title: Regression from Upper One-side Labeled Data
abstract: We address a regression problem from weakly labeled data that are correctly labeled only above a regression line, i.e., upper one-side labeled data.
  The label values of the data are the results of sensing the magnitude of some phenomenon.
  In this case, the labels often contain missing or incomplete observations whose values are lower than those of correct observations and are also usually lower than the regression line. It follows that data labeled with lower values than the estimations of a regression function (lower-side data) are mixed with data that should originally be labeled above the regression line (upper-side data).
  When such missing label observations are observed in a non-negligible amount, we thus should assume our lower-side data to be unlabeled data that are a mix of original upper- and lower-side data.
  We formulate a regression problem from these upper-side labeled and lower-side unlabeled data. We then derive a learning algorithm in an unbiased and consistent manner to ordinary regression that is learned from data labeled correctly in both upper- and lower-side cases. Our key idea is that we can derive a gradient that requires only upper-side data and unlabeled data as the equivalent expression of that for ordinary regression. We additionally found that a specific class of losses enables us to learn unbiased solutions practically. In numerical experiments on synthetic and real-world datasets, we demonstrate the advantages of our algorithm.
authors: Takayuki Katsuki
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