# Order Optimal Regret Bounds for Sharpe Ratio Optimization under Thompson Sampling

arXiv: https://arxiv.org/abs/2508.13749 · HTML: https://mlsanity.com/paper/2508.13749/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 40 references that are in the map.
- First posted on arXiv 19 Aug 2025.

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

- [Introduction to Multi-Armed Bandits](https://mlsanity.com/paper/1904.07272/) (2019)
- [Learning to Optimize Via Posterior Sampling](https://mlsanity.com/paper/1301.2609/) (2013)
- [An Information-Theoretic Analysis of Thompson Sampling](https://mlsanity.com/paper/1403.5341/) (2014)
- [Explore First, Exploit Next: The True Shape of Regret in Bandit Problems](https://mlsanity.com/paper/1602.07182/) (2016)
- [Thompson Sampling for 1-Dimensional Exponential Family Bandits](https://mlsanity.com/paper/1307.3400/) (2013)
- [An Information-Theoretic Analysis for Thompson Sampling with Many Actions](https://mlsanity.com/paper/1805.11845/) (2018)
- [Thompson Sampling Algorithms for Mean-Variance Bandits](https://mlsanity.com/paper/2002.00232/) (2020)
- [Exploration vs Exploitation vs Safety: Risk-averse Multi-Armed Bandits](https://mlsanity.com/paper/1401.1123/) (2014)
- [Optimal Thompson Sampling strategies for support-aware CVaR bandits](https://mlsanity.com/paper/2012.05754/) (2020)
- [Concentration bounds for empirical conditional value-at-risk: The unbounded case](https://mlsanity.com/paper/1808.01739/) (2018)
- [Risk-Constrained Thompson Sampling for CVaR Bandits](https://mlsanity.com/paper/2011.08046/) (2020)
- [A Survey of Risk-Aware Multi-Armed Bandits](https://mlsanity.com/paper/2205.05843/) (2022)
- [Sharpe Ratio-Guided Active Learning for Preference Optimization in RLHF](https://mlsanity.com/paper/2503.22137/) (2025)
- [Adaptive Portfolio by Solving Multi-armed Bandit via Thompson Sampling](https://mlsanity.com/paper/1911.05309/) (2019)
- [The Sharpe predictor for fairness in machine learning](https://mlsanity.com/paper/2108.06415/) (2021)
- [Optimizing Sharpe Ratio: Risk-Adjusted Decision-Making in Multi-Armed Bandits](https://mlsanity.com/paper/2406.06552/) (2024)
- [Variance-Optimal Arm Selection: Misallocation Minimization and Best Arm Identification](https://mlsanity.com/paper/2505.11985/) (2025)

## Built on by (most cited first, 1 of 1)

- [Asymptotic Optimality of Thompson Sampling for Risk-Averse Bandits with Sub-Gaussian Rewards](https://mlsanity.com/paper/2606.09191/) (2026)

## Abstract (arXiv)

In this paper, we study sequential decision-making for maximizing the Sharpe ratio (SR) in a stochastic multi-armed bandit (MAB) setting. Unlike standard bandit formulations that maximize cumulative reward, SR optimization requires balancing expected return and reward variability. As a result, the learning objective depends jointly on the mean and variance of the reward distribution and takes a fractional form. To address this problem, we propose the Sharpe Ratio Thompson Sampling textttSRTS, a Bayesian algorithm for risk-adjusted exploration. For Gaussian reward models, the algorithm employs a Normal-Gamma conjugate posterior to capture uncertainty in both the mean and the precision of each arm. In contrast to additive mean-variance (MV) formulations, which often require different algorithms across risk regimes, the fractional SR objective yields a single sampling rule that applies uniformly across risk tolerances. On the theoretical side, we develop a regret decomposition tailored to the SR objective and introduce a decoupling approach that separates the contributions of mean and variance uncertainty. This framework allows us to control the interaction between the Gaussian mean samples and the Gamma precision samples arising in the posterior. Using these results, we establish a finite-time distribution-dependent O(log n) upper bound on the expected regret. We further derive a matching information-theoretic lower bound using a change-of-measure argument, showing that the proposed algorithm is order-optimal. Finally, experiments on synthetic bandit environments illustrate the performance of textttSRTS and demonstrate improvements over existing risk-aware bandit algorithms across a range of risk-return settings.