs an exponential number of samples to learn on product distributions, instead of a polynomial. This leads to the conclusion that a product distribution by itself does not make a learning problem easier -- an algorithm designed specifically for product distributions is needed.