# Group Orthogonalized Policy Optimization:Group Policy Optimization as Orthogonal Projection in Hilbert Space

arXiv: https://arxiv.org/abs/2602.21269 · HTML: https://mlsanity.com/paper/2602.21269/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 24 references that are in the map.
- First posted on arXiv 24 Feb 2026.

## Builds on (most cited first, 16 of 16)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Deep reinforcement learning from human preferences](https://mlsanity.com/paper/1706.03741/) (2017)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Trust Region Policy Optimization](https://mlsanity.com/paper/1502.05477/) (2015)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [SimPO: Simple Preference Optimization with a Reference-Free Reward](https://mlsanity.com/paper/2405.14734/) (2024)
- [A General Theoretical Paradigm to Understand Learning from Human Preferences](https://mlsanity.com/paper/2310.12036/) (2023)
- [f-GAN: Training Generative Neural Samplers using Variational Divergence Minimization](https://mlsanity.com/paper/1606.00709/) (2016)
- [A Divergence Minimization Perspective on Imitation Learning Methods](https://mlsanity.com/paper/1911.02256/) (2019)
- [R'enyi Divergence Variational Inference](https://mlsanity.com/paper/1602.02311/) (2016)
- [ADPO: Anchored Direct Preference Optimization](https://mlsanity.com/paper/2510.18913/) (2025)
- [APO: Alpha-Divergence Preference Optimization](https://mlsanity.com/paper/2512.22953/) (2025)
- [Orthogonalized Policy Optimization:Policy Optimization as Orthogonal Projection in Hilbert Space](https://mlsanity.com/paper/2601.12415/) (2026)

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

- [When KL Regularization Misfires in Group Policy Optimization](https://mlsanity.com/paper/2610.12161/) (2026)

## Abstract (arXiv)

We present Group Orthogonalized Policy Optimization (GOPO), a new alignment algorithm for large language models derived from the geometry of Hilbert function spaces. Instead of optimizing on the probability simplex and inheriting the exponential curvature of Kullback-Leibler divergence, GOPO lifts alignment into the Hilbert space L2(pi_k) of square-integrable functions with respect to the reference policy. Within this space, the simplex constraint reduces to a linear orthogonality condition <v, 1> = 0, defining a codimension-one subspace H0. Minimizing distance to an unconstrained target u_star yields the work-dissipation functional J(v) = <g, v> - (mu / 2) ||v||^2, whose maximizer follows directly from the Hilbert projection theorem. Enforcing the boundary v >= -1 produces a bounded Hilbert projection that induces exact sparsity, assigning zero probability to catastrophically poor actions through a closed-form threshold. To connect this functional theory with practice, GOPO projects from infinite-dimensional L2(pi_k) to a finite empirical subspace induced by group sampling. Because group-normalized advantages sum to zero, the Lagrange multiplier enforcing probability conservation vanishes exactly, reducing the constrained projection to an unconstrained empirical loss. The resulting objective has constant Hessian curvature mu I, non-saturating linear gradients, and an intrinsic dead-zone mechanism without heuristic clipping. Experiments on mathematical reasoning benchmarks show that GOPO achieves competitive generalization while maintaining stable gradient dynamics and entropy preservation in regimes where clipping-based methods plateau.