Mediated Uncoupled Learning: Learning Functions without Direct Input-output Correspondences
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Ordinary supervised learning is useful when we have paired training data of input X and output Y. However, such paired data can be difficult to collect in practice. In this paper, we consider the task of predicting Y from X when we have no paired data of them, but we have two separate, independent datasets of X and Y each observed with some mediating variable U, that is, we have two datasets S_X = {(X_i, U_i)} and S_Y = {(U'_j, Y'_j)}. A naive approach is to predict U from X using S_X and then Y from U using S_Y, but we show that this is not statistically consistent. Moreover, predicting U can be more difficult than predicting Y in practice, e.g., when U has higher dimensionality. To circumvent the difficulty, we propose a new method that avoids predicting U but directly learns Y = f(X) by training f(X) with S_X to predict h(U) which is trained with S_Y to approximate Y. We prove statistical consistency and error bounds of our method and experimentally confirm its practical usefulness.
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