# Tailoring to the Tails: Risk Measures for Fine-Grained Tail Sensitivity

arXiv: https://arxiv.org/abs/2208.03066 · HTML: https://mlsanity.com/paper/2208.03066/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 97 references that are in the map.
- First posted on arXiv 5 Aug 2022.

## Builds on (most cited first, 26 of 26)

- [Implicit Quantile Networks for Distributional Reinforcement Learning](https://mlsanity.com/paper/1806.06923/) (2018)
- [Fairness Without Demographics in Repeated Loss Minimization](https://mlsanity.com/paper/1806.08010/) (2018)
- [Statistics of Robust Optimization: A Generalized Empirical Likelihood Approach](https://mlsanity.com/paper/1610.03425/) (2016)
- [Variance-based regularization with convex objectives](https://mlsanity.com/paper/1610.02581/) (2016)
- [Risk-sensitive Reinforcement Learning](https://mlsanity.com/paper/1311.2097/) (2013)
- [The Heavy-Tail Phenomenon in SGD](https://mlsanity.com/paper/2006.04740/) (2020)
- [Tilted Empirical Risk Minimization](https://mlsanity.com/paper/2007.01162/) (2020)
- [Fairness risk measures](https://mlsanity.com/paper/1901.08665/) (2019)
- [Policy Gradient for Coherent Risk Measures](https://mlsanity.com/paper/1502.03919/) (2015)
- [Distributionally Robust Optimization: A Review](https://mlsanity.com/paper/1908.05659/) (2019)
- [Coping with Label Shift via Distributionally Robust Optimisation](https://mlsanity.com/paper/2010.12230/) (2020)
- [Risk-Averse Offline Reinforcement Learning](https://mlsanity.com/paper/2102.05371/) (2021)
- [Learning with Average Top-k Loss](https://mlsanity.com/paper/1705.08826/) (2017)
- [Loss minimization and parameter estimation with heavy tails](https://mlsanity.com/paper/1307.1827/) (2013)
- [Adaptive Sampling for Stochastic Risk-Averse Learning](https://mlsanity.com/paper/1910.12511/) (2019)
- [Empirical risk minimization for heavy-tailed losses](https://mlsanity.com/paper/1406.2462/) (2014)
- [Improving Robustness via Risk Averse Distributional Reinforcement Learning](https://mlsanity.com/paper/2005.00585/) (2020)
- [Subspace Embedding and Linear Regression with Orlicz Norm](https://mlsanity.com/paper/1806.06430/) (2018)
- [Statistical Learning with Conditional Value at Risk](https://mlsanity.com/paper/2002.05826/) (2020)
- [Minimizing Spectral Risk Measures Applied to Markov Decision Processes](https://mlsanity.com/paper/2012.04521/) (2020)
- [Estimation of Spectral Risk Measures](https://mlsanity.com/paper/1912.10398/) (2019)
- [Risk Measures and Upper Probabilities: Coherence and Stratification](https://mlsanity.com/paper/2206.03183/) (2022)
- [Spectral risk-based learning using unbounded losses](https://mlsanity.com/paper/2105.04816/) (2021)
- [Efficient Symmetric Norm Regression via Linear Sketching](https://mlsanity.com/paper/1910.01788/) (2019)
- [Supervised Learning with General Risk Functionals](https://mlsanity.com/paper/2206.13648/) (2022)
- [General risk measures for robust machine learning](https://mlsanity.com/paper/1904.11707/) (2019)

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

- [Orlicz regrets to consistently bound statistics of random variables with an application to environmental indicators](https://mlsanity.com/paper/2310.05168/) (2023)
- [Criterion Collapse and Loss Distribution Control](https://mlsanity.com/paper/2402.09802/) (2024)

## Abstract (arXiv)

Expected risk minimization (ERM) is at the core of many machine learning systems. This means that the risk inherent in a loss distribution is summarized using a single number - its average. In this paper, we propose a general approach to construct risk measures which exhibit a desired tail sensitivity and may replace the expectation operator in ERM. Our method relies on the specification of a reference distribution with a desired tail behaviour, which is in a one-to-one correspondence to a coherent upper probability. Any risk measure, which is compatible with this upper probability, displays a tail sensitivity which is finely tuned to the reference distribution. As a concrete example, we focus on divergence risk measures based on f-divergence ambiguity sets, which are a widespread tool used to foster distributional robustness of machine learning systems. For instance, we show how ambiguity sets based on the Kullback-Leibler divergence are intricately tied to the class of subexponential random variables. We elaborate the connection of divergence risk measures and rearrangement invariant Banach norms.