Fine-grained Generalization Analysis of Vector-Valued Learning

Many fundamental machine learning tasks can be formulated as a problem of with vector-valued functions, where we learn multiple scalar-valued functions together. Although there is some generalization analysis on different specific algorithms under the empirical risk minimization principle, unifying regularization framework still lacking. In this paper, initiate regularized by presenting bounds mild dependency output dimension and fast rate sample size. Our discussions relax existing assumptions restrictive constraint hypothesis spaces, smoothness loss low-noise condition. To understand interaction between optimization learning, further use our results to derive first for stochastic gradient descent functions. We apply general multi-class classification multi-label classification, which yield logarithmic extreme Frobenius regularization. As byproduct, Rademacher complexity bound function classes defined in terms strongly convex function.