# Risk level dependent Minimax Quantile lower bounds for Interactive Statistical Decision Making

arXiv: https://arxiv.org/abs/2510.05808 · HTML: https://mlsanity.com/paper/2510.05808/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 17 references that are in the map.
- First posted on arXiv 7 Oct 2025.

## Builds on (most cited first, 7 of 7)

- [Minimax Regret Bounds for Reinforcement Learning](https://mlsanity.com/paper/1703.05449/) (2017)
- [Is Q-learning Provably Efficient?](https://mlsanity.com/paper/1807.03765/) (2018)
- [The Statistical Complexity of Interactive Decision Making](https://mlsanity.com/paper/2112.13487/) (2021)
- [Refined Lower Bounds for Adversarial Bandits](https://mlsanity.com/paper/1605.07416/) (2016)
- [High-probability minimax lower bounds](https://mlsanity.com/paper/2406.13447/) (2024)
- [Assouad, Fano, and Le Cam with Interaction: A Unifying Lower Bound Framework and Characterization for Bandit Learnability](https://mlsanity.com/paper/2410.05117/) (2024)
- [Model-Free Reinforcement Learning with the Decision-Estimation Coefficient](https://mlsanity.com/paper/2211.14250/) (2022)

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

- [Minimax Quantile Lower Bounds for Interactive Statistical Decision Making with Privacy](https://mlsanity.com/paper/2606.23096/) (2026)
- [A stochastic subgradient method with optimal failure exponent](https://mlsanity.com/paper/2609.37425/) (2026)

## Abstract (arXiv)

Minimax risk and regret focus on expectation, missing rare failures critical in safety-critical bandits and reinforcement learning. Minimax quantiles capture these tails. Three strands of prior work motivate this study: minimax-quantile bounds restricted to non-interactive estimation; unified interactive analyses that focus on expected risk rather than risk level specific quantile bounds; and high-probability bandit bounds that still lack a quantile-specific toolkit for general interactive protocols. To close this gap, within the interactive statistical decision making framework, we develop high-probability Fano and Le Cam tools and derive risk level explicit minimax-quantile bounds, including a quantile-to-expectation conversion and a tight link between strict and lower minimax quantiles. Instantiating these results for the two-armed Gaussian bandit immediately recovers optimal-rate bounds.