# Supervised Learning with General Risk Functionals

arXiv: https://arxiv.org/abs/2206.13648 · HTML: https://mlsanity.com/paper/2206.13648/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 67 references that are in the map.
- First posted on arXiv 27 Jun 2022.

## Builds on (most cited first, 17 of 17)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Rethinking the Inception Architecture for Computer Vision](https://mlsanity.com/paper/1512.00567/) (2015)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [ShuffleNet V2: Practical Guidelines for Efficient CNN Architecture Design](https://mlsanity.com/paper/1807.11164/) (2018)
- [Learning Models with Uniform Performance via Distributionally Robust Optimization](https://mlsanity.com/paper/1810.08750/) (2018)
- [Tilted Empirical Risk Minimization](https://mlsanity.com/paper/2007.01162/) (2020)
- [Cumulative Prospect Theory Meets Reinforcement Learning: Prediction and Control](https://mlsanity.com/paper/1506.02632/) (2015)
- [Learning Bounds for Risk-sensitive Learning](https://mlsanity.com/paper/2006.08138/) (2020)
- [Off-Policy Risk Assessment in Contextual Bandits](https://mlsanity.com/paper/2104.08977/) (2021)
- [Minimizing Spectral Risk Measures Applied to Markov Decision Processes](https://mlsanity.com/paper/2012.04521/) (2020)
- [Generalized Risk-Aversion in Stochastic Multi-Armed Bandits](https://mlsanity.com/paper/1405.0833/) (2014)
- [High-Confidence Off-Policy (or Counterfactual) Variance Estimation](https://mlsanity.com/paper/2101.09847/) (2021)
- [Spectral risk-based learning using unbounded losses](https://mlsanity.com/paper/2105.04816/) (2021)
- [X-Armed Bandits: Optimizing Quantiles, CVaR and Other Risks](https://mlsanity.com/paper/1904.08205/) (2019)

## Built on by (most cited first, 6 of 6)

- [Conformal Risk Training: End-to-End Optimization of Conformal Risk Control](https://mlsanity.com/paper/2510.08748/) (2025)
- [Tailoring to the Tails: Risk Measures for Fine-Grained Tail Sensitivity](https://mlsanity.com/paper/2208.03066/) (2022)
- [OrderGrad: Optimizing Beyond the Mean with Order-Statistic Policy Gradient Estimation](https://mlsanity.com/paper/2606.06096/) (2026)
- [Uniform Pessimistic Risk and its Optimal Portfolio](https://mlsanity.com/paper/2303.07158/) (2023)
- [Risk-Entropic Flow Matching](https://mlsanity.com/paper/2512.03078/) (2025)
- [Occupancy-based Quantile Risk Control](https://mlsanity.com/paper/2609.03104/) (2026)

## Abstract (arXiv)

Standard uniform convergence results bound the generalization gap of the expected loss over a hypothesis class. The emergence of risk-sensitive learning requires generalization guarantees for functionals of the loss distribution beyond the expectation. While prior works specialize in uniform convergence of particular functionals, our work provides uniform convergence for a general class of H"older risk functionals for which the closeness in the Cumulative Distribution Function (CDF) entails closeness in risk. We establish the first uniform convergence results for estimating the CDF of the loss distribution, yielding guarantees that hold simultaneously both over all H"older risk functionals and over all hypotheses. Thus licensed to perform empirical risk minimization, we develop practical gradient-based methods for minimizing distortion risks (widely studied subset of H"older risks that subsumes the spectral risks, including the mean, conditional value at risk, cumulative prospect theory risks, and others) and provide convergence guarantees. In experiments, we demonstrate the efficacy of our learning procedure, both in settings where uniform convergence results hold and in high-dimensional settings with deep networks.