# Riemannian Dueling Optimization

arXiv: https://arxiv.org/abs/2603.00023 · HTML: https://mlsanity.com/paper/2603.00023/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 25 of its 39 references that are in the map.
- First posted on arXiv 3 Feb 2026.

## Builds on (most cited first, 25 of 25)

- [Obfuscated Gradients Give a False Sense of Security: Circumventing Defenses to Adversarial Examples](https://mlsanity.com/paper/1802.00420/) (2018)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Mitigating Adversarial Effects Through Randomization](https://mlsanity.com/paper/1711.01991/) (2017)
- [Stochastic Activation Pruning for Robust Adversarial Defense](https://mlsanity.com/paper/1803.01442/) (2018)
- [First-order Methods for Geodesically Convex Optimization](https://mlsanity.com/paper/1602.06053/) (2016)
- [A Perceptual Measure for Deep Single Image Camera Calibration](https://mlsanity.com/paper/1712.01259/) (2017)
- [A Unified Gradient Regularization Family for Adversarial Examples](https://mlsanity.com/paper/1511.06385/) (2015)
- [Averaging Stochastic Gradient Descent on Riemannian Manifolds](https://mlsanity.com/paper/1802.09128/) (2018)
- [Zeroth-Order Optimization Meets Human Feedback: Provable Learning via Ranking Oracles](https://mlsanity.com/paper/2303.03751/) (2023)
- [Learning Trajectory Preferences for Manipulators via Iterative Improvement](https://mlsanity.com/paper/1306.6294/) (2013)
- [Horizon Lines in the Wild](https://mlsanity.com/paper/1604.02129/) (2016)
- [A Second look at Exponential and Cosine Step Sizes: Simplicity, Adaptivity, and Performance](https://mlsanity.com/paper/2002.05273/) (2020)
- [Stochastic Three Points Method for Unconstrained Smooth Minimization](https://mlsanity.com/paper/1902.03591/) (2019)
- [Riemannian Optimization via Frank-Wolfe Methods](https://mlsanity.com/paper/1710.10770/) (2017)
- [Scalable Hyperbolic Recommender Systems](https://mlsanity.com/paper/1902.08648/) (2019)
- [Trajectory Improvement and Reward Learning from Comparative Language Feedback](https://mlsanity.com/paper/2410.06401/) (2024)
- [A One-bit, Comparison-Based Gradient Estimator](https://mlsanity.com/paper/2010.02479/) (2020)
- [Accelerated Gradient Methods for Geodesically Convex Optimization: Tractable Algorithms and Convergence Analysis](https://mlsanity.com/paper/2202.02036/) (2022)
- [Zeroth-order Riemannian Averaging Stochastic Approximation Algorithms](https://mlsanity.com/paper/2309.14506/) (2023)
- [A Fashion Item Recommendation Model in Hyperbolic Space](https://mlsanity.com/paper/2409.02599/) (2024)
- [Minibatch Stochastic Three Points Method for Unconstrained Smooth Minimization](https://mlsanity.com/paper/2209.07883/) (2022)
- [Acceleration Exists! Optimization Problems When Oracle Can Only Compare Objective Function Values](https://mlsanity.com/paper/2402.09014/) (2024)
- [On the Almost Sure Convergence of the Stochastic Three Points Algorithm](https://mlsanity.com/paper/2501.13886/) (2025)
- [DumbleDR: Predicting User Preferences of Dimensionality Reduction Projection Quality](https://mlsanity.com/paper/2105.09275/) (2021)
- [Nesterov's method of dichotomy via Order Oracle: The problem of optimizing a two-variable function on a square](https://mlsanity.com/paper/2409.11077/) (2024)

## Abstract (arXiv)

Dueling optimization considers optimizing an objective with access to only a comparison oracle of the objective function. It finds important applications in emerging fields such as recommendation systems and robotics. Existing works on dueling optimization mainly focused on unconstrained problems in the Euclidean space. In this work, we study dueling optimization over Riemannian manifolds, which covers important applications that cannot be solved by existing dueling optimization algorithms. In particular, we propose a Riemannian Dueling Normalized Gradient Descent (RDNGD) method and establish its iteration complexity when the objective function is geodesically L-smooth or geodesically (strongly) convex. We also propose a projection-free algorithm, named Riemannian Dueling Frank-Wolfe (RDFW) method, to deal with the situation where projection is prohibited. We establish the iteration and oracle complexities for RDFW. We illustrate the effectiveness of the proposed algorithms through numerical experiments on both synthetic and real applications.