Vicinal risk minimization

For labeled observations(x_i,y_i), choose vicinity distributions P_i over inputs near each x_i. The paper’s basic construction replaces each empirical input point mass with P_i while retaining the point label y_i, giving P_est=(1/n)sum_i P_i(x)delta_y_i(y). Minimize R_vic(f)=(1/n)sum_i integral loss(f(x),y_i)dP_i(x) over the chosen hypothesis class F. More generally, the paper allows an estimated joint input-target distribution rather than requiring pointwise kernels. The vicinity distribution and the function class both shape the learning problem. Gaussian neighborhoods are one example, not a defining requirement. Replacing them with exact point masses recovers empirical-risk minimization. However, sending a neighborhood-width parameter to zero after minimizing the vicinal risk need not select every empirical-risk minimizer: the paper discusses a hard-margin limit for a particular linearly separable linear-classifier construction. This principle specifies an objective and modeling choices, not one universal numerical optimizer.