Fine-grained generalization analysis of vector-valued learning

• Main Authors: WU, Liang, LEDENT, Antoine, LEI, Yunwen, KLOFT, Marius
• Format: text
• Language: English
• Published: Institutional Knowledge at Singapore Management University 2021
• Subjects: Statistical Learning Theory; Multi-label Learning; Stochastic Gradient Descent; Artificial Intelligence and Robotics; Theory and Algorithms
• Online Access: https://ink.library.smu.edu.sg/sis_research/7203; https://ink.library.smu.edu.sg/context/sis_research/article/8206/viewcontent/vector_val.pdf
• Institution: Singapore Management University

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