# Mediated Uncoupled Learning: Learning Functions without Direct Input-output Correspondences

arXiv: https://arxiv.org/abs/2107.08135 · HTML: https://mlsanity.com/paper/2107.08135/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 18 of its 38 references that are in the map.
- First posted on arXiv 16 Jul 2021.

## Builds on (most cited first, 18 of 18)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Improved Techniques for Training GANs](https://mlsanity.com/paper/1606.03498/) (2016)
- [Improved Training of Wasserstein GANs](https://mlsanity.com/paper/1704.00028/) (2017)
- [Show, Attend and Tell: Neural Image Caption Generation with Visual Attention](https://mlsanity.com/paper/1502.03044/) (2015)
- [Sharpness-Aware Minimization for Efficiently Improving Generalization](https://mlsanity.com/paper/2010.01412/) (2020)
- [Improving Variational Inference with Inverse Autoregressive Flow](https://mlsanity.com/paper/1606.04934/) (2016)
- [Why ReLU networks yield high-confidence predictions far away from the training data and how to mitigate the problem](https://mlsanity.com/paper/1812.05720/) (2018)
- [Classification with Asymmetric Label Noise: Consistency and Maximal Denoising](https://mlsanity.com/paper/1303.1208/) (2013)
- [Fine-Tuning DARTS for Image Classification](https://mlsanity.com/paper/2006.09042/) (2020)
- [RMDL: Random Multimodel Deep Learning for Classification](https://mlsanity.com/paper/1805.01890/) (2018)
- [Integrating overlapping datasets using bivariate causal discovery](https://mlsanity.com/paper/1910.11356/) (2019)
- [Uplift Modeling from Separate Labels](https://mlsanity.com/paper/1803.05112/) (2018)
- [Clustering via Mode Seeking by Direct Estimation of the Gradient of a Log-Density](https://mlsanity.com/paper/1404.5028/) (2014)
- [Learning from Indirect Observations](https://mlsanity.com/paper/1910.04394/) (2019)

## Built on by (most cited first, 1 of 1)

- [BrokenBind: Universal Modality Exploration beyond Dataset Boundaries](https://mlsanity.com/paper/2602.06451/) (2026)

## Abstract (arXiv)

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 SX = (Xi, Ui) and SY = (U'j, Y'j). A naive approach is to predict U from X using SX and then Y from U using SY, 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 SX to predict h(U) which is trained with SY to approximate Y. We prove statistical consistency and error bounds of our method and experimentally confirm its practical usefulness.