Mediated Uncoupled Learning: Learning Functions without Direct Input-output Correspondences
This addresses a data collection challenge in machine learning where paired input-output data is unavailable, offering a novel solution for scenarios like medical or social sciences with incremental improvements over naive approaches.
The paper tackles the problem of predicting outputs from inputs without paired data, using separate datasets linked by a mediating variable, and proposes a method that avoids predicting the mediator directly, proving statistical consistency and showing practical usefulness in experiments.
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.