# Asymptotic Optimality of Thompson Sampling for Risk-Averse Bandits with Sub-Gaussian Rewards

arXiv: https://arxiv.org/abs/2606.09191 · HTML: https://mlsanity.com/paper/2606.09191/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 16 references that are in the map.
- First posted on arXiv 8 Jun 2026.

## Builds on (most cited first, 6 of 6)

- [Thompson Sampling Algorithms for Mean-Variance Bandits](https://mlsanity.com/paper/2002.00232/) (2020)
- [Optimal Thompson Sampling strategies for support-aware CVaR bandits](https://mlsanity.com/paper/2012.05754/) (2020)
- [A Unifying Theory of Thompson Sampling for Continuous Risk-Averse Bandits](https://mlsanity.com/paper/2108.11345/) (2021)
- [A Survey of Risk-Aware Multi-Armed Bandits](https://mlsanity.com/paper/2205.05843/) (2022)
- [Thompson Sampling for Gaussian Entropic Risk Bandits](https://mlsanity.com/paper/2105.06960/) (2021)
- [Order Optimal Regret Bounds for Sharpe Ratio Optimization under Thompson Sampling](https://mlsanity.com/paper/2508.13749/) (2025)

## Abstract (arXiv)

We prove that ρ-NPTSSG, an anchor-free nonparametric Thompson Sampling algorithm for risk-averse bandits, achieves regret matching the instance-dependent lower bound to leading order in log n, establishing it as asymptotically optimal for any continuous risk functional ρ (CVaR, mean-variance, Sharpe ratio, distortion risk measures, and more) on the class of distributions with bounded density and sub-Gaussian tails, including Gaussian arms. Both this result and its bounded-support counterpart require only continuity of ρ: strictly weaker than the dominance condition of prior parametric Thompson Sampling results, and strictly weaker than the Lipschitz condition of UCB-type algorithms, yielding the first instance-optimal guarantees for non-Lipschitz functionals such as the Sharpe ratio without parametric reward assumptions. The bounded-support case is developed first as a stepping stone sharing the same proof structure. The key technical contributions are a discretisation lemma (bounded support) and a truncated discretisation lemma (sub-Gaussian tails), each projecting the growing-alphabet Dirichlet posterior onto a fixed grid via the Dirichlet aggregation property, holding all polynomial prefactors at fixed degree independent of sample size and breaking the super-exponential barrier that blocked prior proofs.