# Trained Model in Supervised Deep Learning is a Conditional Risk Minimizer

arXiv: https://arxiv.org/abs/2202.03674 · HTML: https://mlsanity.com/paper/2202.03674/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 21 references that are in the map.
- First posted on arXiv 8 Feb 2022.

## Builds on (most cited first, 12 of 12)

- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [U-Net: Convolutional Networks for Biomedical Image Segmentation](https://mlsanity.com/paper/1505.04597/) (2015)
- [Score-Based Generative Modeling through Stochastic Differential Equations](https://mlsanity.com/paper/2011.13456/) (2020)
- [Diffusion Models Beat GANs on Image Synthesis](https://mlsanity.com/paper/2105.05233/) (2021)
- [What Uncertainties Do We Need in Bayesian Deep Learning for Computer Vision?](https://mlsanity.com/paper/1703.04977/) (2017)
- [A Closer Look at Memorization in Deep Networks](https://mlsanity.com/paper/1706.05394/) (2017)
- [Noise2Noise: Learning Image Restoration without Clean Data](https://mlsanity.com/paper/1803.04189/) (2018)
- [Understanding and Utilizing Deep Neural Networks Trained with Noisy Labels](https://mlsanity.com/paper/1905.05040/) (2019)
- [Deep Learning is Robust to Massive Label Noise](https://mlsanity.com/paper/1705.10694/) (2017)
- [Noise2Score: Tweedie's Approach to Self-Supervised Image Denoising without Clean Images](https://mlsanity.com/paper/2106.07009/) (2021)
- [Deep Bayesian Inversion](https://mlsanity.com/paper/1811.05910/) (2018)
- [AR-DAE: Towards Unbiased Neural Entropy Gradient Estimation](https://mlsanity.com/paper/2006.05164/) (2020)

## Built on by (most cited first, 2 of 2)

- [Measurement-conditioned Denoising Diffusion Probabilistic Model for Under-sampled Medical Image Reconstruction](https://mlsanity.com/paper/2203.03623/) (2022)
- [Diffusion Model for Generative Image Denoising](https://mlsanity.com/paper/2302.02398/) (2023)

## Abstract (arXiv)

We proved that a trained model in supervised deep learning minimizes the conditional risk for each input (Theorem 2.1). This property provided insights into the behavior of trained models and established a connection between supervised and unsupervised learning in some cases. In addition, when the labels are intractable but can be written as a conditional risk minimizer, we proved an equivalent form of the original supervised learning problem with accessible labels (Theorem 2.2). We demonstrated that many existing works, such as Noise2Score, Noise2Noise and score function estimation can be explained by our theorem. Moreover, we derived a property of classification problem with noisy labels using Theorem 2.1 and validated it using MNIST dataset. Furthermore, We proposed a method to estimate uncertainty in image super-resolution based on Theorem 2.2 and validated it using ImageNet dataset. Our code is available on github.