# Explicit and Non-asymptotic Query Complexities of Rank-Based Zeroth-order Algorithm on Stochastic Smooth Functions

arXiv: https://arxiv.org/abs/2512.19104 · HTML: https://mlsanity.com/paper/2512.19104/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in Reward modeling and RLHF, one of 10 directions in LLM alignment / RLHF / preference optimisation.
- Builds on 10 of its 25 references that are in the map.
- First posted on arXiv 22 Dec 2025.

## Placed in

- LLM alignment / RLHF / preference optimisation / [Reward modeling and RLHF](https://mlsanity.com/alignment/reward-modeling-and-rlhf/)

## Builds on (most cited first, 10 of 10)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Stochastic First- and Zeroth-order Methods for Nonconvex Stochastic Programming](https://mlsanity.com/paper/1309.5549/) (2013)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [Principled Reinforcement Learning with Human Feedback from Pairwise or K-wise Comparisons](https://mlsanity.com/paper/2301.11270/) (2023)
- [CMA-ES for Hyperparameter Optimization of Deep Neural Networks](https://mlsanity.com/paper/1604.07269/) (2016)
- [Zeroth-Order Optimization Meets Human Feedback: Provable Learning via Ranking Oracles](https://mlsanity.com/paper/2303.03751/) (2023)
- [Regret Analysis for Continuous Dueling Bandit](https://mlsanity.com/paper/1711.07693/) (2017)
- [Explicit and Non-asymptotic Query Complexities of Rank-Based Zeroth-order Algorithms on Smooth Functions](https://mlsanity.com/paper/2512.16200/) (2025)

## Abstract (arXiv)

Zeroth-order (ZO) optimization with ordinal feedback has emerged as a fundamental problem in modern machine learning systems, particularly in human-in-the-loop settings such as reinforcement learning from human feedback, preference learning, and evolutionary strategies. While rank-based ZO algorithms enjoy strong empirical success and robustness properties, their theoretical understanding, especially under stochastic objectives and standard smoothness assumptions, remains limited. In this paper, we study rank-based zeroth-order optimization for stochastic functions where only ordinal feedback of the stochastic function is available. We propose a simple and computationally efficient rank-based ZO algorithm. Under standard assumptions including smoothness, strong convexity, and bounded second moments of stochastic gradients, we establish explicit non-asymptotic query complexity bounds for both convex and nonconvex objectives. Notably, our results match the best-known query complexities of value-based ZO algorithms, demonstrating that ordinal information alone is sufficient for optimal query efficiency in stochastic settings. Our analysis departs from existing drift-based and information-geometric techniques, offering new tools for the study of rank-based optimization under noise. These findings narrow the gap between theory and practice and provide a principled foundation for optimization driven by human preferences.