## Machine Learning from Weak Supervision: An Empirical Risk Minimization ApproachFundamental theory and practical algorithms of weakly supervised classification, emphasizing an approach based on empirical risk minimization.Standard machine learning techniques require large amounts of labeled data to work well. When we apply machine learning to problems in the physical world, however, it is extremely difficult to collect such quantities of labeled data. In this book Masashi Sugiyama, Han Bao, Takashi Ishida, Nan Lu, Tomoya Sakai and Gang Niu present theory and algorithms for weakly supervised learning, a paradigm of machine learning from weakly labeled data. Emphasizing an approach based on empirical risk minimization and drawing on state-of-the-art research in weakly supervised learning, the book provides both the fundamentals of the field and the advanced mathematical theories underlying them. It can be used as a reference for practitioners and researchers and in the classroom. The book first mathematically formulates classification problems, defines common notations, and reviews various algorithms for supervised binary and multiclass classification. It then explores problems of binary weakly supervised classification, including positive-unlabeled (PU) classification, positive-negative-unlabeled (PNU) classification, and unlabeled-unlabeled (UU) classification. It then turns to multiclass classification, discussing complementary-label (CL) classification and partial-label (PL) classification. Finally, the book addresses more advanced issues, including a family of correction methods to improve the generalization performance of weakly supervised learning and the problem of class-prior estimation. |

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algorithm analysis approach approximated assume assumption binary bound called chapter class-prior classification method classification risk clustering collect comparison computed Conference consider consistency convergence convex correction datasets defined denotes density deviation discussed distribution empirical risk Epoch et al example expectation experiments expressed figure fixed formulation function function class given holds independent introduce labels linear loss loss function Machine Learning matrix means measure minimizer MNIST multi-class negative Neural Note obtain optimization overfitting pairwise parameter pattern Pconf performance points positive practice probability probability at least problem Processing Proof PU classification Rademacher complexity RecUU regularization replaced respect risk estimator risk minimizer samples satisfies semi-supervised similar solve Specifically Sugiyama supervised classification Table theorem tion training data unbiased unlabeled data weakly supervised