# Estimation and Applications of Quantiles in Deep Binary Classification

arXiv: https://arxiv.org/abs/2102.06575 · HTML: https://mlsanity.com/paper/2102.06575/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 19 of its 50 references that are in the map.
- First posted on arXiv 9 Feb 2021.

## Builds on (most cited first, 19 of 19)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- ["Why Should I Trust You?": Explaining the Predictions of Any Classifier](https://mlsanity.com/paper/1602.04938/) (2016)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [Deep Inside Convolutional Networks: Visualising Image Classification Models and Saliency Maps](https://mlsanity.com/paper/1312.6034/) (2013)
- [Learning Important Features Through Propagating Activation Differences](https://mlsanity.com/paper/1704.02685/) (2017)
- [Deep Neural Networks are Easily Fooled: High Confidence Predictions for Unrecognizable Images](https://mlsanity.com/paper/1412.1897/) (2014)
- [Robust Loss Functions under Label Noise for Deep Neural Networks](https://mlsanity.com/paper/1712.09482/) (2017)
- [Lipschitz regularity of deep neural networks: analysis and efficient estimation](https://mlsanity.com/paper/1805.10965/) (2018)
- [Simple and Principled Uncertainty Estimation with Deterministic Deep Learning via Distance Awareness](https://mlsanity.com/paper/2006.10108/) (2020)
- [To Trust Or Not To Trust A Classifier](https://mlsanity.com/paper/1805.11783/) (2018)
- [A modern maximum-likelihood theory for high-dimensional logistic regression](https://mlsanity.com/paper/1803.06964/) (2018)
- [Deep Neural Networks for Estimation and Inference](https://mlsanity.com/paper/1809.09953/) (2018)
- [Gradients of Counterfactuals](https://mlsanity.com/paper/1611.02639/) (2016)
- [Cross-Entropy Loss and Low-Rank Features Have Responsibility for Adversarial Examples](https://mlsanity.com/paper/1901.08360/) (2019)
- [A Framework for Democratizing AI](https://mlsanity.com/paper/2001.00818/) (2020)
- [Parsimonious Computing: A Minority Training Regime for Effective Prediction in Large Microarray Expression Data Sets](https://mlsanity.com/paper/2005.08442/) (2020)
- [LALR: Theoretical and Experimental validation of Lipschitz Adaptive Learning Rate in Regression and Neural Networks](https://mlsanity.com/paper/2006.13307/) (2020)

## Built on by (most cited first, 3 of 3)

- [LogGENE: A smooth alternative to check loss for Deep Healthcare Inference Tasks](https://mlsanity.com/paper/2206.09333/) (2022)
- [QuantProb: Generalizing Probabilities along with Predictions for a Pre-trained Classifier](https://mlsanity.com/paper/2304.12766/) (2023)
- [Quantile LSTM: A Robust LSTM for Anomaly Detection In Time Series Data](https://mlsanity.com/paper/2302.08712/) (2023)

## Abstract (arXiv)

Quantile regression, based on check loss, is a widely used inferential paradigm in Econometrics and Statistics. The conditional quantiles provide a robust alternative to classical conditional means, and also allow uncertainty quantification of the predictions, while making very few distributional assumptions. We consider the analogue of check loss in the binary classification setting. We assume that the conditional quantiles are smooth functions that can be learnt by Deep Neural Networks (DNNs). Subsequently, we compute the Lipschitz constant of the proposed loss, and also show that its curvature is bounded, under some regularity conditions. Consequently, recent results on the error rates and DNN architecture complexity become directly applicable. We quantify the uncertainty of the class probabilities in terms of prediction intervals, and develop individualized confidence scores that can be used to decide whether a prediction is reliable or not at scoring time. By aggregating the confidence scores at the dataset level, we provide two additional metrics, model confidence, and retention rate, to complement the widely used classifier summaries. We also the robustness of the proposed non-parametric binary quantile classification framework are also studied, and we demonstrate how to obtain several univariate summary statistics of the conditional distributions, in particular conditional means, using smoothed conditional quantiles, allowing the use of explanation techniques like Shapley to explain the mean predictions. Finally, we demonstrate an efficient training regime for this loss based on Stochastic Gradient Descent with Lipschitz Adaptive Learning Rates (LALR).