Chaining Bounds for Empirical Risk Minimization

This paper extends the standard chaining technique to prove excess risk upper bounds for empirical risk minimization with random design settings even if the magnitude of the noise and the estimates is unbounded. The bound applies to many loss functions besides the squared loss, and scales only with the sub-Gaussian or subexponential parameters without further statistical assumptions such as the bounded kurtosis condition over the hypothesis class. A detailed analysis is provided for slope constrained and penalized linear least squares regression with a sub-Gaussian setting, which often proves tight sample complexity bounds up to logartihmic factors.