Quantile-Scaled Bayesian Optimization Using Rank-Only Feedback

TL;DR

The paper tackles optimization when absolute objective values are unavailable and only rank information can be observed. It introduces Quantile-Scaled Bayesian Optimization (QS-BO), which maps ranks to heteroscedastic Gaussian targets via a quantile-scaling pipeline, enabling standard Gaussian process surrogates and acquisitions. Empirical results on synthetic 1D and 2D benchmarks (including Sinusoidal-Quadratic, Forrester, and Branin) show QS-BO consistently achieves lower final values and tighter performance distributions than Random Search, with statistical significance at the 1% level. The work demonstrates a practical framework for rank-only feedback with potential applications in preference learning, recommendation, and human-in-the-loop optimization, while outlining avenues for scalability, broader baselines, and theoretical guarantees.

Abstract

Bayesian Optimization (BO) is widely used for optimizing expensive black-box functions, particularly in hyperparameter tuning. However, standard BO assumes access to precise objective values, which may be unavailable, noisy, or unreliable in real-world settings where only relative or rank-based feedback can be obtained. In this study, we propose Quantile-Scaled Bayesian Optimization (QS-BO), a principled rank-based optimization framework. QS-BO converts ranks into heteroscedastic Gaussian targets through a quantile-scaling pipeline, enabling the use of Gaussian process surrogates and standard acquisition functions without requiring explicit metric scores. We evaluate QS-BO on synthetic benchmark functions, including one- and two-dimensional nonlinear functions and the Branin function, and compare its performance against Random Search. Results demonstrate that QS-BO consistently achieves lower objective values and exhibits greater stability across runs. Statistical tests further confirm that QS-BO significantly outperforms Random Search at the 1\% significance level. These findings establish QS-BO as a practical and effective extension of Bayesian Optimization for rank-only feedback, with promising applications in preference learning, recommendation, and human-in-the-loop optimization where absolute metric values are unavailable or unreliable.

Figures (4)

• Figure 1: Bayesian Optimization process showing the Gaussian Process posterior mean and variance, the acquisition function below, and selected observation points. The arrow (green line) indicates the next sampling location suggested by the acquisition function.
• Figure 2: TPE Approach showing how the $y^*, \ell(x), g(x)$ as well as the next point selected by $\ell(x)/g(x)$ changes with iteration. The iterations goes on till the allocated budget is exhausted.
• Figure 3: Plots of 1D functions. The latent Gaussian process model (dotted lines) closely follows the shape of the true function (blue lines), and the evaluated points become increasingly concentrated around the minimum values of $f$.
• Figure 4: Boxplots comparing the performance of Random Search and QS-BO across multiple runs on three benchmark functions: (a) the 1D sinusoidal–quadratic function, (b) the Forrester function, and (c) the Branin function. In all cases, QS-BO achieves consistently lower function values than Random Search, highlighting its effectiveness.